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TABLE OF CONTENTS
Contenido 1. NÚMEROS NATURALES .....................................................................................................................3
2. DIVISIBILIDAD ............................................................................................................................... 11
3. SISTEMA MÉTRICO DECIMAL ............................................................................................................ 19
4. POLÍGONOS Y CIRCUNFERENCIAS ....................................................................................................... 27
5. FRACCIONES ................................................................................................................................. 39
6. NÚMEROS DECIMALES .................................................................................................................... 45
7. NÚMEROS ENTEROS ....................................................................................................................... 51
8. PROPORCIONALIDAD ...................................................................................................................... 57
9. LENGUAJE ALGEBRAICO ................................................................................................................... 69
10. FIGURAS PLANAS ........................................................................................................................... 75
11. FUNCIONES Y GRÁFICAS .................................................................................................................. 85
12. ESTADÍSTICA ................................................................................................................................. 91
13. PROBABILIDAD .............................................................................................................................. 99
14. Cuerpos Geométricos y Volúmenes ......................................................................................... 105
3
1. NÚMEROS NATURALES
Los números naturales son los que utilizamos para contar las cosas. Su símbolo
matemático es . Son el 0, 1, 2, 3, 4,…
Operaciones con números:
Multiplicar: es sumar muchas veces el mismo número. Para escribir una multiplicación
usaremos un punto “·” en vez de una cruz “x”. Por ejemplo: 2+2+2+2+2+2+2 = 2 · 7 = 14. En
este caso, los factores son 2 y 7, y su producto 14.
Dividir: se usa siempre que hay que repartir en partes iguales. Recuerda que
dividendo es igual a divisor por cociente más resto.
Potencia: es multiplicar muchas veces el mismo número. Ejemplo: 2 · 2 · 2 · 2 · 2 = 25 = 32.
Aquí, la base es 2 y el exponente 5.
Raíz cuadrada: hay que encontrar un número que al cuadrado sea el radicando (el número
del que queremos sacar la raíz). Por ejemplo, , porque 32 = 9.
Orden: Usamos los símbolos < y > . Diremos que 3 < 5 y que 10 > 2 .
Cuadrado perfecto: llamamos así a los números naturales al cuadrado. Por ejemplo, 22=4,
32=9, 42=16, 52=25, 62=36, 72=49, 82=64, 92=81, etc.
Ejercicio: Calcula y memoriza los siguientes cuadrados perfectos.
112 = 122 = 132 = 142 = 152 = 252 =
Ejercicio: Calcula las raíces:
Orden de las operaciones (jerarquía de las operaciones):
No siempre se hacen las cuentas de izquierda a derecha. Tenemos que hacerlo en este orden:
1) Paréntesis.
2) Potencias y raíces.
3) Multiplicaciones y divisiones
4) Sumas y restas.
4
Propiedades de la suma
Conmutativa: Podemos sumar en cualquier orden. Por ejemplo: 3 + 5 = 5 + 3 = 8.
Asociativa: al sumar varios números, podemos agruparlos como queramos.
Ejemplo: 3+5+2 = (3+5)+2 = 3+(5+2) = 10.
Elemento neutro: Al sumar 0 a un número, se obtiene el mismo número.
Propiedades de la multiplicación
Conmutativa: Podemos multiplicar en cualquier orden; “el orden de los factores no altera el
producto”. Por ejemplo: 3 · 5 = 5 · 3 = 15.
Asociativa: al multiplicar varios números, podemos agruparlos como queramos. Por
ejemplo: 3 · 5 · 2 = (3 · 5) · 2 = 3 · (5 · 2) = 30.
Elemento neutro: Al multiplicar un número por 1, se obtiene el mismo número.
Distributiva: Al multiplicar un número por una suma, podemos multiplicar primero cada
sumando. Por ejemplo: 2 · (3 + 4) = 2 · 3 + 2 · 4 = 6 + 8 = 14.
Propiedades de las potencias
Producto de potencias:
- Misma base: se deja la base y se suman los exponentes. Ejemplo: 32·34=32+4=36.
- Mismos exponentes: se multiplican las bases y se deja el exponente. Ejemplo: 23·53=103.
Cociente de potencias: (es casi lo mismo)
- Misma base: se deja la base y se restan los exponentes. Ejemplo: 36·34=36-4=32.
- Mismos exponentes: se dividen las bases y se deja el exponente. Ejemplo: 103:53=23.
Potencia de potencia: se multiplican los exponentes. Ejemplo: (102)3=106 y ((102)3)5=1030.
Exponente 0: el resultado es 1. Ejemplo: 230=1, 30=1.
Exponente 1: se deja solo la base. Ejemplo: 231=23 y 31=3.
5
NATURAL NUMBERS
Natural numbers are those we use for counting things. Their mathematical symbol is .
They are 1, 2, 3, 4,… If we include number 0, they are called whole numbers.
Operations with numbers:
Multiplication: it is the result of the repeated addition of the same number. For writing a
multiplication we shall use a dot “·” instead of a cross “x”. For instance: 2+2+2+2+2+2+2 =
2 · 7 = 14. In this case, factors are 2 and 7, and their product 14.
Division: it is used any time that distributing in equal parts is needed. Remember that:
dividend equals divisor times quotient plus remainder.
Exponentiation: is the result of the repeated multiplication of the same number. For
instance: 2 · 2 · 2 · 2 · 2 = 25 = 32. Here, the base is 2 and the exponent 5. (it can be read as 2
raised to the 5th power, or 2 to the 5, or 2 raised to the exponent [of] 5)
Square root: it is to find a number such that squared equals to the radicand (the number
we want to take the root). For instance, , since 32 = 9.
Order: We will use symbols < and > .
We shall say that 3 < 5 (3 is less than 5) and 10 > 2 (10 is greater than 2).
Perfect square: Any natural number that is the square of another one. For instance, 22=4,
32=9, 42=16, 52=25, 62=36, 72=49, 82=64, 92=81, etc.
Exercise: calculate and memorize the following perfect squares.
112 = 122 = 132 = 142 = 152 = 252 =
Exercise: calculate roots:
Order of operations (precedence rules):
Accounts are not always performed from the left side to the right. We must follow this order:
1) Parentheses (brackets).
2) Exponents and roots.
3) Multiplication and divisions
4) Addition and subtraction.
6
Addition properties
Commutative: The order in which two numbers are added does not matter. For instance:
3 + 5 = 5 + 3 = 8.
Associative: when adding several numbers, the order in which additions are performed does
not matter. Example: 3+5+2=(3+5)+2=3+(5+2)=10
Identity element: when adding zero to any number, the quantity does not change.
Properties of multiplication
Commutative: one can multiply in any order; “the order of the factors doesn’t change the
product”. For instance: 3 · 5 = 5 · 3 = 15.
Associative: When three or more numbers are multiplied, the product is the same regardless
of the grouping of the factors. For example: 3 · 5 · 2 = (3 · 5) · 2 = 3 · (5 · 2) = 30
Identity element: anything multiplied by one is itself.
Distributive: when multiplying a number by an addition, one can multiply first each addend.
For instance: 2 · (3 + 4) = 2 · 3 + 2 · 4 = 6 + 8 = 14.
Exponent Properties
Product of powers:
- Same bases: add the exponents and keep the common base. Example: 32·34=32+4=36.
- Same exponents: multiply the bases and keep the exponent. Example: 23·53=103.
Quotient of powers:
- Same bases: subtract the exponents and keep the common base. Example: 36·34=36-4=32.
- Same exponents: divide the bases and keep the exponent. Example: 103:53=23.
Power to a power:: multiply the exponents. Example: (102)3=106 and ((102)3)5=1030.
Zero exponent: any number raised to the zero power is equal to 1. Example: 230=1, 30=1.
First power: just keep the base. Example: 231=23, 31=3.
7
NATURAL NUMBERS – PRACTICE
1. Calculate:
a) = b) =
c) = d) =
e) = f) =
g) = h) =
i) 114:112-10 :104-1+2· 3- - = j) =
k) = l) - - =
2. Calculate
a) = b) =
c) = d) =
e) = f) =
g) = h) =
i) - - = j) =
k) = l) =
Solutions: 1. [a] 5 [b] 13 [c] 11 [d] 23 [e] 29 [f] 15
[g] 28 [h] 1 [i] 24 [j] 4 [k] 2 [l] 20.
2. [a] 12 [b] 10 [c] 0 [d] 6 [e] 7 [f] 11
[g] 1 [h] 1 [i] 5 [j] 20 [k] 9 [l] 14.
8
Natural Numbers - Word Problems
1. A lorry is carrying a car and a van. The car weighs 3,582kg and the van 893 kg more than the car. How
much weight is the lorry carrying?
2. A cinema sells 458 tickets for the afternoon show and 137 less for the evening show. How many
tickets are sold in total?
3. Pepe has €142,3 4 in the bank, and he wins €34,023 in the lottery. How much more money does he
need to buy a house that costs €271,000? (*) Remember: comma is NOT the decimal point.
4. A salesman gets an order for 35 televisions that cost €20 each. How much is the invoice (factura) for?
5. A dumping lorry carries 8,500kg of sand each time it makes a trip. How
many kg does it carry in a total of 15 trips?
6. A factory produces 205 cars every day. If it sells each car for €1 ,205,
how much money does it make over 30 days?
7. 36 sugar cubes are split evenly (a partes iguales) between four sugar pots (cafetera). How many cubes
are in each pot?
8. If you have a total of 247 kg of potatoes, how many 5kg sacks can be filled with potatoes? How many
kilos of potatoes are left over?
9. Four partners split €7,52 of profits (beneficios) from their business. How much does each get?
10. A lorry is carrying 12 pallets that have a combined weight of 2,496kg. How much does each weigh?
11. On a farm, 547 eggs are collected and put into cartons of one dozen. How many cartons are filled?
How many eggs are left over?
12. A company pays € , 40 to rent an office for six months (180 days). How much does the company pay
each day?
13. When Juan opened his shop this morning he had € 5 in the till (caja). When he closed it this evening
he had €2, 5. How many Euros worth of goods (euros de mercancías) did he sell during the day?
14. A warehouse worker buys 4,500kg of oranges and uses them to fill cartons that hold 15kg each. How
many cartons does he fill?
15. A bottling factory fills 225 soft drink (refresco) bottles per minute. How many bottles are filled in a
quarter of an hour?
16. Last night 586 people spent the night in a hotel. Today, 147 new clients arrived and 208 of the old
clients left. How many clients will sleep in the hotel tonight?
17. A lorry is carrying 8,600kg of flour packed into 50kg sacks. How many sacks is the lorry carrying?
18. A butcher sells 5 kg of beef (carne de vaca) at €13 per kilo and 3 kilos of pork (carne de cerdo) at €7
per kilo. How much money does he get in total?
9
1st ESO Mathematics Exam (trial exam) IES Extremadura
Bilingual section
Surname ______________________________________________ Mark
Name __________________________________
NATURAL NUMBERS
1. (0.75 pts.) The square of a village has 78 tiles (baldosas) long and 67 width. How many tiles does it
have altogether (en total)?
Answer:
2. (0.75 pts.) A factory of sheets of paper will give out (repartir) 42321 packages among 23 high
schools. How many of them will receive each? How many will be spared (de sobra)? Check it.
Answer:
3. (1 pt.)
- Write an example of the commutative property of the addition:
- Calculate, using the distributive property:
a) 5·(3+4)= b) (6 + 3+1)·2=
4. (3.5 pts.) Calculate, considering the order of operations:
a) 2 + 3 · (7 - 2 · 3 + 5) – 1 + 4 = b) =
c) =
5. (2 pts.) Use the properties of powers:
a) b) =
c) = d)
6. (1 pt.) Approximate the following square roots, (writing how many digits they have, and the first one).
a)
b)
c)
d)
7. (1 pt.) Calculate the following square root and its remainder. Check the result: .
11
2. DIVISIBILIDAD
Múltiplos (de un número): son los que se obtienen al multiplicar el número por algún otro.
Ejemplo: 6 es múltiplo de 3 porque 6 = 3 · 2. Podemos escribir 6 .
También, diremos que 3 es divisor de 6, o que 6 es divisible por 3 (la división es exacta).
Número primo: si sólo es divisible entre 1 y entre sí mismo. Ejemplo: 5 es primo ya que
sólo se puede dividir entre 1 y 5.
Criterio de la raíz: para ver si un número es primo, basta probar si es divisible entre alguno
de los primos menores que su raíz cuadrada. Si es menor que 100, basta probar con 2,3,5,7.
Número compuesto: el que tiene varios divisores. Ejemplo: 6 es compuesto porque 3 es
divisor suyo; esto es, porque es múltiplo de 3. El 1 no es primo ni compuesto.
Factorizar un número: es escribirlo como producto de números primos.
MCD y MCM
El máximo común divisor (MCD) de varios números es el MAYOR de sus divisores (comunes).
Si tenemos la descomposición en números primos, se calcula multiplicando los factores
comunes elevados a los menores exponentes. Si no los hay, el MCD es 1, y son primos entre sí.
El mínimo común múltiplo (MCM) es el MENOR de los divisores (comunes).
Si tenemos la descomposición en números primos, se calcula multiplicando todos los factores (comunes o no comunes) elevados a los mayores exponentes.
Criterios de divisibilidad.
Al comprobar la divisibilidad hay que ver si la división es exacta. A veces podemos cambiar
primero el número por otro más sencillo. Los criterios dicen por qué número cambiarlo.
Divisible por Criterio
2 La última cifra (debe ser par: 0, 2, 4, 6, 8).
3 La suma de las cifras.
4 Nos quedamos con las dos últimas cifras. De esas dos, si la primera es par,
podemos quitarla también, y si no, cambiarla por un 1.
5 La última cifra (0 ó 5).
6 Que sea y .
7 Multiplicamos por 2 la última cifra y se resta con lo que queda del número.
9 La suma de las cifras.
11 Sumamos las cifras en lugar par. Luego las de lugar impar, y los restamos..
Para el criterio del 10, 100, etc., basta mirar si acaba en 0, en 00, etc.
12
DIVISIBILITY
Multiple (of a number): is the product of any quantity and the number. Example: 6 is
multiple of 3 since 6 = 3 · 2. We may write 6 .
We also say 3 that is a divisor of 6, or 6 is divisible by 3 (the division leaves no remainder).
Prime number: is a natural number that has exactly two distinct natural number divisors: 1
and itself. Example: 5 is prime as only 1 and 5 divide it.
Root test: to check if a number is prime, it is enough to check if it is divisible by any prime
number less than its square root. If it is less than 100, just try with 2, 3, 5 and 7.
Composite number: a number having several divisors. Example: 6 is composite since 3 is its
divisor; that is to say, since it is multiple of 3. Number 1 is neither prime nor composite.
Factoring a number is the decomposition into a product of powers of prime numbers.
GCD y LCM (of several numbers)
The greatest common divisor (GCD) is the LARGEST of their (common) divisors.
It can be found using prime factorizations; multiplying all primes common to them raised to
the smallest exponents. If there are none, GCD is 1, and numbers are coprimes.
The least common multiple (LCM) is the SMALLEST number multiple of all them.
It can be found using prime factorizations; multiplying all primes (common or not) raised to the largest exponents.
Divisibility rules.
To check divisibility one must verify that the division is exact. Sometimes we can replace it
first for a simpler number. Divisibility rules tell us which number.
Divisor Divisibility condition
2 The last digit is even (0, 2, 4, 6, or 8).
3 Sum the digits.
4 Examine the last two digits. If the first of them is even, we can remove it. If
it is odd, replace it by 1.
5 The last digit is 0 or 5.
6 It is divisible by 2 and by 3
7 Subtract 2 times the last digit from the rest.
9 Sum the digits.
11 Add even digits on the one hand and odd digits on the other hand. Then,
substract both quantities.
Rules for 10, 100, etc., just check if it ends in a 0, a 00, etc.
13
PRIME NUMBERS - PRACTICE
1. Use the divisibility rule for number 4: 376, 1268, 504, 9530, 756
2. Use the divisibility rule for number 6: 186, 1798, 105, 770, 522
3. Use the divisibility rule for number 7: 184, 399, 686, 286, 7147
4. Use the divisibility rule for number 11: 5293, 925078, 18634, 66825, 15925.
5. Complete the following chart using the divisibility conditions:
Number 2 3 4 5 6 7 9 11
324 Yes Yes No Yes No
1155 Yes
455 No No No
330 Yes No
1287 No Yes No No No Yes
2156 Yes No No Yes No
6. Use the root test to say if 73 is a prime number
7. Use the root test to say if 109 is a prime number
Solutions
1) 376 , 1268 , 504 , 9530 , 756
2) 186 , 1798 , 105 , 770 , 522
3) 376 , 399 , 686 , 286 , 7147
4) 5293 , 925078 , 18634 , 66825 , 15925
5)
6) It is prime since it is less than 102, and it is neither divisible by 2, 3, 5 not 7.
7) It is prime since it is less than 112, and it is neither divisible by 2, 3, 5, 7 nor 11.
Number 2 3 4 5 6 7 9 11
324 Yes Yes Yes No Yes No Yes No
1155 No Yes No Yes No Yes No Yes
455 No No No Yes No Yes No No
330 Yes Yes No Yes Yes No No Yes
1287 No Yes No No No No Yes Yes
2156 Yes No Yes No No Yes No Yes
14
8. Say if 247 is a prime number.
9. Use Erathostenes’ Sieve (Criba deEratóstenes) to find all the prime numbers less than 120.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40 41 42 43 44 45
46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
61 62 63 64 65 66 67 68 69 70 71 72 73 74 75
76 77 78 79 80 81 82 83 84 85 86 87 88 89 90
91 92 93 94 95 96 97 98 99 100 101 102 103 104 105
106 107 108 109 110 111 112 113 114 115 116 117 118 119 120
10. Factorize (write as a product of prime numbers):
12= 36= 50= 75= 2500= 2025=
245= 648= 341= 5005= 1089= 64=
625= 27= 180= 1400= 495= 363=
343= 3773= 70= 84= 150= 154=
388= 231= 286= 294= 308= 385=
418= 483= 490= 292= 726= 1008=
1372= 2975= 4356= 37268= 427= 747=
329= 3087= 13860= 570= 630= 1610=
159= 6279= 2310= 2730= 1331= 47124=
Solutions
8) It is NOT prime, since it is divisible by 13; 247:13=19.
9) 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113.
10) 12=22·3 36=22·32 50=2·52 75=3·52 2500=22·54 2025=34·52
245=5·72 648=23·34 341=11·31 5005=5·7·11·13 1089=32·112 64=26
625=54 27=33 180=22·32·5 1400=2
3·52·7 495=32·5·11 363=3 112
343=73 3773=7
3·11 70=2·5·7 84=2
2·3·7 150=2·3·5
2 154=2·7·11
388=22·97 231=3·7·11 286=2·11·13 294=2·3·72 308=22·11·7 385=5·7·11
418=2·11·19 483=3·7·23 490=2·5·72 292=22·73 726=2·3·112 1008=24·32·7
1372=22·73 2975=52·7·17 4356=22·32·112 37268=22·11
3·7 427=7·61 747=32·83
329=7·47 3087=32·7
3 13860=2
2·3
2·5·7·11 570=2·3·5·19 630=2·5·3
2·7 1610=2·5·7·23
159=3·53 6279=3·7·13·23 2310=2·3·5·7·11 2730=2·3·5·7·13 1331=113
47124 = 22·3
2·7·11·17
15
11. Use the remainder of the division to answer the following questions:
a) Is 1000 multiple of 50? Why?
b) Find the multiple of 50 previous to 1000 and the next one.
c) Find all the multiples of 50 between 1000 and 1200.
12. Use the remainder of the division to answer the following questions:
a) Is 498 multiple of 18? Why?
b) Find the multiple of 18 previous to 498 and the next one.
c) Find all the multiples of 18 between 500 and 600.
13. Use the remainder of the division to answer the following questions:
a) Is 13 divisor of 385? Why?
b) Find the multiple of 13 previous to 385 and the next one.
c) Find all the multiples of 13 between 400 and 450.
14. Find the Least Common Multiple (LCM) and the Greatest Common Divisor (GCD)
a) 20, 35. b) 20, 30 c) 40, 21 d) 12, 24 e) 12, 19 f) 72, 84
g) 90, 120 h) 24, 50 i) 63, 48 j) 42, 60 k) 36, 45 l) 60, 588
m) 46, 98 n) 105, 135 o) 270, 234 p) 315, 420 q) 48, 52 r) 12, 20
s) 24, 18 t) 45, 144 u) 75, 36 v) 63, 27 w) 14, 56 x) 33, 110.
15. Find the Least Common Multiple (LCM) and the Greatest Common Divisor (GCD)
a) 180, 252, 594 b) 924, 1000, 1250 c) 140, 325, 490
d) 725, 980, 1400 e) 2420, 441, 350 f) 952, 1771, 350.
Solutions:
3. a) Yes, since the remainder is 0. b) 1000-50=950, and 1000+50=1050. c) 1000, 1050, 1100, 1150, 1200.
4. a) No, since the remainder is 12. b) 498-12=486, and 486+18=504. c) 504, 522, 540, 558, 576, 594.
5. a) No, since the remainder is 8. b) 385-8=377, and 377+13=390. c) 403, 416, 429, 442.
6. a) 22·5·7; 5 b) 22·3·5; 2·5 c) 23·3·5·7; 1 d) 2
3·3; 22·3 e) 22·3·19; 1 f) 2
3·32·7; 22·3
g) 23·3
2·5; 2·3·5 h) 2
3·3·5
2; 2 i) 2
4·3
2·7; 3 j) 2
2·3·5·7; 2·3 k) 2
4·3
2·5; 3
2 l) 2
2·3·5·7
2; 2
2·3
m) 2·72·23; 2 n) 3
3·5·7; 3·5 o) 2·3
3·5·13; 2·3
2 p) 2
2·3
2·5·7; 3·5·7 q) 2
4·3·12; 2
2 r) 2
2·3·5; 2
2
s) 23·3
2; 2·3 t) 2
4·3
2·5; 3
2 u) 2
2·3
2·5
2; 3 v) 3
3·7; 3
2 w) 2
3·7; 2·7 x) 2·3·5·11; 11
7. a) 22·33·5·7·11; 2·32 b) 2
3·3·54·7·11; 2 c) 22·52·72·13; 5
d) 23·5
2·7
2·29; 5 e) 2
2·3
2·5
2·7
2·11
2; 1 f) 2
3·5
2·7·11·17·23; 7.
16
DIVISIBILITY WORD PROBLEMS
(*) Set these problems up using either the LCM or the GCD.
1. The Foro Theatre and the Trajano Theatre run movies continuously, and each starts its first feature at
1:00 P.M. (at 13:00 hours) If the movie shown at the Foro lasts 80 minutes and the movie shown at
the Trajano lasts 2 hours, when will the two movies start again at the same time? How many of each
film will be shown?
2. Ana has 450 football cards and 840 basketball cards. She wants to place them in stacks on a table so
that each stack has the same number of cards, and no stack has different types of cards within it.
What is the largest number of cards that she can have in each stack? How many stacks will she have?
3. Juan has just moved to a new town and is celebrating a party for his neighbours. He buys 120 cookies
(galletas) and 90 brownies (pastel de chocolate y nueces) to share, and wants to split (repartir) them
equally among the plates with no food left over. What is the greatest number of plates he can make to
share? How many of each will be on each plate?
4. Jeffrey has 36 packets of sugar and 48 packets of artificial sweetener (edulcorante). He wants to divide
them into identical groups, with no packets left over, so that the groups can be distributed to some
tables at the restaurant where he works. What is the greatest number of groups Jeffrey can make?
5. Luis goes to the grocery store (tienda “de ultramarinos”) every 14 days and visits the gym every 6
days. If he did both errands (recado) today, how many days will pass before he does both on the same
day again? How many times will he have done each errand?
6. Gustavo's Sports sells golf tees (taco de golf) in packs of 15. Meanwhile, Travis's Gear (equipamiento)
sells them in packs of 10. If both shops sold the same number of golf tees this week, what is the
smallest number of tees each could have sold? How many packs have sold each of them?
7. A committee organizing a marathon has 42 jugs of water and 21 jugs of sports drink. The committee
would like to set up a number of refreshment stations along the marathon course, with the same
combination of jugs of water and jugs of sports drink at each station, with no beverages (bebidas) left
over. What is the greatest number of refreshment stations that can be set up? How many of each jug
will be at each station?
8. Antonio's Bath Shop sells bars of soap in boxes of 18 bars and bottles of soap in boxes of 10 bottles.
An employee (empleado) is surprised to discover that the shop sold the same number of bars and
bottles last week. What is the smallest number of each type of soap that the shop could have sold?
How many boxes of each type?
9. Robert has two pieces of rope (cuerda), one 420 dm long and the other 168 dm long. He wants to cut
them up to produce many pieces of rope that are all of the same length, with no rope left over. What
is the greatest length, in dm, that he can make them? How many pieces will he have in total?
10. Elliot tells his mother that the secret number he is thinking of is divisible by 22 and tells his father that
it is divisible by 20. If Elliot is telling the truth to both of his parents, what is the smallest secret
number that Elliot could be thinking of? (*) Hint: divisible by 22 is the same as multiple of 22
17
1st ESO Mathematics Exam (trial exam) IES Extremadura
Bilingual section
Surname ______________________________________________ Mark
Name __________________________________
REVISION (4 pts.)
1) (1 pt.) Use the properties of exponents:
a) 212·512·812 = b) (122)5:210= c) 64·65: 69=
2) (2 pts.) Calculate:
a) 056 195:54·409 b) 40055:3021 25
3) (1 pt.) Calculate 3821 and check the result:
Root: Remainder:
DIVISIBILITY (6 pts.)
4) (1 pt.) Complete the following chart using the divisibility conditions:
Number 2 3 4 5 7 9 10 11
294 Yes No No No
1400 Yes Yes No
495 No
5) (1 pt.) Is 131 a prime number? (Explain why using a sentence)
Answer:
6) (3 pts.) Calculate the LCM (MCM) and GCD (MCD) of the following numbers:
a) 72 and 100. b) 175 and 605. c) 100, 98 and 1573.
7) (1 pt.) Ana is buying nuts (tuercas) and bolts (tornillos-pernos) at a local hardware store (ferretería).
The store sells nuts in packs of 12 and bolts in packs of 10.
If Ana wishes to buy the same number of nuts and bolts, what is the smallest number of nuts that she
can buy?
Answer:
19
3. SISTEMA MÉTRICO DECIMAL
Magnitud: cantidad que se puede medir con números, usando unidades de medida.
Forma compleja: cuando se usan varias unidades para expresar la medida. Ej. 3km 200m.
UNIDADES:
Longitud: Utilizaremos el metro (m).
Múltiplos: decámetro (dam), hectómetro (hm), kilómetro (km).
Submúltiplos: centímetro (cm), decímetro (dm), milímetro (mm).
Masa: el gramo (g). Además, se usan la tonelada t=1000kg, y el quintal 1q=100kg.
Capacidad: el litro (l). (y a veces el miriagramo mag=10kg)
Superficie: el metro cuadrado (m2). Los múltiplos y submúltiplos van de 100 en 100.
- Unidades agrarias: área a=100m2, hectárea ha=10.000m2, centiárea ca= 1m2.
Volumen: el metro cúbico (m3). Los múltiplos y submúltiplos van de 1.000 en 1.000.
RELACIÓN ENTRE UNIDADES: 1l =1dm3 (1ml =1cm3). Un litro de agua destilada pesa 1kg.
SISTEMA SEXAGESIMAL
Se usa para medir ángulos (grados, minutos y segundos) y tiempo (horas, minutos y segundos).
Ejemplos: Un ángulo puede medir 30º 20’ 15’’, y una película durar 1h 20m 10s.
Los múltiplos y submúltiplos van de 60 en 60.
Tipos de ángulo:
Agudo: mide menos de 90º. Obtuso: más de 90º. Recto: 90º. Llano: 180º.
Posición relativa de dos ángulos:
- Complementarios: suman un ángulo recto.
- Suplementarios: suman un ángulo llano.
- Consecutivos: tienen el mismo vértice y un lado en común.
- Adyacentes: son suplementarios y consecutivos.
- Opuestos por el vértice: tienen el mismo vértice y sus lados están sobre las mismas rectas.
Los ángulos opuestos siempre miden igual.
km
Hm
dam
M
Dm
cm
mm
h
m
S
10 10
10 10
10 10
:10 :10
:10 :10
:10 :10
:60 :60
60 60
20
METRIC DECIMAL SYSTEM
Magnitude: amount that can be measured with numbers, using units of measurement.
Mixed units: When we use multiple units to express the quantity. Ex. 3km 200m.
UNITS:
Length: We use the metre (m).
Multiples: decametre (dam), hectometre (hm), kilometre (km).
Submultiples: centimetre (cm), decimetre (dm), millimetre (mm).
Mass: gram (g). They are also used the ton t=1000kg, and quintal 1q=100kg.
Capacity: the litre (l). (and sometimes the myriagram mag=10Kg)
Surface: square metre (m2). Multiples and submultiples are counted by 100ths.
- Agrarian units: are a=100m2, hectare ha=10.000m2, centiare ca= 1m2.
Volume: cubic metre (m3). Multiples and submultiples are counted by 1000ths.
RELATIONSHIPS BETWEEN THEM: 1l =1dm3 (1ml =1cm3). One litre of distilled water weighs 1kg.
SEXAGESIMAL SYSTEM
It is used to measure angles (degrees, minutes and seconds) and time (hours, minutes and seconds).
Examples: An angle can measure 30º 20’ 15’’, and a film last 1h 20m 10s.
Multiples and submultiples are counted by sixtieths.
Kind of angles:
Acute: measures less than 90º. Obtuse: greater than 90º. Right: 90º. Straight: 180º.
Relative position of a pair of angles:
- Complementary: if the sum of them is 90º.
- Supplementary: if they sum a straight angle.
- Consecutives: they have a common vertex and a common arm.
- Adjacents: supplementary and consecutives.
- Vertically opposites: they have a common vertex and their arms lie on the same lines.
Opposite angles always measure equal.
km
Hm
dam
M
dm
cm
mm
h
M
S
10 10
10 10
10 10
:10 :10
:10 :10
:10 :10
:60 :60
60 60
21
METRIC DECIMAL SYSTEM WORD PROBLEMS
1. Ethan lives at one end of Park Avenue. Brian lives at the other end of the avenue. It is 5.8
kilometres from one end of Park Avenue to the other. If Ethan walks 2.79 kilometres toward
Brian's house, how many meters does he have to walk to get there?
2. Aaron and Noah wanted to have a contest (competición) to see which of their paper
airplanes could fly the longest distance. Aaron's plane flew four meters. Noah's plane only
flew seventy-nine centimetres. How much further did Aaron's plane fly?
3. Benjamin has five aquariums at home. The tanks hold eighty-seven litres, forty-four litres,
sixty-eight litres, seventy-nine litres, and thirty-four litres of water. How much water do all of
the tanks hold in all?
4. Anthony has a pet spider and a pet guinea pig. The spider is 10 mm long and the guinea pig is
35 cm long. How much longer is the guinea pig than the spider (in mm)?
5. There are eight aluminium cans sitting on the shelf. Each can contains two hundred
sixty-four millilitres of soup. How many litres of soup are there in total?
6. Tamara has got a ten-litre jug of orange soda. She has already drunk three thousand, six
hundred-fifty millilitres. How much soda is left in the jug, in ml?
7. The cost of shipping (enviar) a package is $3.2 for kilogram shipped. Juan, Rosa and Samuel,
each have one package to ship. Each package is a different weight (6kg, 8 kg 22 g and 404 g).
Compute, the weight of each package in kg, and the cost to ship the package.
8. Antonio is going to the river near the school. If Antonio walks to his friend house and then to
the river, it would be five thousand, six hundred sixty-six meters. If he walks straight to the
river, it will only be four kilometres.
How many more meters does Anthony need to walk if he walks to his friend house first
instead of (en lugar de) walking directly to the river?
9. A naughty (travieso) boy heard someone on television talking about how good a milk bath is
for one’s skin. He decides to try it. He goes to the refrigerator and got out all the milk. The
bathtub (bañera) holds twenty-seven litres of liquid. If there are 75 centilitres of milk in each
bottle, how many bottles will he have to use to fill up the bathtub?
10. The volume of the tank in a factory is 6m315dm3500cm3. How many litres is that?
11. A plane took off (despegó) at 12h45m and landed at 15h35m. How much did the flight last?
22
12. Our property has five sides. They are 7,2 decametres, one 1,1 hectometres,
1hm5dam8m, eighty-five meters and 1hm3dam. What is the perimeter of
our property in meters?
13. If one feather (pluma) weighs seven hundred milligrams, how much will thirty-seven
identical feathers weigh? (Specify your answer in grams)
14. Matthew keeps track of his weight on a calendar. On April 1 he weighed forty-six kilograms.
On May 1 he weighed nine hundred grams more. By June 1 he had gained another two
kilograms. How much was his weight, in kilograms, on the first of June?
15. Eric wants to fill up his car's gas (gasoline) tank (depósito). The tank holds 21 litres and
currently has 5l 8dl. How many litres of gas will it take to fill the tank?
16. Sydney's boyfriend gave her a heart-shaped box of candy (dulces) with 7dag9g of chocolate
candy in it. Elizabeth was jealous (celosa) because her boyfriend only gave her 3dag of
chocolate candy. How many grams more did Sydney get?
17. Katherine was given a tiny lamb (cordero pequeñito) by her uncle. The lamb was a runt (el
más pequeño de la camada) and had to be fed (alimentado) by hand. Every three hours
Katherine gives it two hundred ninety millilitres of formula. How much formula, in litres, will
the lamb get in 24 hours?
18. There is a jar in refrigerator. If Julia pours two hundred eight millilitres of water in the jar six
times to fill it, how many litres of water does it take to fill the jar?
19. A lorry is carrying 8.5 tons load (carga) and unloads 1q 20kg and then another 2t 500kg.
What is the load remaining in the lorry? If the lorry unloads 1,750kg more and later on loads
a weigh of 28.3q, what is the load in the lorry now?
20. We have 2.5ha32a planted with sunflowers (girasol). We learn (nos enteramos) that lands
larger than 30,000m2 can ask for a subsidy (pedir una subvención). How much land is left to
plant to ask for the subsidy?
21. A clock gains (se adelanta) 3m 3s every day. How much will gain in a month (30 days)?
22. A box of matches (cerillas) has 40cm3 volume. How many boxes can we keep in a 1.8dm3
drawer (cajón)?
23. Marta has run 8km in 1h30m12s. How long did it take her to run each km?
23
Revision Exercises
METRIC DECIMAL SYSTEM
1) Calculate using kg: 23.4t 5q 40.9kg – 2t 62300hg =
2) Find the perimeter of a field whose sides measure: 123m 40cm, 423.5cm, 0.31km and
2.4hm.
How much is it left (cuánto le falta) to be 7hm? (use mixed units)
3) The flowerbed (parterre, arriate) of my house has the following
measures:
height = 4m, width = 50cm, depth = 2dm.
a) Compute its volume in m3. V=
b) Compute its capacity in litres. C=
[*] Volume is computed as HeightWidthDepth.
4) How many hectares are in 52,321.8 m2? Say why. (Dí por qué)
SEXAGESIMAL SYSTEM
5) Write using seconds: 4h 20m 30s=
6) Write using mixed units: 35.12º =
7) Look at the figure. If angle A=50º 1 ’ 4 ’’,
B= ..... because .....
C=...
D= .... because ......
Solutions:
[1] 15,710.9kg [2] Perimeter: 6hm 7dam 3m 4dm. It is left 2dam 6m 6dm to be 7hm.
[3] V=4·0.5·0.2=0.4m3. C=0.4·1000= 400litres.
[4] Solution: 52,321.8m2 = 523.218m2= 5.23218ha, because 1ha = 10,000 m2.
[5] : 15630s [6] 35º 7’ 12’’.
[7] B=50º1 ’4 ’’ because it is the same as A. C=180-50º1 ’4 ’’=12 º43’12’’ because it is adjacent to A.
D=12 º43’12’’:3 = 43º14’24’’, because it is one third of C.
24
1º ESO Mathematics (Worked out Trial exam) – [Solutions on the next page] IES Extremadura
Bilingual section
Surname ______________________________________________ Mark
Name __________________________________
DECIMAL METRIC SYSTEM (6 pts.)
1) (1 pt.) Write as metres: 2hm 9 dam 21dm + 34 dam 7m 9 cm =
2) (1 pt.) a) The reservoir (embalse) of Proserpina has a capacity of 3’4hm3.
How many litres of water is it?
b) Write using mixed units (forma compleja): 2315442’21m3=
3) (1 pt.) Write as kilograms: 2’05 t 3’1 q 40 kg 200 g =
4) (1 pt.) How many dam2 are in 8 hectares?
5) (2 pts.) A factory has a surface of 3’12hm2 14’6m2.
How much is it missing to have 5 ha? Answer:
SEXAGESIMAL SYSTEM (4 pts.)
6) (1 pt.) Write using seconds: 3º 5’ 12’’=.
7) (1 pt.) Write using mixed units: 2’31h=
8) (1 pt.) Given angles A=20º25’ 32’’ and B=30º 44’ 2 ’’, compute A+B=
9) (1 pt.) Look at the figure:
Fill the chart
Measure Reason why (motivo) Measure Reason
B C
D E
A=40º B
C
D E
25
DECIMAL METRIC SYSTEM AND SEXAGESIMAL SYSTEM [ANSWER KEY]
1) Write as metres: 2hm 9 dam 21dm + 34 dam 7m 9 cm =639’19m.
hm dam m dm cm 2 9 2 1 0 3 4 7 0 9 6 3 9 1 9
2) (1 pt.) The reservoir of Proserpina has a capacity of 3’4hm3.
a) How many litres of water is it? It is 3’4hm3=3400 000 000 dm3=3,4001000,000l of water.
b) Write using mixed units: 2315442’21m3=2hm3 315dam3 442m3 210dm3.
hm3 dam3 m3 dm3 002 315 442 210.
3) (1 pt.) Write as kilograms: 2’05 t 3’1 q 40 kg 200 g =2400’200kg.
t q (mag) kg hg dag g 2 0 5 0 0 0 0 3 1 0 0 0 0 4 0 2 0 0 2 4 0 0 2 0 0
4) (1 pt.) How many dam2 are in 8 hectares? There are 8ha=80000m2 = 800dam2.
5) (2 pts.) A factory has a surface of 3’12hm2 14’6m2.
How much is it missing to have 5 ha? Answer: It is 5-3’12146=1’87854 ha missing.
hm2 dam2 m2 dm2 (It has 03 12 14 60 = 31214’6m2=3’12146ha of surface).
6) (1 pt.) Write using seconds: 3º 5’ 12’’=185’12’’=11112’’.
3º = 180’; 185’ = 11100’’.
7) (1 pt.) Write using mixed units: 2’31h=2h18m36s.
0’31h=18’6m, and 0’6m=36s.
8) (1 pt.) Given angles A=20º25’ 32’’ and B=30º 44’ 2 ’’, compute A+B=51º10’.
A+B = 50º69’60’’ = 50º70’0’’ = 51º10’.
9) (1 pt.) Look at the figure:
Fill the chart
Measure Reason Measure Reason
B 180º It is adjacent to A=40º C 40º It is vertically opposite to A
D 180º It is the same angle as B E 40º It is the same angle as A
A=40º B
C
D E
26
1st ESO Mathematics Trial Exam IES Extremadura
Bilingual section
Surname ______________________________________________ Mark
Name __________________________________
METRIC DECIMAL SYSTEM (6 pts.)
1) (1 pt.) Write as metres: 3hm 4 dam 21dm + 34 dam 7m 9 cm =
2) (1 pt.) Write as litres: 2 0000dal - 1000l 25000 dl =
3) (0.5 pts) Write using mixed units (forma compleja): 1837’02 g =
(0.5 pts) Write as kilograms: 0’32 t 1’5 q 17 kg 16 g =
4) (0.5 pts) How many dam2 are 6 hectares?
(0.5 pts) How many hectares are 2km2?
5) (2 pts.) We want to fence in (vallar) a squared field of 3dam 50 cm of side.
a. How many metres of fence do we have to buy?
b. If each metre of fence costs 15’50€, how much does it cost fencing the field in?
SEXAGESIMAL SYSTEM (4 pts.)
6) (1 pt.) Write using hours, minutes and seconds: 18930s=.
7) (1 pt.) Write using seconds: 10º 20’’=
8) (1 pt.) Given angles A=20º35’ 15’’ and B=30º 24’ 50’’, compute A+B=
9) (1 pt.) Look at the figure:
a. Which angles are vertically opposites? nWhich ones are adjacents?
b. How much does angle C measure? And angle B?
B A=30º
C D E
27
4. POLÍGONOS Y CIRCUNFERENCIAS
Polígono: figura cerrada y plana limitada por segmentos, llamados lados, que se cortan en
los vértices. Se llaman triángulo, cuadrilátero, pentágono, hexágono, etc.
Si todos los ángulos interiores son convexos, el polígono es convexo. Si alguno es cóncavo, el
polígono es cóncavo. Si todos los lados y ángulos son iguales, el polígono es regular.
- Polígono inscrito: cuando sus vértices están en una circunferencia. Si el polígono es
regular, el segmento que une el centro con el lado se llama apotema.
Los ángulos de todo triángulo suman 180º, y los de un polígono convexo, 180 · (lados-2).
TRIÁNGULOS. Clasificación
Equilátero: todos iguales. Isósceles: dos iguales. Escaleno: todos distintos.
Rectángulo: hay uno recto. Acutángulo: todos agudos. Obtusángulo: uno obtuso.
Rectas y puntos notables:
Medianas: unen cada vértice con el punto medio del lado opuesto. Se cortan en el
baricentro (G) (centroide o centro de Gravedad).
Mediatrices: perpendiculares a los lados que pasan por el punto medio. Se cortan en el
circuncentro (O), que es el centro de la circunferencia circunscrita.
Alturas: perpendicular desde cada vértice al lado opuesto. Se cortan en el ortocentro (H).
Recta de Euler: une el baricentro, circuncentro y ortocentro.
Bisectrices: dividen cada ángulo a la mitad. Se cortan en el
incentro, que es el centro de la circunferencia inscrita.
CUADRILÁTEROS.
- Los ángulos suman 360º.
Paralelogramo: todos los lados paralelos, dos a dos. Puede ser:
Cuadrado: lados iguales y ángulos rectos Rectángulo: ángulos rectos,
Rombo: lados iguales Romboide: lados y ángulos iguales dos a dos.
Trapecio: sólo dos lados paralelos. Puede ser rectángulo, isósceles o escaleno.
Trapezoide: no tienen lados paralelos.
Ángulos
Lados
28
CIRCUNFERENCIAS.
- Curva en la que todos los puntos distan lo mismo de otro llamado centro.
- Radio: segmento que une el centro con un punto de la circunferencia.
- Cuerda: segmento que une dos puntos de la circunferencia.
- Diámetro: cuerda que pasa por el centro.
- Arco: parte de la circunferencia que hay entre dos puntos.
Posiciones relativas.
(*) Tangencia en un punto: cuando dos figuras son “casi” iguales “cerca” de ese punto.
- Recta y circunferencia: pueden ser secantes, tangentes o exteriores. Cuando son tangentes,
el radio es perpendicular a la recta.
- Circunferencias: pueden ser concéntricas, tangentes (interiores o exteriores), secantes,
interiores o exteriores.
29
POLYGONS AND CIRCLES
Polygon: closed and plane figure bounded by segments, called edges or sides that meet at
the vertices (vertex, corners). They are called triangle, quadrilateral, pentagon, hexagon, etc.
If all interior angles are convex, the polygon is said to be convex. If there is any concave, it is
said to be concave. If all angles and sides are equals, the polygon is regular.
- Inscribed polygon: when all the vertices lie on a circumference. If the polygon is regular,
the segment uniting the centre and any side is called apothem.
The sum of angles in a triangle is 180º, and in a convex polygon is: 180 · (sides-2).
TRIANGLES. Classification
Equilateral: all equal. Isosceles: two equal. Scalene: all unequal.
Right: one is right. Acute: all are acute. Obtuse: one is obtuse.
Lines and points associated:
Median: line through a vertex and the midpoint of the opposite side. They intersect in the
barycentre (G) (centroid or centre of mass).
Side Bisector: perpendicular line passing through the midpoint of the side. They meet in the
circumcentre (O), which is the centre of the circumcircle.
Altitude: line through a vertex and perpendicular to the opposite side. They connect in the
orthocentre (H). It is the height of the triangle.
Euler’s line: unites baricentre, circumcentre and orthocentre.
Angle Bisector: line through a vertex which cuts the
corresponding angle in half. The angle bisectors intersect in
the incentre, the centre of the triangle's incircle.
QUADRILATERALS.
- The interior angles of a simple quadrilateral add up to 360º.
Parallelograms: quadrilateral with two pairs of parallel sides. They include:
Square: sides of equal length and right angles Rectangle: right angles,
Rhombus (diamond): all sides are of equal length Rhomboid: two pairs of equal sides and angles.
Trapezium: one pair of opposite sides are parallel. Can be right, Isosceles or Scalene.
Trapezoid: no sides are parallel.
Angles
Sides
30
CIRCUMFERENCES AND CIRCLES
(In English they prefer to talk about circles instead of circumferences).
- Curve where all points are at the same distance of one called centre.
- Radius: segment from its centre to any point of the circle.
- Chord: line segment whose endpoints both lie on the circumference of the circle.
- Diameter: chord that passes through the centre.
- Arc: portion of the circumference of a circle between two points.
Relative positions.
(*) Tangency at a point: when two figures are “almost” equal “near” the point.
- Line and a circle: they can be secant, tangent or exterior. When they are tangents, the
radius is perpendicular to the line.
- Circles: they can be concentric, tangents (internal or external), secants, internal or
externals.
31
INSCRIBED POLIGONS - PRACTICE
16. Try and draw the circumcircle for the following polygons. If it is not possible, show why.
17. Try and draw the incircle for the following polygons. If it is not possible, show why.
32
Solutions:
1)
2)
(It is not possible for this one)
33
TRIANGLES EXERCISES
1. Determine if there can be a triangle with the following side measures. Classify them
a=6 cm b=7.2 cm
c=5.5 cm
a=6.3cm b=2.9 cm
c=2.6 cm
a=1 cm
b=6 cm
c=7.3 cm
a=2.8 cm
b=6.7 cm
c=7.3 cm
a=1.9 cm
b=4.5 cm
c=5.6 cm
a=4.4 cm
b=3.5 cm
c=4.4 cm
a
a
c c
b b
34
2. Draw the circumcircle (circunferencia circunscrita) for the following triangle. Compare it with the
solution.
3. Draw the circumcentre (O), altitudes and orthocentre (H), barycentre (G) and Euler’s line for this
triangle. Compare it with the solution.
4. Draw the angle bisectors (bisectrices), incentre (incentro) and incircle (circunferencia inscrita) for
this triangle. Compare it with the solution.
35
5. Draw the circumcircle (circunferencia circunscrita) for the following triangle. Compare it with the
solution.
6. Draw the circumcentre (O), orthocentre (H), medians and barycentre (G) and Euler’s line for this
triangle. Compare it with the solution.
7. Draw the angle bisectors (bisectrices), incentre (incentro) and incircle (circunferencia inscrita) for
this triangle. Compare it with the solution.
36
8. Draw the circumcircle (circunferencia circunscrita) for the following triangle. Compare it with the
solution.
9. Draw the circumcentre (O), orthocentre (H), medians and barycentre (G) and Euler’s line for this
triangle. Compare it with the solution.
10. Draw the angle bisectors (bisectrices), incentre (incentro) and incircle (circunferencia inscrita)
for this triangle. Compare it with the solution.
37
1st ESO Mathematics Trial Exam IES Extremadura
Bilingual section
Surname ______________________________________________ Mark
Name __________________________________
POLYGONS AND CIRCLES
1. (2 pt.) Say if the following figures are polygons. If they are not, say why, and if they are, write
their name and if they are either concave or convex.
a) b) c) d)
2. (1,5 pt). How much do the angles of this polygon sum?
A measures: B measures:
3. (1 pt.) Can there be a triangle having the following sides measures?
a. 3cm, 8cm and 5cm: (Say why)
b. 6cm, 4cm and 5cm: (Say why)
4. (1 pt.) Write the classification of parallelograms. Draw an example of each type.
5. (2 pts.) Fill the chart with the relative positions:
(Exercise 6 on the back)
C1 & C2
C2 & C3
C3 & P
C2 & r
C3 & r
C2 & Q
C3 & Q
C3 & C4
A
B
P
C1
C2
C3
Q
C4 r
38
6 (2.5 pts)
a) Match concepts related:
Euler’s Line
Barycentre Side Bisectors (Mediatrices)
Circumcentre Altitude
Orthocentre Circumcircle (circunferencia circunscrita)
Incentre Angle Bisector (Bisectrices)
Incircle (circunferencia inscrita)
Medians
b) Draw the circumcircle and the Euler’s line for the following triangle:
39
5. FRACCIONES
Usaremos las fracciones para escribir divisiones donde no necesitemos calcular el cociente.
Ejemplo:
. El conjunto de todas las fracciones se llama números racionales. Su símbolo es .
- Fracción propia: si el numerador es menor que el denominador. Ej.:
.
- Fracción impropia: si el numerador es mayor que el denominador. Ej.:
.
- Cualquier número es una fracción. Basta poner “1” como denominador. Ej.: 3=
.
Propiedad fundamental: dos fracciones son equivalentes (el mismo número) si al multiplicar
en cruz se obtiene lo mismo.
Operaciones con fracciones:
Amplificación: multiplicar numerador y denominador por el mismo número.
.
Reducir a común denominador es amplificarlas para que el denominador sea el MCM.
Simplificación: dividir numerador y denominador por el mismo número.
. Cuando
una fracción no puede simplificarse más, es irreducible.
Orden: reducimos a común denominador y comparamos los numeradores.
(*) Si tienen el mismo numerador, es mayor la de menor denominador.
Suma y resta: reducimos a común denominador, y se suma o restan los numeradores.
Multiplicación: se multiplica en línea.
.
(*) Para calcular una parte de un número, se multiplica. Ej.
de 20€ =
División: se multiplica en cruz.
.
Después de hacer una cuenta, siempre hay que SIMPLIFICAR el resultado.
·2
:3
40
FRACTIONS
We use fractions for writing a division where we do not need to calculate the quotient.
Example:
, which is read as three over four, three fourths or three quarters. The set of all
fractions is called rational numbers. It is represented by the symbol .
(*) A line usually separates the numerator and denominator. If the line is slanting (¾) it is called a
solidus or forward slash. If the line is horizontal, it is called a vinculum or, informally, a “fraction bar”.
- Proper fraction: if the numerator is less than the denominator. Ex.:
.
- Improper fraction: if numerator is greater than denominator. Ex.:
.
- Any number is a fraction. Just place “1” as denominator. Ex.: 3=
.
Fundamental property: we can test if two fractions are equivalents (have the same value) by
cross-multiplying their numerators and denominators (results must be equals). This is also
called taking the cross-product.
Operations with fractions:
Convert: multiply numerator and denominator by the same number.
.
Reduce to common denominator: convert them to turn their denominator into the LCM.
Reduce: divide numerator and denominator by the same number.
. When a fraction
cannot be reduced any more, it is said to be irreducible, or in lowest terms.
Order: reduce to common denominator. The larger fraction is the one with the larger
numerator.
(*) If two fractions have the same numerator, then the fraction with the smaller denominator
is the larger fraction.
Addition and subtraction: reduce to common denominator, and add or subtract
numerators.
Multiplication: multiply across the top and bottom.
.
(*) To calculate a fraction of a number, multiply them. Ex.
of 20€ =
Division: cross-multiply.
.
After a calculation, you need to SIMPLIFY, whenever possible.
·2
:3
41
OPERATIONS INVOLVING FRACTIONS – PRACTICE
1. Order, from the least to the greatest:
2. Simplify: a)
= b)
= c)
= d)
= e)
=
3. Simplify before multiply:
a)
= b)
= c)
= d)
=
e)
= f)
= g)
= h)
:
=
4. Calculate. Remember to simplify, whenever it is possible:
a)
= b)
=
c)
= d)
=
e)
= f)
=
g)
= h)
=
i)
= j)
=
k)
= l)
=
5. Work out
a)
= b)
=
c)
= d)
=
e)
= f)
=
Solutions:
1.
.
2 [a]
[b]
[c]
[d]
[e]
3. [a]
[b] 3 [c]
[d] 2 [e] 4 [f] 1
[h] 2.
4. [a]
[b]
[c]
[d]
[e]
[f]
[h]
[i]
[j]
[k]
[l] 18.
5. [a]
[b]
[c]
[d]
[e]
[f]
.
42
Fractions - Word Problems
1. A kilo of steak fillets (filetes) costs €12. What is the price of one quarter of a kilo?
2. There are 24 students in our class. Two thirds voted for Marta as class representative. How
many votes did Marta get?
3. Only two fifths of the contents of a 15l container of oil are left. How many litres are left?
4. A quarter of a kilo of salami cost me €2. How much does one kilo cost?
5. A quarter of a kilo of ham (jamón) costs € . How much does one kilo cost?
6. Three quarters of a kilo of salami cost € . How much does one kilo cost?
7. Iván cut his pie into 6 pieces and ate 2. Raquel cut her pie into 9 pieces and ate 3. Who has
eaten a larger portion?
8. One fifth of the sheep in a flock (rebaño) are black and the rest are white. What fraction of
the sheep are white? [Sol: 4/5 of the sheep are white]
9. A tank of petrol was full but 3/10 of it was used this morning and another 1/10 this
afternoon. What fraction of the warehouse (almacén) remains full? [Sol: 3/5]
10. Roberto filled 1/4 of his garden with tomatoes and 1/6 with beans (judías). What fraction
of the garden is in use? [Sol: 5/12 is in use]
11. Last week Lucia read 1/3 of an adventure book and this week she read another 1/2. What
fraction of the book does she have left to read? [Sol: She’s read 5/ ; she has 1/ left]
12. A farmer sows (sembrar) half a field with wheat (trigo) and one third of the remaining half
with barley (cebada). What fraction of the field is sown with barley?
13. Half a litre has been used to fill four cups. How much does each cup hold?
14. A 5-litre jug (jarra) is used to fill 30 cups. What fraction of a litre fits in each cup?
15. Marta has gone for a walk and has covered three quarters of the total route so far. If she
still has 2km to walk, how long is the total route? [Hint: set the problem up using a drawing]
€2
43
16. Pedro goes out and spends one fifth of his money at the cinema and one fourth on snacks.
What fraction of his money does he spend? If he goes out with €40, how much does he
have left? [He spends /20. Out of 40€ it is 1 €; he’s got 22€ left]
17. Half of the passengers on a plane are European, one fourth African and the rest American.
What fraction of them are Americans? If the plane is carrying 200 passengers, how many
Americans are on the plane? [Sol: 1/4 are Americans; they are 50 passengers]
18. Augusto Theater Company sold out 2/3 of the performances it held. Trajano Company sold
out 1/10 of the performances it put on. Which theatre company sold out a greater fraction
of its performances? [Sol: Augusto sold a greater fraction]
19. We bought half-kilo block of butter (mantequilla) and used a third of it to make a cake.
What fraction of a kilo of butter does the cake contain? [Sol: 1/6 kilo]
20. A perfume bottle can cold 1/5 of a litre.
How many bottles can be filled with one litre of perfume? [Sol: 5 bottles]
How many litres of perfume do you need to fill 20 bottles? [4 litres]
How many bottles can be filled with 3 litres of perfume? [15 bottles]
21. On Wednesday the farmers at the Juanito Farm picked 2/3 of a barrel of tomatoes. On
Thursday, the farmers picked 1/2 as many tomatoes as on Wednesday (la mitad que el
miércoles). How many barrels of tomatoes did they pick on Thursday? [1/3 of a barrel]
22. A cookie factory uses 1/4 of a barrel of flour (harina) in each batch (hornada) If the factory
used 1/2 of a barrel of flour yesterday. How many batches did they make? [2 batches]
23. Yesterday a gardener watered (regó) one third of his field and today he watered half of the
remaining land. What is the total fraction of the land watered? [2/3 of the land]
24. Antonio skated for 1 and 2/3 hours last month and 7 and 2/3 hours this month. How many
hours did Antonio skate in all? (Note: Use mixed numbers) [9 and 1/3 hours]
25. A tea shop brewed (elaborar) 7 and 7/8 pitchers (jarras) of chamomile tea, of which it
sold 3 and 1/8 pitchers. How many pitchers remained unsold? [4 and 3/4 pitchers]
26. In one week, Leslie's family drank 2 and 5/8 cartons of regular milk and 1 and 7/8 cartons
of soy (soja) milk. How much milk did they drink in all? [4 and 1/2 cartons]
44
1º ESO Mathematics Trial Exam IES Extremadura
Bilingual section
Surname ______________________________________________ Mark
Name __________________________________
1. (1 pt.) Write the names of these fractions (in English) using the two ways we use in class.
a) 4
9 b)
11
6 c)
22
1 d)
9
5
2. (1 pt.) Say if the following fractions are proper or improper. If any is improper, write it as a “mixed
number” (quotient plus a proper fraction using the remainder).
a) 35
17 b)
13
68 c)
18
13
3. (0.5 pts). Say if they are equivalents: a) 8
6
6
4and b)
5
2
25
10and
4. (1 pt.).Compute a) ”Two fifths of sixty” b) “five ninths of 72”
5. (1 pt.) In a basketball game, Pedro has scored a sixth of the points, Carlos one half and Juan the
rest. What fraction of points did Juan score? Who scored the most?
6. (1 pt.) Simplify: a) 75
225 b)
60
48
7. (1 pt.) Order from least to greatest: 6
5,
4
1,
3
2
8. (1.5 pts.) Compute:
a)
= b)
c)
=
9. (1 pt.) Calculate:
=
10. (1 pt.) Pedro is 5 years old, Luis 8 and Antonio 12.
a) Write the inverse of these numbers.
b) Order those inverse numbers from least to greatest
.
45
6. NÚMEROS DECIMALES
Usaremos unidades decimales (décima, centésima, milésima,…). El conjunto de todos los
números con decimales se llama números reales. Su símbolo es .
La parte entera va antes de la “coma”, y la parte decimal detrás.
Tipos de decimales:
Exacto: vienen de una división exacta. Sólo tienen algunos decimales.
Periódico: vienen de una división no exacta. Un grupo de decimales (periodo) se repite
infinitamente. Son puros si toda la parte decimal es periódica. Si no, son mixtos.
Irracionales ( ): no son fracciones, y tienen infinitos decimales no periódicos.
Operaciones (cómo se hacen):
Orden: primero comparamos la parte entera. Después la decimal, rellenando con ceros.
Aproximación: - Truncar: quitamos las cifras decimales que no necesitemos.
- Redondear: truncamos, pero hay que aumentar en uno la última cifra si la
siguiente cifra era mayor o igual que 5.
Suma y resta: hay que colocar las “comas” encima de las comas.
Multiplicación: en el resultado hay que poner tantos decimales como tuvieran los factores.
División: corremos las comas hasta que el divisor no tenga decimales. Después, cuando
bajamos el primer decimal, se pone la “coma” en el cociente.
46
DECIMAL NUMBERS
We will use decimal units (tenth, hundredth, thousandth,…). The set of all decimal numbers is
called real numbers. It is represented by the symbol .
The integer (integral) part is the part to the left of the point, and the decimal part to the right.
Types of decimal numbers:
Terminating: they come from a terminating decimal fraction. Decimals stop after a few digits.
Periodic (recurring decimal): they come from a recurring decimal fraction. A group of
decimals (period) repeat forever. We call those which start their recurring cycle
immediately after the decimal point purely recurring. Those that have some extra
digits before their cycles are also called mixed recurring (or eventually recurring).
Irrational ( ): any number which does not stop and does not end with a recurring pattern
(thus they are non-fractional numbers).
Operations (how to perform them):
Order: compare first the integer part. Then the decimal, filled with trailing zeros.
Approximation: - Truncating : remove the decimal digits you do not need.
- Rounding: truncate, but increase in one the last digit if the next one was
greater or equal to 5.
Adding and subtracting: line up the terms so that all the decimal points are in a vertical line.
Multiplication: the number of decimal places of the result is the sum of the decimal places in
the factors.
Division: move all decimal points to right to make the divisor a whole number. Put decimal
point in the quotient when you use the first decimal of the dividend.
(to some decimal places)
47
Decimal Numbers - Word Problems
Add and subtract
1. Tamara bought 0.99 pounds (libra=453,6g) of peanuts and 0.67 pounds of raisins (uvas
pasas). How many pounds of snacks did she buy in all?
2. Cesar put 3.43 grams of salt in a recipe of soup. Then he let the soup cook for an hour.
Before he served the soup, Cesar added 5 grams of salt. How much salt did Cesar add to
the soup in all?
3. On Monday, Luisa walked 9.53 kilometres. On Tuesday, she walked 1.48 kilometres less
than she had walked on Monday. How far did Luisa walk on Tuesday?
4. It snowed 0.75 inches on Monday and 0.9 inches on Tuesday. How much did it snow in all
on Monday and Tuesday? [Sol: 1.65 inches]
5. Roger ran 4.68 miles (milla=1.6km). Then biked 5.44 miles. How far did he go? [10.12miles]
6. In March it rained 0.72 inches. It rained 0.62 inches less in April than in March. How much
did it rain in April? [Sol: 0.82inches]
7. On a school trip, a class travels 21.597 kilometres by train and 48.212 kilometres by bus.
How far did the class travel? [Sol: 69.809km]
Multiplication
8. Louis walks 4.9 miles each day. How many miles will Louis walk in 9 days?
9. A photography shop can develop 9.5 photographs per hour. How many photographs can it
develop in 5 hours?
10. Yasmine walks 0.23 miles on each trip to the park. How far will Yasmine walk if she makes 6
trips to the park? [Sol: 1.38miles]
11. Each piece of cardboard is 0.1 centimetres thick. If Tanya stacks 6 pieces of cardboard on
top of one another, how thick will the stack be? [Sol: 0.6cm thick]
12. Haley bought 7 plastic discs. Each disc weighs 2.1 ounces (onza=28.35gr) . How much do
the 7 plastic discs weigh in all? [Sol: 14.7ounces]
48
Division
13. A factory used 165.6 kilograms of tomatoes to make 2 batches (lotes) of pasta sauce. What
quantity of tomatoes did the factory put in each batch?
14. A 2-story building is 20.1 feet tall. How many feet tall is each story?
15. Deborah's Taffy (AmE: caramelo masticable) Shop made 898.2 kilograms of
taffy in 8 days. How much taffy, on average, did the shop make per day?
16. Daphne bought a package of 5 oatmeal cookies. The total weight of the cookies was 0.2
ounces. How much did each cookie weigh? [Sol: 0.04ounces]
17. A pasta factory made 1.3 pounds of pasta in 2 minutes. How much pasta, on average, did
the factory make each minute? [Sol: 0.65punds]
Combined word problems
18. In a lift, we load (cargar) 5 bags that weigh 12.745kg each. Then two people who weigh
65kg and 85.7kg go up. The maximum load of the lift is 350kg. Can a person who weighs
86.7kg go up? [Yes, since there are 214.425kg loaded. We can load up to 135.575kg]
19. Jaime is carrying a basket that weighs 1.5kg. He buys two oranges bags that weigh 3.4kg
each. How much weigh is Jaime carrying? [Sol: 8.3kg]
20. How much money does Bob need to buy 8 limousine rides
and 3 helicopter rides? [Sol: £724.04]
21. A Caribbean vacation package costs €1200.55. How much
money does Marie need to buy 8 Caribbean vacation packages? [Sol: € 04.40]
22. Lidia bought 9 “pineapple-upside-down” cakes (pastel de
piña) that each cost the same amount. She spent €112.50 in
all. How much did each cake cost? [Sol: €12.50 each]
23. Miguel and Estrella decided to earn money walking dogs
(ganar dinero sacando a pasear perros). Miguel charges
(cobra) €6.50 per day, and walked a neighbour's poodle (caniche) for 5 days. Estrella
charges €7.20 per day and walked a Dalmatian for 8 days. How much money did they earn
in all? [Sol: € 0.10]
49
1st ESO Mathematics Trial Exam IES Extremadura
Bilingual section
Surname ______________________________________________ Mark
Name __________________________________
1. (1.5 pt.) Fill the chart using the correct number, and write the fractions as a decimal number:
1 Terminating 2. Purely recurring 3. Eventually recurring 4. Irrational
After some decimal digits, there is a group of decimals repeating forever. Decimals stop after a few digits. Decimals have no recurring pattern A group of decimals repeat forever, immediately after the decimal point
5.2431431431431...
2.1234567891011121314...
2. (0.5 pts.) Write as a rational number (fraction) in lowest terms: 1’56=
3. (1 pt.) Order from GREATEST to LEAST. Write in English the name of the least one.
3’8 ; 6’32 ; 3’82 ; 6’235 ; 3’823.
4. (0.5 pt.). Round to thousandths: a) 2.34567= b) 5.234321=
5. (4 pts.) Compute:
a) 32’65+28’326-25’62= b) 86’4-22’563+180=
c) (82’56-20) · 83’25= d) 53'3288'3
17'179
6. (1 pt.) Laura is making 43’5kg of bagels (rosquillas) and wants to pack them in boxes of 0’250kg.
How many boxes does she need?
7. (1.5 pts.) Work out using two decimal places. Check the result 21'5
51
7. NÚMEROS ENTEROS
Los números enteros son todos los naturales (0, 1, 2,…) y sus opuestos (-1,-2,-3,…). Su
símbolo matemático es . Pueden representarse en la recta numérica:
Operaciones:
Orden: Usamos los símbolos < y > . Un número es mayor que otro si está más a la derecha
en la recta numérica. Diremos que 3 < 5 y que -3 > -5.
Suma: sumamos positivos por una parte y negativos por otra. Dos números con distinto
signo se restan, y se deja el signo del mayor. Ejemplo: 3 + 2 – 4 – 5 + 3 = 8 – 9 = -1.
Multiplicación, división: se usa la y paréntesis
Quitar paréntesis: (usando la propiedad distributiva)
Signo “+”: se quita el paréntesis y se deja todo como está. Ej.: +(-5+1)= -5+1.
Signo “-“: se cambia cada signo. Ej.: -(-5+1)= +5 -1.:
Potencia de base negativa (entre paréntesis): si el exponente es impar dejamos el signo “-“
y si es par se quita. Por ejemplo: (-3)2=32=9 y (-2)3 = - 23= - 8.
Mucho cuidado con - 32 = - 9. ¡Si la base NO está entre paréntesis, no se quita el signo!
Raíz cuadrada: No hay raíces cuadradas de números negativos.
Por ejemplo, , porque 32 = 9, pero no existe.
Valor absoluto: es el número sin el signo. |-2| = 2, y |+2| = 2.
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
Regla de los signos
+ · + = +
+ · - = -
- · + = -
- · - = +
52
INTEGER NUMBERS
The Integers are formed by the natural (or whole) numbers (0, 1, 2,…) and their opposites.
Their mathematical symbol is . We may depict the integers on the number line:
Operations:
Order: We will use symbols < and > . A bigger number will always be found to the right
of a smaller number. We shall say that 3 < 5 and 10 > 2 .
Sum: We add positives on the one side and negatives on the other. Integers with different
signs: subtract them and keep the sign of the largest. Example: 3 + 2 – 4 – 5 + 3 = 8 – 9 = -1.
Multiplication, division: one uses the and parenthesis
Eliminate parentheses
“+” sign: remove the parentheses and leave everything as it is. Ex.: +(-5+1)= -5+1.
“-“ sign: change the signs. Ex.: -(-5+1)= +5-1.
Power having negative base (in parentheses): if the exponent is odd, then leave “-“ sign,
but if the exponent is even, remove it. For example: (-3)2=32=9 y (-2)3 = - 23= - 8.
Watch out!: - 32 = - 9. If the base is NOT in parentheses, do not remove the sign!
Square root: There is no square root for negative numbers.
For instance, , because 32 = 9, but does not exist.
Absolute value: it is the number without its sign. |-2| = 2, and |+2| = 2.
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
Rules of signs
+ · + = +
+ · - = -
- · + = -
- · - = +
53
INTEGER NUMBERS - WORD PROBLEMS
1. To play a game on a game board, Juan puts his game piece on START. On his first turn, he
moves his game piece ahead 7 spaces. On his second turn, Juan moves his game piece back 4
spaces. How many spaces away from START is his game piece now?
2. Ana’s game piece is on square 24 of a game board. She draws a card (sacar una carta) that
says, “Move back 4 spaces.” Then she draws a card that says, “Move back 2 spaces.” On
which square is Ana’s game piece now?
3. The temperature outside is 0ºF. If the temperature drops (descender) 14º overnight (por la
noche), what was the overnight low temperature?
4. The temperature outside is -16ºF. Then the temperature rises (aumentar-ascender)
20 degrees. What is the current outdoor temperature?
5. Monarch butterflies travel an average (media) of about 15 feet off the ground. One butterfly
flies to a height of 22 feet. Tell how much higher it flies than average. [*] 1 foot=30.48cm
6. A transaction register is used to record money deposits and withdrawals (sacar dinero) from
a checking account. It shows how much money Vanessa, a college student, had in her
account as well as the 6 checks she has written so far (hasta ahora).
a) Find each balance.
b) Which check did Vanessa write that made her account overdrawn?
c) Vanessa called home and asked for a loan (préstamo). Her parents let her borrow $400.
What is her balance now?
d) After her parents let her borrow the $400, Vanessa wants to spend $200 on clothes and
$150 on decorations for her dorm room. Does she have enough money in the bank?
Express her balance with an integer if she buys these items.
7. Jim was recovering in the shade (a la sombra) from a walk in the hot desert. His temperature
dropped 2ºF each hour for 2 hours. What was the total change in his temperature?
Check No. Date Description of Transaction Payment Deposit Balance
9/04 spending money from parents $500 $500
1 9/07 college bookstore — textbooks $291
2 9/13 graphing calculator $99
3 9/14 Luckily found money! $40
4 9/16 bus pass $140
5 9/24 Charlie’s Pizza $24
6 9/26 School trip $30
54
8. For each kilometre above Earth’s surface, the temperature decreases 7ºC. If the
temperature at Earth’s surface is 20, what will be the temperature 7 kilometres above the
surface?
9. Maria was penalized a total of 12 points in 6 baking contests for not starting on time.
Suppose she was penalized an equal number of points at each competition. Write an integer
that describes the penalty during each contest.
10. The temperature dropped 32°C in 4 hours. Suppose the temperature dropped by an equal
amount each hour. What integer describes the change?
11. Judges in some figure skating competitions must give a mandatory (obligatorio) 5-point
deduction for each jump missed (fallido) during the technical part of the competition. Marisa
has participated in 5 competitions this year and has been given a total of -20 points for
jumps missed. How many jumps did she miss?
12. Katherine is very interested in cryogenics (the science of very low temperatures). With the
help of her science teacher she is doing an experiment on the affect of low temperatures on
bacteria. She cools one sample of bacteria to a temperature of -51°C and another to -76°C.
What was the temperature difference in the two experiments?
13. On Tuesday the mailman delivers 3 checks for $5 each and 2 bills (factures) for $2 each. If
you had a starting balance of $25, what is the ending balance?
14. You owe $225 on your credit card. You make a $55 payment and then purchase $87 worth of
clothes at Dillards. What is the integer that represents the balance owed on the credit card?
15. If it is -25ºF in Rantoul and it is 75ºF in Honolulu, what is the temperature difference
between the two cities?
16. It was a very freaky weather (clima peculiar) day. The temperature started out at 9°C in the
morning and went to -13°C at noon (mediodía). It stayed at that temperature for six hours
and then rose 7°C. How far below the freezing point (0°C) was the temperature at 6 p.m.?
17. A submarine dove (se sumergió) 836 ft. It rose at a rate of 22 ft per minute. What was the
depth (profundidad) of the submarine after 12 minutes? [*] 1 foot=30.48cm
18. The mailman delivered (repartió) a $22 check and 3 -$14 bills today. He also took back 1 -$5
bill. What is the total in the mailbox?
55
INTEGER NUMBERS – REVISION EXERCISES
1. Calculate:
a) = b) =
c) = d) =
e) = f) =
g) = h) =
i) - - - - - - - = j)
k) = l) - - - - - - - =
2. Calculate
a) = b) =
c) = d) =
e) = f) =
g) = h) =
i) - - - - - = j) =
k) = l) =
Solutions: 1. [a] -10 [b] -14 [c] 5 [d] 0 [e] 6 [f] 26
[g] 5 [h] 10 [i] -8 [j] 8 [k] 2 [l] 15.
2. [a] 12 [b] 5 [c] 8 [d] -7 [e] 24 [f] 12
[g] 16 [h] -1 [i] 6 [j] 8 [k] -7 [l] -3.
56
1st ESO Mathematics Trial Exam IES Extremadura
Bilingual section
Surname ______________________________________________ Mark
Name __________________________________
INTEGER NUMBERS
1. (1 pt.) Order from LEAST to GREATEST.
-2, +3, 0, 7, +2, -5, -1, 4, +6.
2. (1 pt.) Compute: a) (-2)·(-3)·(+5)= b) (+35):(-7):(-5)=
3. (4 pts.). Work out
a) (0.5 pts.) -3 + (-2) + 7 - (-4) + 2 -8+3=
b) (0.5 pts.) [-3+7]-[9-(-2)]-7=
c) (1 pt.) 3 - 2· [3 - 4·(-3 - 1)] - (-7)=
d) (1 pt.)
e) (1 pt.) 10 9310)5)·(3(2·3 + =
4. (2 pts.) Apply the properties of exponents and signs
a) (-6)5·63·(-6)4= c) ((-5)2)4·5=
b) (-4)8:45·(-2)3= d) (-2)3·(-2)5:26·30=
5. (1 pt.) I owe (debo) 500€ to a bank. Then, I pay in (ingreso) 300€, spend (gasto) 150€, and last I
pay in 270€. Do I still (todavía) owe money? How much?
6. (1 pt.) Inside a car, the temperature is 24ºC, and outside -2ºC. What is the difference of
temperature?
57
8. PROPORCIONALIDAD
- Razón entre dos magnitudes: es su cociente.
Ejemplo:
. Además, diremos que 8 es el antecedente y 10 el consecuente.
- Proporción: es la igualdad entre dos razones. El número obtenido se llama constante de
proporcionalidad. Ejemplo:
es una proporción y la constante es 2.5. Además, 5 y 10 se
llaman extremos, y 2 y 25 medios.
Relación entre dos magnitudes:
Directa: si una aumenta, la otra también. Si además son proporcionales, para calcular una se
multiplica la constante por la otra, y al doble de una le corresponde el doble de la otra, etc.
Inversa: si una aumenta, la otra disminuye. Si además son proporcionales, para calcular una
se divide la constante por la otra, y al doble de una le corresponde la mitad de la otra, etc.
Cálculo del cuarto proporcional:
Para resolver algo como
, se usa la propiedad fundamental: ( 5·x=1 · 20 )
“El producto de los medios es igual al producto de los extremos”.
Resolución de problemas. Se pueden hacer de dos formas:
1ª: calcular la constante de proporcionalidad “reducción a la unidad”.
2ª: hacer el planteamiento usando la regla de tres.
- Para el Reparto Proporcional, la cantidad a repartir se corresponde con el total.
PORCENTAJES:
Cálculo: se multiplica (el porcentaje en forma de número). Ejemplo: 5% de 20 = 0.05 · 20=1.
Problemas: se pueden hacer de dos formas.
1ª: Por regla de tres. Siempre es directamente proporcional. El total es el 100%.
2ª: Se calculan los porcentajes y luego se hacen las operaciones.
58
PROPORTIONALITY
- Ratio between magnitudes: it is their quotient.
Example:
. Furthermore, we will say 8 is the antecedent and 10 the consequent.
- Proportion: It is a statement that two ratios are equal. The number obtained is called
constant of proportionality. Example:
is a proportion and the constant is 2.5.
Furthermore, 5 and 10 are called extremes, and 2 and 25 means.
Relationship between magnitudes (types of variation):
Direct: the greater the 1st, the greater the 2nd. If they are also proportional, to calculate one,
multiply the constant by the other. If one doubles, the other will also double, etc.
Inverse: the greater the 1st, the smaller the 2nd. If they are also proportional, to calculate
one, divide the constant by the other. If one doubles, the other will become half as large, etc.
Finding the fourth proportional:
To solve something like
, one uses the fundamental property: ( 5·x=1 · 20 )
“the product of the means is equal to the product of the extremes”.
Problem solving. There are two ways:
1st: find the constant of variation “unitary method”.
2nd: set the problem up using the rule of three.
- For Proportional Distribution, the amount to share out corresponds to the total.
PERCENTAGES:
Calculation: multiply (with the percentage as a number). Example: 5% of 20 = 0.05 · 20=1.
Problems: there are two ways.
1st: Rule of three. Variation is always direct. The total is 100%.
2nd: Calculate percentages first and then perform the calculations.
59
PROPORTIONALITY - WORD PROBLEMS
1. Compute the following ratios (using a sentence and a fraction in lowest terms):
a) Pyramids to total figures. b) Pyramids to cylinders.
c) Cylinders to total figures d) Cylinders to cubes.
e) Cubes to pyramids. f) Cubes to total figures.
2. Compute the following ratios (using a sentence and a fraction in lowest terms):
a) Puppies (perritos) to total pictures. b) Puppies to kitties.
c) Kitties (gatitos) to total figures d) Bear cubs (oseznos) to puppies.
e) Bear cubs to kitties. f) Bear cubs to total pictures.
3. Are these ratios equivalent?
a) “ large rubber bands to 3 small rubber bands” and “12 large
rubber bands to small rubber bands”
b) “8 kilograms in 6 minutes” and “4 kilograms in 3 minutes”
c) “12 employees every 18 months” and “2 employees every 5 months”
d) ”3 large clipboards to 4 medium clipboards” and “15 large clipboards to
20 medium clipboards”
4. Do these ratios form a proportion?
a) “4 children for every 9 adults” and “8 children for every 18 adults”
b) “14 houses to 6 apartments” and “7 houses to 4 apartments”
c) “3 elementary school students for every 15 middle school students”
and “4 elementary school students for every 11 middle school students”
d) ” 14 crates (cajón de embalar) every 20 minutes” and “21 crates every
30 minutes”
e) “30€ per 5 employees” and “110€ per 20 employees”
60
Direct proportionality
5. The ratio of boys to girls in history class is 4 to 5. How many girls are in the class if there are
12 boys in the class?
6. A factory produces 6 motorcycles in 9 hours. How many hours does it take to produce 16
motorcycles?
7. James read 4 pages in a book in 6 minutes. How long would you expect him to take to read 6
pages?
8. According to a recipe (receta) we can make 3 pies with 7 cups of flour (harina). How many
pies can be made with 28 cups of flour?
9. Recently, frogs have been disappearing from the wild due to a contagious fungus. To
monitor the current population in a certain region, Pepe collects 100 frogs, marks them, and
releases them. Later, he collects 770 frogs and finds that 35 of them are marked. To the
nearest whole number, what is the best estimate for the frog population?
10. Sara can type 90 words in 4 minutes. About how many words would you expect her to type
in 10 minutes?
Inverse Proportionality
11. If a school kitchen has enough food for 234 pupils for 37 days, how long will the same food
last 73 pupils ?
12. You can exactly fit 7 volumes of an encyclopaedia each 5.7cm thick (grueso) on a shelf
(estantería). How many volumes each 6.7cm thick fit on the same shelf?
13. If 7 electricians can wire (cablear) some new houses in 17 days , how many electricians
would be required to do the job in 9 days? [*] Sol:13’22, so we take on 14 electricians.
14. Jane can type at 6o words a minute and took 35minutes to complete a letter. How long
would John take who types at 43 words a minute ? [*] Sol: 48.8min=48m48s.
15. A car travelling at 45km/hr takes 33 minutes for a journey. How long does a car travelling at
55km/hr take for the same journey ? [*] Sol: 27 min.
61
Proportionality
16. The Lakewood Wildcats won 5 of their first 7 games this year. There are 28 games in the
season. About how many games would you expect the Wildcats to win this season?
17. A farmer has enough cattle (ganado) feed to feed 64 cows for 2 days. How long would the
same food last 32 cows? [*] Sol: 4 days.
18. Two slices of Dan’s Famous Pizza have 230 Calories. How many Calories would you expect to
be in 5 slices of the same pizza?
19. Andy paid 1.40€ for 4 grapefruits (pomelo). How many of them can he purchase for 2.10€?
20. It takes 12 workers 28 hours to leaflet (repartir folletos) an area of houses. How long would
21 workers take to do the same work ? [*] Sol: 16 hours.
21. A car travelling at an average speed of 67km/hr takes 14 hours to complete a journey. How
long would the same journey take at 42km/hr ? [*] Sol: 22.4h=22h24m.
22. Leslie is monitoring the population of wild camels in the deserts of Australia. Leslie catches
46 camels, tags them, and releases them. A little while later, a helicopter pilot takes an aerial
photograph of 180 camels, 23 of which are marked. To the nearest whole number, what is
the best estimate for the camel population?
23. 160 markers (corrector de exámenes) take 3 hours to complete marking their examination
scripts (examines escritos). How many markers will complete it in 48 hours?[*] Sol: 10 markers.
24. Pansy, is going to estimate how many catfish (pez gato-bagre) are in
a local river. First, she marks 240 catfish with tags. Later, she
captures 910 catfish, of which 24 had tags. To the nearest whole
number, what is the best estimate for the catfish population?
25. A farmer has enough cattle feed to feed 300 hens for 20 days. If he buys 100 more hens,
how long would the same amount of feed feed the total hens? [*] Sol: 15 days.
26. Marta grew 10 plants with 5 seed packets. With 8 seed packets, how many total plants can
Marta have in her backyard (patio trasero)? [*] Sol: 16 plants.
27. 5 taps (grifo) can fill a container in 10 hours. If we use 8 taps, how long will it take to fill the
container?
62
Percentages:
28. Lydia is making a collage using 100 photographs arranged in a
square pattern. The shaded area in the model indicates the part of
the collage already covered by photos. What percent of the collage
is finished?
29. In the school chorus, 52% of the girls sing soprano and 44% sing
alto. Make a model to show this percents.
30. Write the following ratios as percent, fraction and as a decimal.
a) There are 4 trombones out of 25 instruments in our town band.
b) A company has a 4% return rate (tasa de retorno=beneficios) on its products.
c) At Ben’s Burger Palace, 45% of the customers order large soft drinks (refresco grande).
d) In a basketball mach, a player made 60% of his field goals.
e) In Janie’s class, 7 out of 25 students have blue eyes.
f) Michael answered
questions correctly on his test.
g) On Saturday afternoon, 41/50 telephone calls taken at “The Overlook” restaurant were
for dinner reservations.
h) According to a census, 76% of the workers commute to work (ir al trabajo) by driving alone.
i) China accounts for 0.207 of the world’s population.
j) When asked to choose their favourite sport, 27% of Merida adults who follow sports
selected professional football.
k) I have downloaded 0.58 of a file.
l) In the last elections, 39% of eligible (con derecho a voto) adults voted.
31. There are 520 students at Extremadura High School. 80% of these students take the bus.
How many students take the bus?
32. Theresa is 60% as old as her sister Mala, who is 20 years old. How old is Theresa?
33. Charlie wants to leave a 15% tip (propina) for a meal that costs $40. How much should
Charlie leave for a tip?
34. Charmaine wants to buy a shirt for $15. If the sales tax (impuesto) is 4% of $15, how much
will she pay in sales tax?
63
35. Juana had both chocolate cake and coconut cake at her birthday party. There were 40
people at the party, and 8 of them picked chocolate cake. What percentage of the people at
the party picked chocolate cake? [*] Sol: 20%.
36. Marshall's Cafe has regular coffee and decaffeinated coffee. This morning, the cafe served 75
coffees in all, 40% of which were regular. How many regular coffees did the cafe serve?
37. Thomas finished the race in 120 minutes. James took 95.5% as long as Thomas to finish the
race. How long did it take James to finish the race?
38. Port Lanshine Auto Repair originally paid its mechanics $20 per hour. Now that there is a
new owner, mechanics make $25 per hour. What was the percent of increase in pay?
39. A while ago, Justin bought shares of stock in a high-tech firm. At that time, the selling price
was $16 per share (acción). According to today's newspaper, the shares are now worth
(valen) $28 per share. What is the percent of increase in the value of the stock?
40. A DVD player that normally costs $160 is on sale for 70% of its normal price. What is the sale
price of the DVD player?
41. At Westside High School, 24% of the 225 sixth grade students walk to school. About how
many of the sixth grade students walk to school?
42. The concession stand at a football game served 178 customers. Of those, about 52% bought
a hot dog. About how many customers bought a hot dog?
43. Max has completed 39% of his reading assignment. If there are 303 pages in the assignment,
about how many pages has Max read?
44. A recent study shows that people spend about 31% of their time asleep. If a person has slept
for 24.18 years, how old is him?
45. The human body is 72% water, on average. If someone claims (afirma) to have 60.48l of
water in his body, how much does he weigh? [*] Sol: He weighs 84kg.
46. The sales tax rate in Mérida is 18%. If we pay 45€ of tax when buying a game console, what is
its price? [*] Sol: Its price is 250€.
47. 80% of the students in a class are going to a school trip. In all, they are 20 students on the
trip. How many students are there in the class? [*] Sol: There are 25 students.
64
PROPORTIONALITY - WORD PROBLEMS
Solve the following problems finding a formula first (unitary method).
1. Carlos makes $28 in 4 hours. [Sol: dollars=7·hours]
a) How much does he make in 1 hour? [Sol: 7$]
b) How much will he make in 20 hours? [Sol: 140$]
c) How long will take him to make 84 dollars? [Sol: 12·hours]
2. Janet's car uses 9 gallons (1 galón=3.785litros) to go 270 miles (1 milla=1.61km). [Sol: miles=30·gallons]
a) How far can she drive with 36 gallons? [Sol: 1080miles]
b) How many gallons will she use to go 450 miles? [Sol: 15·gallons]
3. Bertha went on a diet and lost 6 pounds (1 libra=0.456kg) in 2 weeks. How long will it take her to
lose 15 pounds? [Sol: pounds=3·weeks. It will take her 5 weeks]
4. 2 bottles cost 3.00€. [Sol: price=1.50 bottles]
a) How much will 7 bottles cost? [Sol: 10.50€]
b) How many bottles can you buy with $16.50? [Sol: 11·bottles]
5. 12 pupils put 10€ each to buy a present for their teacher, [Sol: pupils·euros=120; euros=
]
a) How much does the present cost? [Sol: 120 euros]
b) How much should put each pupil if they were 15 pupils? [Sol: 8 euros]
6. It takes 3 hours to go from Badajoz to Madrid, driving at 120km/h.[Sol: time·speed=360; speed=
]
a) How far is Madrid from Mérida? [Sol: 360 km]
b) How long will it take driving at 100km/h [Sol: 3.6hours=3h36m]
7. In a ship, they have meals enough to feed 150 passengers for 8 days. [Sol:·days =
]
a) How many meals do they have? [Sol: 1200 meals]
b) How many days can they feed 40 passengers? [Sol: 30 days]
c) How many passengers can they feed in 20 days? [Sol: 60 passengers]
8. We can storage our wine production using 8 barrels of 200 litres each. [Sol: barrels =
]
a) How many litres of wine do we have? [Sol: 1600 litres]
b) What should be the capacity of the barrels if we want to use 32 barrels? [Sol: 50 litres each]
9. If 5 men can paint a house in 36 hours: [Sol: hours·men=180; hours =
]
a) How long would it take 1 man to paint the house? [Sol: 180 hours]
b) How many men would be needed to paint the house in 12 hours? [Sol: 15 men]
65
PROPORTIONAL DISTRIBUTION - WORD PROBLEMS
1. Ana, Luisa and Pedro win 550€ in the lottery. Ana put 5€, Luisa 2€ and Pedro 3€. How much
should receive each of them? [Sol: Ana 275€, Luisa 110€, Pedro 165€].
2. Luis and Juani earn (ganan) 40€ baby-sitting (haciendo de canguro). Luis worked for 2 hours
and Juani for 3 hours. How much should receive each of them? [Sol: Luis 16€, Juani 24€].
3. Distribute 9 digital-board among 3 high-schools, proportionally to the pupils they have.
High-School A: 125 pupils, High-School B: 375, High-School C: 625. [Sol:A: 1, B:3 and C:5 boards ].
4. A Maths exam lasts 50 minutes. Distribute the time spent with each exercise proportionally
to their value. Exercise 1: 4 points, Exercise 2: 3 points, Exercise 3: 2 points, Exercise 4: 1
point. [Sol:E1: 20min, E2: 15min, E3: 10min and E4: 5min.].
5. Distribute 70€ between two brothers directly to their final marks. Víctor: 8 points. Amanda 6
points. [Sol: Víctor 40€, Amanda 30€].
6. A hospital is sending 20 patients to other hospitals. Distribute the patients proportionally to
the available (disponible) beds. Hospital A: 20 beds, Hospital B: 24 beds, Hospital C: 36 beds.
[Sol: Hospital A: 5 patients, Hospital B: 6 patients, Hospital C: 9 patients].
7. A company is giving €3 5 gratification to three of their typist. Ana typed 250 pages, Juan 240
pages, and María 280. Distribute it proportionally. [Sol: Ana: 125€, Juan 120€, María 140€].
8. Luis and Sonia got 27 points in a trivia competition. Luis answered correctly 24 questions and
Sonia 30 questions. Distribute the points proportionally. [Sol: Luis: 12, Sonia 15].
9. A company is going to distribute 1,035€ among three employees (proportionally to the years
worked). Juan: 35 years; Ana, 24; and Luis, 10. [Sol: Juan 525€, Ana 360€, Luis 150€].
10. For the Carnival Parade (desfile de carnaval), the city council (ayuntamiento) is going to
distribute 35000 candies among 4 krewes (comparsas), proportionally to the number of their
members. 1st krewe: 30 members; 2nd krewe: 45, 3rd krewe 50, 4th krewe: 30. How many
corresponds to each? [Sol: 1st
: 7500, 2nd
: 11250, 3rd
: 12500, 4th
: 7500].
67
1st ESO Mathematics Trial Exam IES Extremadura
Bilingual section
Surname ______________________________________________ Mark
Name __________________________________
PROPORTIONALITY
1. (1 pt.) In order to choose the date for a Maths exam, 9 out of the 30 pupils of a class voted for next
Monday. What percentage of the pupils voted for Monday?
2. (1 pt.) Do they form a proportion? “14 customers in 4 hours” and “21 customers in hours”
3. (1.5 pts) 1ºA-end-of-course picnic will cost € if it has 11 attendees (asistentes). If there are 25
attendees, how much will the picnic cost? Write a sentence in English explaining if the relationship
is direct or inverse.
4. (1.5 pts) Luis wants to go in a travel that takes him 6 hours if he drives at 90 km/h. How long
would it take him if he drives at 60 km/h? Write a sentence in English explaining if the relationship
is direct or inverse.
5. (1.5 pts.) Write as a percentage, fraction and decimal number:
a. 25%=
b. 0.34=
c. 8
3
6. (1 pt.) Compute 22% of 1200:
7. (1 pt.) A USB memory card costs 15€, and now it is offered with a 20% off. What is the price now?
8. (1.5 pts) In a class, 14 pupils pass Maths exam. If we know it is the 70% of the pupils. How many
pupils are there in the class?
69
9. LENGUAJE ALGEBRAICO
- Expresión algebraica: números y letras “variables” unidos por operaciones matemáticas.
- Valor numérico: resultado de sustituir las letras por números en una expresión algebraica.
- Monomio: producto de un número “coeficiente” por varias letras “parte literal” (se admiten
exponentes en las letras pero no raíces de letras, o letras en el denominador, etc.). El grado
del monomio es la suma de los exponentes.
Suma y resta de monomios semejantes (con la misma parte literal): se suman o restan los
coeficientes. Ej.: 2x+5x=7x, y 2x-5x=-3x.
- Polinomio: suma y/o resta de varios monomios “términos”.
Término independiente: un número que aparece sin letras. Su grado es 0.
Quitar paréntesis: (usando la propiedad distributiva)
Signo “+”: se quita el paréntesis y se deja todo como está. Ej.: +(-5x+1)= -5x+1.
Signo “-“: se cambia el signo de cada término. Ej.: -(-5x+1)= +5x -1.
Producto por un número: se usa la propiedad distributiva. Ej.: 2·(-5x+1)= -10x+2.
ECUACIONES
- Ecuación: Igualdad entre expresiones algebraicas que sólo se cumple para algunos valores de
las letras, que se llaman solución. Resolver la ecuación es encontrar las soluciones.
Las letras se llaman incógnitas. El grado es el mayor grado de los términos.
La parte de la izquierda primer miembro, y la de la derecha, segundo miembro.
Resolución de ecuaciones de primer grado (ecuaciones lineales):
1. Quitar paréntesis.
2. Quitar denominadores.
3. Transponer y agrupar términos
semejantes.
4. Despejar la incógnita.
(reduciendo primero a común denominador, o si es una igualdad de fracciones, se multiplica en cruz).
(se les cambia el signo al cambiarlos de miembro).
(Si un número multiplica a un miembro, pasa al otro
dividiendo, y viceversa)
70
ALGEBRAIC LANGUAGE
- Algebraic expression: numbers and letters “variables” joined by mathematical operations.
- Numerical value: the result of replacing letters by numbers in an algebraic expression.
- Monomial: algebraic expression where a number “coefficient” is multiplied by letters “literal
part” (exponents are allowed for letters but not square roots, nor a letter in a denominator,
etc.). The degree of a monomial is the sum of the exponents.
Addition and subtraction of like monomials (with the same literal part): add or subtract
coefficients. For instance: 2x+5x=7x and 2x-5x=-3x.
- Polynomial: addition and/or subtraction of several monomials “terms”.
Independent term: a number without letters. Its degree is 0.
Ex: 3x2-5x+1: second degree trinomial, principal coefficient 3 and independent term 1.
Operations with polynomials: (with the polynomial in parentheses)
“+” sign: remove the parentheses and leave everything as it is. Ex.: +(-5x+1)= -5x+1.
“-“ sign: change the sign of the terms. Ex.: -(-5x+1)= +5x-1.
Product by a number: one uses the distributive property. Ex.: 2·(-5x+1)= -10x+2.
EQUATIONS
- Equation: Equality of two algebraic expressions that is correct for only certain values of the
letters, called solution. To solve an equation is to find its solutions.
Letters are called unknowns. The degree is the greatest of the terms’ degrees.
The left side is the first member, and the right side, the second member.
Solving first degree equations (linear equations):
5. Eliminate parentheses.
6. Eliminate fractions.
7. Transpose and combine like terms.
8. Get the unknown by itself.
(reduce first to common denominator, or if it is an equality of fractions, cross multiply.).
(changing the sign when changing to the other side).
(If a number multiplies on one member, goes to the
other dividing it, and vice versa)
71
ALGEBRAIC LANGUAGE - WORD PROBLEMS
1. Let a represent Marta’s age. Write an expression showing the age of each member of her
family:
a. Marta’s mother is three times older than Marta
b. Her father is two years older than her mother.
c. Marta’s elder sister is four years older than Marta
d. Marta’s little brother is one year younger than Marta.
2. Translate the following expressions into algebraic language.
a. Roberto uses a scale to weigh himself.
b. Ana weighs four kilos more than Roberto
c. Luisa weighs twice as much as Ana
d. Rosa weighs four kilos less than Luisa.
3. The difference between “9 times some number” and “six more than twice the number”.
4. Let n represent a number. Write an expression for each of the following descriptions:
a. The number plus seven.
b. One more than the number.
c. One less than the number.
d. Twice (dos veces) the number.
e. The number times two plus three.
f. Half of the number.
5. A number and the same number minus one add up to 77. What are the two numbers?
6. If you add fifteen to two times a number, you obtain a result of 25. What is the number?
[Sol: The number is 5]
7. Jason and Megan are brother and sister. Jason is 4 years older than Megan. If Jason is 16
years old, write and solve an equation to find Megan’s age.
8. One number is 2 more than another. Their total sum is 8. Find both numbers. [Sol: 3 and 5]
9. Oscar is saving money to buy a jacket that costs $47. He has already saved $25. Write and
solve an equation to find how much more money Oscar needs to save.
72
10. Bonnie has 27 more cans than Jackie. If she has 56 cans, write and solve an equation to
find how many cans Jackie has.
11. Virginia’s mother gave her marbles (canicas) for her birthday. Virginia lost 13 of them. If
she has 24 marbles left, write an equation to find how many her mother gave her.
12. Claudio went for a walk. While he was walking, $1.35 fell out of his pocket. When he
returned home, he counted his money and had $2.55 left. Write and solve an equation to
find how much money was in Claudio’s pocket when he started his walk.
13. Aida was hang gliding. After losing 35 feet in altitude, she was gliding at 125 feet. Write
and solve an equation to find her height when she started hang gliding.
14. Paz has 3 times as much money in her wallet as in her pocket. There is $18 in her wallet.
Write and solve an equation to find how much money is in her pocket.
15. In his home aquarium, Elisa has 12 times as many guppies as he has
goldfish. Elisa just counted 72 guppies. Write and solve an equation to
find how many goldfish (pez de colores) he has.
16. The Virgin Islands were acquired by the United States in 1927. This is 29 years after Puerto
Rico was acquired. Write an equation to model this situation. [Sol: x+29=1927; x=1898]
17. A number and the previous one add up 63. Which are those numbers? [Sol: 32 and 31]
18. One forest has twice as many trees as another one. Between them, they have 120000
trees. How many trees are in each forest? [Sol: 4000 and 8000]
19. Rafael spent half of his money in the cinema and one fifth in afternoon snacks. He’s got
3 € left. How much money did he have? [Sol: 120€]
20. Considering hens and rabbits on a farm, they add 30 heads and 90 legs. How many hens
and how many rabbits are there on the farm? [Sol: 15 rabbits and 15 hens]
21. Ana posed (poner un problema) the following riddle (acertijo): “half my age, plus one sixth
of my age, plus one fourth of my age add up my age minus six”. How old is she? [Sol: 72y.]
73
POLYNOMIALS AND EQUATIONS – REVISION EXERCISES
1. Calculate:
a) - - - = b) - - - =
c) = d) - - - =
e) - - = f) - - - =
2. Solve and check the result
a) b) - - - -
c) - - - d) - - - -
e) f) - - - - - - -
3. Solve and check the result
a)
b)
4 +1
3- = c)
d)
e)
8
5-3 =
1
2 +3 f)
g)
h) -x+5
i)
4. Solve
a)
b)
1-
15-4+3
3=
-3-6
5 c)
d)
2x-5 e)
x-1 f)
g)
h)
i)
j)
=x+1 k)
+1 l)
+2
Solutions:
1. [a] - . [b] - [c] - - [d] y+5 [e] - [f] - .
2. [a] x=4 [b] y=-1 [c] x=5 [d] z=3 [e] m=5 [f] x=6.
3. [a] x=5 [b] x=2 [c] y=6 [d] x=3 [e] y=-1 [f] y=-2
[g] x=-3 [h] x=10 [i] x=6.
4. [a] x=2 [b] x=5 [c] x=5 [d] x=3 [e] x= 1 [f] x=2
[g] x=-2 [h] x=3 [i]x=2 [j]x=4 [k] x=3 [l] x=-3.
74
1st ESO Mathematics Trial Exam IES Extremadura
Bilingual section
Surname ______________________________________________ Mark
Name __________________________________
POLYNOMIALS AND EQUATIONS
1. (1.5 pts.) Say if they are monomials. For the monomials, name their parts (including the
degree).
a) 5x2 b) 2xy3 c) d)
2 (0.5 pts.) Say if they are polynomials. For the polynomials, say their degree and how many
terms they have.
a) 3x2-9xy+2 b) 4- +3y
3 .Solve the following equations (1 pt. each)
a) 3·(x-2)+2 (x+1)=6.
b) -6x=3·(5x+8)-3
c)
d)
4 (2 pts.) Solve:
5. Word problems (1 pt. each). Set the problem up using an equation and solve it.
a) Twice a number minus five is eleven. What is the number?
b) Find a number such that the number and the following number add up twenty three.
75
apotema
10. FIGURAS PLANAS
- Teorema de Pitágoras: en todo triángulo rectángulo, el cuadrado de la hipotenusa es igual a
la suma de los cuadrados de los catetos.
Terna Pitagórica: si a, b, c son enteros. Ej: (3,4,5); (5,12,13); (7,24,25); ( ,15,17),…
Perímetro de una figura: suma de las longitudes de los lados.
ÁREAS
Paralelogramos: base por altura A=b·h.
- Para el rombo y la cometa, A =
(D y d son las diagonales).
Triángulo: A =
.
Trapecio: A =
Polígono regular: A =
. (apotema: distancia del centro al lado).
FIGURAS CIRCULARES
- Longitud de la circunferencia de radio r: L= 2r. Para un arco de a grados: L =
· 2r.
- Área del círculo: A=r2. Para un sector circular de a grados: A =
r2.
- En general, hay que descomponer la figura en otras de áreas conocidas.
(*) Polígonos regulares: Cálculo de radio, lado y apotema usando proporcionalidad (no hay que saberse la tabla):
nº Lados Conocido el lado Conocida la apotema Conocido el radio
5 a=0,69·lado r=0,85·lado lado=1,45·a r=1,24·a a=0,81·r lado=1,18·r
6 a=0,87·lado r = lado lado=1,15·a r=1,15·a a=0,87·r lado = r
7 a=1,04·lado r=1,15·lado lado=0,96·a r=1,11·a a=0,90·r lado=0,87·r
8 a=1,21·lado r=1,31·lado lado=0,83·a r=1,08·a a=0,92·r lado=0,77·r
9 a=1,37·lado r=1,46·lado lado=0,73·a r=1,06·a a=0,94·r lado=0,68·r
10 a=1,54·lado r=1,62·lado lado=0,65·a r=1,05·a a=0,95·r lado=0,62·r
a2 = b2 + c2
D
d
b h
B
b
h
76
apothem
PLANE FIGURES
- Pythagorean Theorem: In a right triangle the square of the hypotenuse is equal to the sum
of the squares of the legs (the other two sides).
Pythagorean Triple: when a, b, c are integers. Ex: (3,4,5); (5,12,13); (7,24,25); ( ,15,17),…
Perimeter of a figure: the sum of its sides’ lengths.
AREAS
Parallelograms: base times height A=b·h.
- For rhombus (diamond) and kite, A =
(D and d are the diagonals).
Triangle: A =
. (Half the product of the base and height)
Trapezium: A =
Regular Polygon: A =
. (apothem: distance from centre to sides).
CIRCULAR FIGURES
- Circumference of a circle of radius r: L= 2r. For an arc of a degrees: L =
· 2r.
- Area of a circle: A=r2. For a circular sector of a degrees: A =
r2.
- Usually, one has to break the figure into others of known areas.
(*) Regular polygons: Computing radius, side and apothem using proportionality. (please do not learn this chart by heart!):
No. sides Knowing the side Knowing the apothem Knowing the radius
5 a=0,69·side r=0,85·side side=1,45·a r=1,24·a a=0,81·r side=1,18·r
6 a=0,87·side r = side side=1,15·a r=1,15·a a=0,87·r side = r
7 a=1,04·side r=1,15·side side=0,96·a r=1,11·a a=0,90·r side=0,87·r
8 a=1,21·side r=1,31·side side=0,83·a r=1,08·a a=0,92·r side=0,77·r
9 a=1,37·side r=1,46·side side=0,73·a r=1,06·a a=0,94·r side=0,68·r
10 a=1,54·side r=1,62·side side=0,65·a r=1,05·a a=0,95·r side=0,62·r
a2 = b2 + c2
b h
D
d
B
b
h
77
PLANE FIGURES – REVISION EXERCISES
1. Find the areas, using the formulas: [*] 1inch=2.54cm; 1foot=30.48cm; 1yard=91.44cm.
a) b) c) d)
Sol: A=22.5in2 A=132cm2 A=0.91m2 A=14000ft2
e) f) g)
Sol:144ft2 A=0.5m2 A=2.3in2.
h) i) j)
Sol: 17.5in2 A=186cm2 A=13.95ft2.
k) l) m)
Sol: A=11.8cm2 A=200-10=190cm
2
n) o) p) q)
Sol: A=621-42=579 ft2
r) s) t)
[Sol: A=8.725m2]
u) v)
1.3m 1.5m
2.5m 3.5m 4m
78
2. Find the length of the missing side, or measure the distance between points.
a) b) c) d)
Sol: c = 10 cm c=13 ft c=17units c=3.9cm
e) f) g) h)
Sol: a=12cm a=3ft a=3 feet a=6m
3. Calculate the length of the lines (and the perimeter in d)
a) b) c) d)
Sol: 3.6 units 4.24 units 5.1 units 2·5+2·1.41=12.82 units
4. Find the area and perimeter. [Hint: you may need the Pythagorean Theorem].
a) b) c)
Sol: A=108cm2. P=48cm h=15; A=495in
2; P=100in. A=420ft
2; P=90ft.
d) e) f) g)
Sol: A=39.8in2; P=24.85in A=55’16in2; P=3 ’83in A=285.12cm2;P=80.94cm
h) i) j)
A=52.1cm2; P=34.73cm A=56in2; P=36in
79
5. Find the area
a) b) c) d)
e) f) g) h)
i) j) k) l)
m) n) o) p)
6. Find the area and the perimeter
a) b) c) d)
e) f) g) h)
i) j) k) l)
[Sol: A=42in2]
A=130in2
80
7. Calculate the area and the perimeter. [Hint: you may need the Pythagorean Theorem].
a) b) c) d)
Sol: A=22cm2; P=26.48cm
8. Calculate the area and the perimeter of the following figures:
a) b) c) d)
A=5.52cm2;P=11.4cm A=6.15cm
2; P=10.34cm A=30cm
2; P=12.3cm
e) f) g) h)
A=12.9dm2; P=23.13dm A=81.2cm2;P=49.25cm
9. Work out the area and the perimeter
a) b) c)
A=240cm2; P=68.7cm
10. Find the area
a) b) c)
Hint: Say what fraction of a hexagon it is. Hint: Square or Diamond? Sol: A=134.5cm2.
1.5 cm 2 cm
1.4 cm 3 cm
36 cm
15 cm
6 cm 6 cm 16 cm
6cm
4 dm
3 dm
8 dm
1.25 m
3 m
4 m
2 cm
2 cm
4 cm
4 cm
5 cm
6 cm
4 cm
4 cm
6 cm
8 cm
9 cm
9 cm
12 cm
4 cm
4 dm
16 cm 20 cm
10 cm
2 cm
81
PLANE FIGURES - WORD PROBLEMS
1. The perimeter of a rectangular cement patio is 86m. The patio is 29m long. How wide is it?
2. The area of a square playground is 100m2. How long is each side of the playground?
3. Harold measures the circular floor of a tepee, and then calculates that it
has a circumference of 18.84 meters. What is the floor's radius?
4. The perimeter of a greeting card (tarjeta de felicitación) is 78
centimetres. It is 15 centimetres wide. How tall is it?
5. Ana’s swimming pool is rectangular in shape with a circular wading pool
(AmE: piscina para niños) at one corner, as shown. Find its surface and
the distance around the edge.
6. Three people are walking on the street. Luisa is straight behind Bea and directly left of
Pepa. If Bea and Luisa are 3 meters apart, and Pepa and Bea are 5 meters apart, what is the
distance between Luisa and Pepa?
7. Rosa is attending a school orchestra concert. She sees her math teacher seated 8 meters
ahead of her and her science teacher seated 15 meters to her right. How far apart are the
two teachers?
8. Juana is deep-sea diving (buceo) with two friends. Cody is floating on the surface 12 meters
above Juana, and Hani is exploring a coral reef 16 meters in front of Juana. How far apart
are Cody and Hani?
9. A pigeon (paloma-pichón) was sitting 9 meters from the base of a telephone pole (palo)
and flew 15 meters to reach the top of the pole. How tall is the telephone pole?
10. Two cars leave the same parking lot (lugar), with one heading (dirigirse hacia) north and
the other heading east. After several minutes, the northbound (en dirección norte) car has
travelled 8km, and the eastbound (en dirección este) car has travelled 6km. Measured in a
straight line, how far apart are the two cars?
11. Maria is carpeting her living room, shown in the diagram on the right.
It is rectangular, but has a bay window (ventanal panorámico).
She has taken the measurements shown on the diagram. What
area of carpet will she need? If each ft2 costs 3.50€, what is its price?
82
12. Whitney is in a hot air balloon that has just taken off and is floating above its launching
point. Russell is standing on the ground, 8 meters away from the launching point. If
Whitney and Russell are 17 metres apart, how high up is Whitney?
13. A farmer needs to buy fencing to go around his garden. The garden is a 20m by 15m
rectangle. How much fencing will he need? If each metre costs 15€, how much is in all?
14. The largest living thing in the world is the “General Sherman
sequoia” in Sequoia National Park, California. It is 83m high, has
a diameter of 11.1 m, and has an estimated weight of 2,150 tons.
Find the sequoia’s circumference to the nearest tenth of a metre.
15. Elisa and Veronica are in charge of making a banner for the volleyball game this Saturday.
How much poster paper will they need for a parallelogram-shaped banner with height 1
metres and base 2 metres?
16. Jose is painting the flag of Brunei (a country in Southeast Asia)
for a geography project at school. How many square inches will
he cover with each paint? [*] 1inch=2.54cm.
17. This is a pattern of a quilting square that Ana will use to make a quilt
(edredón). How much of each fabric (tejido) will she need?
18. Courtney wants to carpet part of her bedroom that is shaped like a
right triangle with base 4.8 meters and height 5.2 meters. How much
carpet (alfombra) will she need?
19. The grass in two fields of a farm needs replacing. One field is 42 and the other is 53 m2.
The turf (césped) costs 2.30€ per square metre. How much will it cost to replace the grass?
20. The Rodríguez family wants to put tile (baldosa) on their bathroom floor. Each tile is 0.25
square meters. Their bathroom is 4 meters by 6 meters. Tiles are sold in packs of 6 tiles.
a) How many tiles will be needed?
b) How many boxes will be needed? Will there be any spare (de sobra) tile?
c) Each box costs 15.20€. How much will they pay?
21. Daniel just bought a used sailboat (barco velero) with two sails (velas)
that need replacing. Each ft2 costs 20€. How much sail fabric will Daniel
need to replace each sail? How much does each cost?
83
12cm
7cm
1st ESO Mathematics Trial Exam IES Extremadura
Bilingual section
Surname ______________________________________________ Mark
Name __________________________________
PYTHAGOREAN THEOREM, AREA AND PERIMETER
1. (3 pts.) For this isosceles triangle, compute:
a. (1 pt.) Its perimeter.
b. (1 pt.) Its height.
c. (1 pt.) Its area.
2. (2 pts.) Find the area and perimeter for
this diamond.
3. (2 pts.) Work out the area and perimeter of this figure,
knowing that for any heptagon, apothem 1.04·side
4. (1 pt.) Find the area.
5. (1 pt.) FInd the area of this circle sector:
6. (1 pt.) How much does it cost to wallpaper (empapelar) a wall of 3.5m of side using a paper that
costs 4€ per m2?
10cm
12cm
10cm
20cm
60cm
30cm
20cm
36º
3 m
85
11. FUNCIONES Y GRÁFICAS
EL PLANO CARTESIANO
Para dar coordenadas a los puntos de un plano usamos dos rectas, que se llaman ejes de
coordenadas. Su punto de corte se llama origen de coordenadas (O).
- Eje de abscisas (X): recta horizontal.
- Eje de ordenadas (Y): recta vertical.
El plano queda dividido en cuatro cuadrantes.
Para cada punto, escribimos sus coordenadas entre
paréntesis. Ejemplo P(3,2).
FUNCIONES
- Una función es una relación entre dos conjuntos (inicial y final), de manera que a cada
elemento del primero le corresponde uno solo del segundo
- Variable independiente (x): los valores del conjunto inicial.
- Variable dependiente (y): los valores del conjunto final.
Formas de expresar una función:
- Lenguaje natural: una frase que describe la relación. Ejemplo: “el doble de cada número”.
- Tabla de valores: en una fila (o columna) colocamos la variable independiente, y en otra la
dependiente. Ejemplo:
- Una gráfica: en el eje de abscisas colocamos la variable
independiente (x) y en el de ordenadas la dependiente (y).
Ejemplo:
- Una fórmula: una ecuación describe cómo están relacionadas las variables. Substituyendo el
valor de “x” en la fórmula, se obtiene el de “y”. Ejemplo: y=2x. Para x=3 resulta y=2·3=6.
[*] Para las rectas y= mx+n, el coeficiente de “x” se denomina pendiente, y mide la inclinación
de la recta. Por ejemplo, y=3x+n tiene pendiente m=3.
x 0 1 2 3 4 5
y 0 2 4 6 8 10
-5 -4 -3 -2 -1 1 2 3 4 5
-5
-4
-3
-2
-1
1
2
3
4
5
x
y
I Primer
Cuadrante
II Segundo
Cuadrante
III Tercer
Cuadrante
IV Cuarto
Cuadrante
-5 -4 -3 -2 -1 1 2 3 4 5
-5
-4
-3
-2
-1
1
2
3
4
5
x
y
86
FUNCTIONS AND GRAPHS
CARTESIAN PLANE
We use a pair or lines to set coordinates for each point of a plane; they are called coordinate
axis. They meet at the origin of coordinates (O).
- Abscissa axis (X): horizontal line.
- Ordinate axis (Y): vertical line.
The plane is divided into four regions called quadrants.
For each point we write its coordinates in parentheses.
Example P(3,2).
FUNCTIONS
- A function is a relationship between two sets (initial/input and final/output), such that each
input is associated to one and only one output.
- Independent variable (x): the input to a function.
- Dependent variable (y): the output.
Ways to give a function:
- Natural language: a sentence describing the relationship. Example: “the double of each
number”.
- Table of values: we put the independent variable in a row (or column) and the dependent in
another. Example:
- A graph: we put the independent variable (x) in the
abscissa axis and the dependent (y) in the ordinate.
Example:
- A formula: and equation describing how the variables are related. By substituting the value
of “x” into the formula, one get “y” value. Example: y=2x. For x=3 results y=2·3=6.
[*] For lines y= mx+n, the “x”coefficient is called slope, and measures the steepness of the
line. For example, y=3x+n has a slope m=3.
x 0 1 2 3 4 5
y 0 2 4 6 8 10
-5 -4 -3 -2 -1 1 2 3 4 5
-5
-4
-3
-2
-1
1
2
3
4
5
x
y
I First
Quadrant
II Second
Quadrant
III Third
Quadrant
IV Fourth
Quadrant
-5 -4 -3 -2 -1 1 2 3 4 5
-5
-4
-3
-2
-1
1
2
3
4
5
x
y
87
FUNCTIONS – REVISION EXERCISES
1. Plot the points, connect them and write and name the quadrant or axis they lie in:
a)
b)
c)
d)
e)
2. For the following functions, construct a table of values, a graph, express them using a sentence and write their slope:
a) y= 2x + 1 b) y=3x-8 c) y=5-4x d) y = x+3
e) y= -2x+4 f) y=
+ 1 g) y=
+ 2 h) y =
3. Write the following statements as functions using a formula. Find the slope.
a) Gabriel knows how to sing 4 songs. He will be learning 2 songs every week he takes voice
lessons.
Variables: weeks and songs.
- How many songs will he know in 5 weeks?
- If he knows how to sing 18 songs, how many weeks have passed?
b) María’s got 65 scented soaps (jabón perfumado) left at her shop. She is assembling gift
baskets, using 3 soaps on each.
Variables: “gift baskets” and “scented soaps left”.
- If she assembles 8 baskets, how many soaps are left?
- If she’s got 32 soaps left, how many baskets has she assembled?
Point P1(1,-2) P2(2,-2) P3(3,-2) P4(3,1) P5(0,3) P6(-2,1) P7(-2,-2) P8(0,-2)
Lies in…
Point P1(2,0) P2(2,1) P3(0,1) P4(-2,0) P5(0,0) P6(0,-4) P7(2,-4) P8(1,0)
Lies in…
Point P1(3,0) P2(1,1) P3(1,3) P4(0,1) P5(-2,1) P6(0,0) P7(0,-2) P8(1,0)
Lies in…
Point P1(2,-4) P2(4,-2) P3(4,0) P4(2,2) P5(0,2) P6(-2,0) P7(-2,-2) P8(0,-4)
Lies in…
Point P1(0,-1) P2(1,0) P3(2,5) P4(0,3) P5(-2,3) P6(-4,5) P7(-3,0) P8(-2,-1)
Lies in…
88
c) We have already had 7 exams. From now on, we are having 3 exams every week.
Variables: “exams” and “weeks”.
- How many exams will be done in 5 weeks?
- When will we have done 19 exams?
d) A plane is flying 1200m high and starts landing descending 50 meters each minute.
Variables: “height” and “minutes elapsed (transcurridos)”.
- How high is it flying in 10 minutes?
- When is it flying 450m high?
e) Ana’s got 45 spools (bobina) of thread left. She uses 4 on each shirt that she sews (coser).
Variables: “spools remaining” and “shirts sewn”.
- How many shirts has she sewn when she’s got spools left?
- If she sews 7 shirts, how many spools are remaining?
f) Borja’s got 10€ in his piggy bank (hucha). He saves 4€ every week.
Variables: money saved and weeks. [Sol: y=4x+10]
- How much does he have after 12 weeks? [Sol: 58€]
- How many weeks have passed if he has saved 0€? [Sol: 20 weeks]
g) A car rental fare (tarifa de alquiler) is $100 per day plus $0.30 per mile travelled.
Variables: price and miles travelled. [Sol: y=0.30x+100]
- How many miles have we travelled if we are charged 2 €? [Sol: 560 miles]
- What is the price for 500 miles travelled? [Sol: 250€]
h) Pedro's got 96€. He is buying some books, each of which costs 3€.
Variables: Money left and books bought. [Sol: y=96-3x]
- If he buys 8 books, how much does he have left? [Sol: 72€]
- If he has 0€ left, how many books has he bought? [Sol: 12 books]
i) Alicia is preparing quiches. She’s got 3 eggs left, and uses 4 eggs on
each quiche.
Variables: Quiches prepared and eggs remaining. [Sol: y=83-4x]
- If she prepares 5 quiches, how many eggs does she have left? [Sol: 63 eggs]
- If there are 15 eggs remaining, how many quiches has she prepared? [Sol: 17 quiches]
89
j) Francisco is on holidays and so far, he has taken 8 pictures. From now on, he is taking 9
pictures every day he spends on holidays.
Variables: Days and pictures. [Sol: y=9x+8]
- If he is 5 days on holidays, how many pictures will he have? [Sol: 53 pictures]
- If he’s got 89 pictures, how many days have passed? [Sol: 9 days]
k) “Construcciones Mirandilla” has 5 crates (cajas) of nails (clavo). They use 3
crates on each house they build.
Variables: Crates remaining and houses. [Sol: y=56-3x]
- If they have 32 crates left, how many houses have they built? [Sol: 8 houses]
- If they build 5 houses, how many crates are left? [Sol: 41 crates]
l) We are observing a plant's growth is directly proportional to time. It measured 2 cm when it
arrived at the nursery (vivero) and 2.5 cm exactly one week later. Suppose that the plant
continues to grow at this rate.
Variables: length and weeks. [Sol: y=2+0.5x]
- What is its size after 4 weeks? [Sol: 4 cm]
- How many weeks have passed it is 3.5 cm long? [Sol: 3 weeks]
m) Belén has already caught 7 fish on her family fishing trip. Suppose she will catch 3 fish
every day she goes out fishing.
Variables: days and fish caught. [Sol: y=3x+7]
- If she’s caught 25 fish, how many days have passed? [Sol: 6 days]
- How many fish will she have in 10 days? [Sol: 37 fish]
n) Pablo has written 30 pages in his notebook. He uses 4 pages for every class.
Variables: no. of classes attended and pages. [Sol: y=4x+30]
- How many pages will he have written in 20 classes? [Sol: 110 pages]
- When he has 78 pages written, how many classes has he attended? [Sol: 12 classes]
91
12. ESTADÍSTICA
- La estadística es la ciencia que se ocupa de recoger, resumir y analizar datos. También de
sacar conclusiones, hacer predicciones y tomar decisiones.
Tabla de Frecuencias
- Frecuencia absoluta: número de veces que se repite el dato.
- Frecuencia relativa: indica qué parte del total es ese dato. Podemos medirlo en porcentaje.
- Frecuencia acumulada: el número de datos que hay menores o iguales que él.
Gráficos Estadísticos
- Diagrama de barras (y de puntos): dibujamos la variable estadística en el eje horizontal y la
frecuencia en el vertical.
- Polígono de frecuencias: unimos las barras del gráfico con segmentos.
- Diagrama de sectores: dividimos un círculo en partes proporcionales a la frecuencia.
- Caja y bigotes: ponemos los valores mínimo, máximo y cuartiles en la recta numérica.
- Diagrama de tallo-hojas: ordenamos los datos en una tabla. El tallo es la primera parte del
número y la hoja la última cifra. Hay una hoja por cada dato (y puede haber hojas iguales).
Medidas de Centralización
- Media ( ): suma de los datos dividido entre el número de datos. Puede ser ponderada,
(multiplicando por unos “pesos”, para dar más importancia a algunos datos).
- Mediana: tras ordenar los datos, el que queda justo en la mitad. Los cuartiles son los que
quedan en el 25%, 50% (la mitad), y el 75%.
- Moda: el dato que tenga la mayor frecuencia (puede haber varias modas).
92
STATISTICS
- Statistics is the science that gathers, summarizes and analyses data. Also, it is dedicated to
find conclusions, make predictions and make decisions.
Frequency tables/charts
- Absolute frequency: the number of times the data value occurs.
- Relative frequency: what part of the total is that data. It can be measured in percentage.
- Cumulative frequency: the number of data less or equal than a given data.
Statistic Graphs
- Bar graph: (and line plot) we put the statistic variable on a horizontal axis. Frequency is put
on the vertical.
- Frequency polygon: join the bars by line segments.
- Circle graph (pie chart): a circular chart divided into sectors, proportional to the frequency.
- Box and whisker: we place the least data, the greatest data and quartiles on the number line.
- Stem-and-leaf plot: the stem tells the first part of the number and the leaf tells the last digit
of the number. There is one leaf for each data (and leaves can be repeated).
Central Tendency measures
- Mean ( ): the sum of the numbers divided by the number of data. It can be weighted,
(multiplying by some “weights”, to give more importance to some data).
- Median: when the numbers are arranged from least to greatest, it is in the middle.
Quartiles split the data into four sections. Each section contains 25% of the data.
- Mode: the number that appears most often. (There can be several modes.)
93
STATISTICS – PRACTICE
1. Calculate the final mark of these students, considering their marks and how much each
subject (asignatura) is worth (valorada).
a) If Maths is worth “2”, Science “1” and English “2”
b) Find their marks if Maths is worth “1”, Science “1.5” and English “2”
c) Find their marks if Maths is worth “2.5”, Science “2” and English “1”
2. A candidate obtains the following percentages in an examination. English 46%,
Mathematics 67%, Science 72%, Ph.E. 58%, Technology 53%. It is agreed to give double
weights to marks in English and Mathematics as compared to other subjects. What is the
weighted mean?
3. The English teacher uses a weighted mean to calculate the final mark of her pupils. The
chart shows the marks of 4 pupils (out of 10) in each section, and the weights in
percentage. Calculate their final marks.
4. In a school, 85 boys and 35 girls appeared in a public examination. The mean mark of boys
was found to be 40% whereas (mientras que) the mean marks of girls was 60%. Determine
the average marks percentage of the school. [Hint: Use a weighted mean] [Sol: 45.83%]
5. A class of 25 students took a science test. 10 students had an average (arithmetic mean)
score of 80. The other students had an average score of 60. What is the average score of
the whole class? [Hint: Use a weighted mean] [Sol: 68]
6. David tracks the number of classmates who ask for permission to go pee. His records are:
a) Draw a line plot, and make a table of frequencies for each subject (asignatura).
b) Find the mean and mode numbers of pupils for each subject.
c) Draw a box-and-whisker plot. What do they tell you about the differences in both
subjects (use min. and max. values, medians and quartiles).
English classes 2 0 1 4 0 2 2 2 3 3 1 5
Maths classes 0 1 2 2 3 3 3 4 5 5 6 7
Pupil Maths Science English Mark
Luis 7 9 5
Juana 8 6 5
Eva 6 10 7
Pupil Writing
(30%)
Listening
(25%)
Reading
(15%)
Exercises
(20%)
Behaviour
(10%)
Final
Mark
Sergio 8 7 7 10 10 =8,20
Elena 10 9 10 5 5 =8,25
Isabel 6 7 5 9 10 =7,10
Alonso 5 5,5 6,5 4 2 =4,85
94
7. These box-and-whisker plots show the ages of people using a public pool (piscina) in two
different days. Compare the differences in ages of the pool-users.
a) Looking at the minimum and maximum values.
b) Looking at the medians
c) Looking at the quartiles
8. This line plot shows the number of goals scored in the soccer
matches played by a school team this year.
a) In how many matches did the team score 0 goals?
b) In how many matches did the teams score 2 or more goals?
c) Draw a box-and-whisker plot and the frequency table.
d) Find the central tendency measures (mean, mode, median).
9. This pictograph shows the ages of cars parked in a parking lot.
a) How many cars are parked? [Sol: 20 cars]
b) Draw a box-and-whisker plot and the frequency table.
c) Find the central tendency measures.
10. Raquel keeps track (lleva la cuenta) of the number of pupils who get bust (pillados)
chewing gum (comiendo chicle) in class every day. The data below shows her records
before the teacher decided to tell pupils off (reñir) and after.
a) Draw a line plot and make a table of frequencies for each set.
b) Find the mean and mode numbers of pupils bust “before” and “after”.
c) Draw a box-and-whisker plot. Use it to compare what happened when the teacher
decided to tell pupils off (use min. and max. values, medians and quartiles).
11. Draw a stem-and-leaf plot for the following sets of data.
a) Draw a back-to-back stem-and leaf plot, and look at its shape to compare the data sets.
b) Group by units and make a table of values for each set.
[Sol: The scores of Class A are fairly evenly spread (igualmente repartidos) between 6.5 and 9.8. The
scores of Class B are clustered tightly (muy agrupados) around the high-8s and 9s].
Before telling off 7 4 5 9 5 10 6 4 7 11 8 9 5 10 5 8
After telling off 6 8 3 4 5 4 6 5 4 6 3 6 5 7 3 8
Class A-Test scores 6.5 6.9 7 7.2 7.8 8.1 8.5 8.7 9 9.8
Class B-Test scores 8.7 8.8 8.8 9 9.2 9.5 9.5 9.5 9.7 9.7
95
12. Use the number of rotten tomatoes per barrel are on a shipment to make a stem-and-leaf,
and box-and-whisker plot: {18, 19, 23, 28, 43, 50, 54, 57, 58, 58, 61, 61}
13. Two holiday companies each organized a trip to visit pyramids in Egypt for people aged over
50. Their ages were
a) Draw a back-to-back stem-and-leaf plot for the data and use it to compare the ages of the
people who travelled with each company.
b) Group each set of data by tens and make the frequency table.
[Sol: The ages of the people who travelled with Company B were fairly evenly spread (repartidos
igualmente) between 52 and 85 (the leaves are all of similar length). The people who travelled with
Company A were typically younger, and were closer together in age –most of them were in their
60s- (This is shown by a very long leaf representing the people in their 60s]
14. Juan has conducted a survey asking the number of TVs in people’s households (hogar). The
data set is shown below.
{2, 0 , 1 , 3 , 1 , 1 , 2 , 2 , 0 , 2, 2 , 2 , 2 , 2 , 1 3 , 1 , 3 , 1 , 3}
Make a table of frequencies and the bar graph. Calculate the central tendency measures.
15. In a survey, some students were asked which of four colours they like best. The results are:
Red, Red, Blue, Red, Yellow, Pink, Yellow, Blue, Green , Yellow
Pink, Blue, Blue, Red, Blue, Green, Blue, Pink, Red, Blue
Make a table of frequencies and statistical graphs.
16. A number of dog owners were asked how many times
they had taken their dog to the veterinarian in the last
month. The results are shown on the pictograph.
Make a table of frequencies and the bar graph.
Calculate the central tendency measures.
17. Twenty students in a class were asked “how many siblings (hermanos/as) live at your
house”. These are their answers
{2, 1, 5, 1, 3, 4, 1, 1, 2, 2, 3, 2, 4, 2, 1, 3, 2, 2, 4, 5}
Use these results to make a frequency table. Calculate the central tendency measures.
18. We ask the students in a class “what season is your birthday in?”. Results are: Spring 7
people; Summer 8; Autumn 9, and winter 11. Make a frequency table and statistical graphs.
Company A 56 57 61 64 65 66 67 68 68 69 70
Company B 52 55 55 56 59 60 63 65 65 66 66
68 71 71 73 79 80 82 83 85
96
19. These box-and-whisker plots show the cholesterol levels of a group of people before and
after they took a certain medicine for two months. What do they show you about
cholesterol levels before and after taking the drug (medicina)?
a) Looking at the minimum and maximum values.
b) Looking at the medians
c) Looking at the quartiles
[Sol: a) The highest values for cholesterol were found “before”, and the lowest, “after” taking the drug. The
range is similar for both sets.
b) The median is much lower in the “after” box, which indicates the typical cholesterol was reduced by
the medicine.
c) The “before” box is much further toward the higher end of the scale, this means that cholesterol
levels were generally higher before the medicine was taken.
Conclusion: There is a general reduction on people’s cholesterol after taking the medicine.]
20. These box-and-whisker plots show the prices of stock in two stores. What do they tell you
about the price differences in the two stores?
a) Looking at the minimum and maximum values.
b) Looking at the medians
c) Looking at the quartiles
[Sol: a) Store B has the lowest priced item ($10). Store A’s
lowest price is much higher ($30). Both stores sell their most expensive item for the same price ($90).
So Store B has a greater range of prices.
b) Store A’s stock has a median price of $70, whereas (mientras que) Store B’s stock has a median
price of $40. So Store B’s items are typically less expensive than Store A’s. (Half of Store A’s stock is
under $70, but half of the Store B’s stock is under $40)
c) The middle 50% of the prices in Store B are much more spread out (esparcidos) than they are in
Store A. They go from $30 to nearly $70. The middle 50% of the prices in Store A are concentrated
more tightly (más concentrados) around the median value of $70.]
21. The box-and whisker plots below show the test scores in two classes. Compare them
a) Looking at the minimum and maximum values.
b) Looking at the medians
c) Looking at the quartiles
[Sol: a) Class A’s test scores had a much larger range than Class B’s. Both the highest and lowest scores
overall were found in Class A.
b) Class B’s scores were generally higher than Class A’s. The median score for Class B was more than
50, but for Class B it was about 43.
c) More than half the students in Class B scored more than 50, whereas in Class A only one-quarter
scored more than 50.]
97
1º ESO Mathematics Trial Exam IES Extremadura
Bilingual section
Surname ______________________________________________ Mark
Name __________________________________
FUNCTIONS AND STATISTICS:
1. (1.25 pts.) Plot the following points and name the quadrant or axis in which they lie. Then, connect
them using segments.
Point Lies in... Point Lies in...
P1(4,3) P5(0,-2)
P2(1,1) P6(1,-2)
P3(-2,4) P7(4,2)
P4(-3,0)
2. (2 pts.) Consider the function y=2x-3.
a) Make a table of values
b) Graph the function.
c) Express the function using a sentence.
3. (2.25 pts.) Juani’s got 0 soup spoons left at her restaurant.
She uses 4 spoons each time she sets a table.
a) Relate “no. of spoons left” and “tables set” using a formula. Say what type of variable are each.
b) If she sets 8 tables, how many spoons are left?
c) If she’s got 32 spoons left, how many spoons has she set?
4. (4 pts.) Final marks of the 20 pupils in a class are:
6, 4, 5, 8, 5, 5, 6, 6, 7, 6, 6, 4, 7, 8, 6, 7, 7, 8, 5, 7.
a) (1.5 pts.) Draw the frequency table and the line plot corresponding to these marks.
b) (1 pt.) Draw the bar chart along with the frequency polygon.
c) (1 pt.) Find the central tendency measures of the marks of the class (mean, mode and median).
d) (1 pt.) Make the box-and-whisker plot
99
13. PROBABILIDAD
Experimento: suceso que podemos repetir bajo condiciones semejantes. Puede ser:
Deterministas: sólo hay un resultado posible.
Aleatorio: si hay varios resultados posibles. Cada uno de estos resultados se llama
suceso elemental. El conjunto de todos los resultados posibles se llama Espacio Muestral (E).
La unión de varios sucesos elementales se llama suceso compuesto.
Experimento compuesto: cuando el experimento puede dividirse en otros más
sencillos. Para calcular el espacio muestral podemos usar un diagrama de árbol.
Ejemplo: “lanzar dos monedas” puede descomponerse en: 1ª moneda y 2ª moneda.
- Si no necesitamos todo el espacio muestral, podemos podar el árbol: no dibujar las ramas
que no nos interesan.
Probabilidad: es el grado de creencia que tenemos de que ocurra un suceso. La medimos en
porcentaje, fracción o número (entre 0 y 1).
Un suceso de probabilidad 100%, se llama suceso seguro, y el de probabilidad 0% se llama
suceso imposible.
Cálculo de probabilidades:
- Regla de LaPlace:
, pero sólo puede usarse cuando podemos afirmar
que todos los casos tienen la misma probabilidad (son equiprobables).
- Diagramas de árbol: la probabilidad de cada suceso se calcula multiplicando la de cada rama.
- Ley de los grandes números: la probabilidad es la frecuencia relativa, siempre que se haya
repetido el experimento muchas veces.
100
PROBABILITY
Experiment: test we can repeat under similar conditions. It can be:
Deterministic: there is only one possible result.
Random: There is more than one possible result. Each of them is called simple event l.
The set of all possible outcomes is called the Sample Space (S).
If an event has more than one elements then it is called a compound event.
Compound Experiment: when the experiment can be break into some more
simple ones. We can use a tree diagram to calculate the sample space.
Example: “tossing two coins” may be broke into: 1st coin and 2nd coin.
- If we do not need the whole sample space we can prune the tree: do not draw the branches
we are not interested in.
Probability: degree of likelihood that something will happen. We can measure it in
percentage, fraction or number (between 0 and 1).
A 100% probable event is called sure event, and a 0% probable event is called impossible
event.
Computing probabilities:
- LaPlace’s rule (rule of Succession):
, it can only be used when we
can state that all outcomes are equally likely (are equiprobable).
- Tree diagrams: the probability of each event is computed multiplying the probability of each
branch.
- Law of Large Numbers: the relative frequency is the probability, as long as we repeat the
experiment many times.
P=0 P=
=50% P=1=100%
101
PROBABILITY - PRACTICE
14. We have a bag containing 1 red marble (canica), 3 green marbles 4 blue marbles and 2
yellow marbles. We pick one marble at random.
a) Find the sample space and the probability of the simple events.
b) Find the probability of choosing a marble that is not green.
15. At a carnival (feria), kids spin the wheel shown to win a prize. Find the
sample space ant the probability of the simple events.
[Vocabulary: linterna= flashlight, imán=magnet]
16. María rolls a 6-sided die (dado). Find the simple space and the probability of the simple
events. List all the possible outcomes that math each of the following events; then find
their probability:
a) Rolling and odd number. b) Rolling a number less than 3.
c) Rolling a number other than 5.
17. Laura tosses a coin (lanza una moneda), and Francisco spins the spinner
shown. Find the sample space. [Vocabulary: cara=Heads, Cruz=Tails]
18. A two-digit number is chosen by picking the “tens place” from Box A
and the “ones place” from Box B.
a) There are 20 possible outcomes. Say why, without writing the
sample space.
b) List all the possible numbers beginning with a 3 that can be drawn from the boxes.
c) Which of the following is NOT a possible outcome? 53 54 57 51.
d) List all the possible outcomes for the event “drawing a number less than 20”.
19. Melisa is choosing what to wear to school. She has three shirts and two pairs of pants. The
probabilities of choosing each are:
Shirts: white (50%), blue (30%) and red (20%). Pants: black (80%) and brown (20%).
a) Draw a tree diagram to show the different outfits (conjunto) Melisa can wear.
b) Write the sample space, and find the probabilities of the simple events.
20. Six chips (fichas) are placed in a bag [3 red, 1 yellow, 2 blue]. We pick three of them.
a) Write the sample space when we do not place the chips back in the bag. Find P(R,B,Y)
b) Answer, without writing the tree: How many possible outcomes are when we replace
the chip back in the bag after it is picked?
102
21. Alicia and Álvaro are playing a game. Alicia rolls a die, and Álvaro tosses a coin.
a) Use a tree diagram to find the sample space for this experiment. [Hint: 12 outcomes]
b) Find the probability of either rolling less than three, or Tossing tails. [Sol: P=8/12=66.7%]
22. Each set of cards shown below is turned over and shuffled (barajado). Then one card is
picked. Fill the chart with probabilities:
23. Juana has a bag containing 100 marbles. There are 50 red marbles, 30 blue marbles and 20
green marbles.
a) She picks out one marble. Write the sample space and find the probability of each
simple event. [Sol: S={R,B,G}. P(R)=50%, P(B)=30%, P(G)=20%]
b) She picks out two marbles (with replacement). What is the probability of picking a red
and a blue marbles? [Hint: tackle it (trata de resolverlo) using a pruned tree] [Sol: P=30%]
c) What is the probability if section (b) were without replacement? [Sol: P=30.3%].
24. The spinner spun once. Find P(even number and less than 7) [Sol P=30%].
Find P(even number or less than 7) [Sol: P=80%].
25. Alicia takes a standard pack of 52 playing cards. [Numbers from 1 to 10, plus 3 faces
for each suit: (red) D=diamonds, (black) C=clubs (tréboles), (red) H=hearts, (black) S=spades (picas)]. She picks one
card at random. Calculate:
a) P(red) b) P(5)= c) P(red or 5) d) P(red and 5)
e) P(face) f) P(black and face) g) P(black or face) h) P(hearts).
Sol: a) P=
b) P=
c) P=
d) P=
e) P=
f) P=
g) P=
h) P=
26. Virginia is decorating for the school dance. She can choose from the list of decoration:
a) How many possible combinations are there using one of each?
b) How many do not include Moons as a Hanging decoration?
c) How many include Clouds? And Blue streamers? And “Clouds or Blue streamers”?
Hanging decorations
Stars
Moons
Planets
Clouds
Table Top
Decorations
Rockets
Airplanes
Streamer (serpentina) colours
Red
Yellow
Blue
Set P(star) P(triangle)
A =
=25% =
=25%
B =0 =
=20%
C =
=100% =0
D =
=40% =
=20%
103
27. Two dice are rolled. Find the probability that the sum of the numbers is: [Use a pruned tree]
a) Equal to 8 b) Over 8. c) Equal to 10. d) Less than 5.
[Sol: a) P(8)=
=13.9%; b) P(>8)=
=27.8%; c) P(10) =
=8.3%; d) P(<6) =
=16.7%].
28. A coin is tossed three times, find the probability of each of the following events:
a) Heads on exactly 2 tosses b) Tails on 2 or more tosses.
c) The coin landing on the same side all 3 tosses d) The 1st and 3rd coins landing on the
[Sol: a) P=37.5% b) P=50% c) P=25% d) P=50%] same side.
29. The weather channel says there is a 30% chance of rain today. What is the probability that
it will not rain today? [Sol: 70%].
30. Cintia picks one card from a standard pack of 52 cards. She makes a note of the suit (palo),
then replaces the card and picks another one.. Find P(A) and P(not A), where “event A” is
Cintia picking: [*] Sometimes it is easier P(not A).
a) At least one heart b) Two cards different suit.
c) Two diamonds d) One heart and one club.
[D=diamonds, C=clubs (tréboles), H=hearts, S=spades (picas)]
31. You roll a die, pick a card, and find the sum.
a) How many different sums are possible? [Sol: 7 possible sums; between 5 and 11]
b) What is the probability of obtaining 6? [Sol: P=28.6%]
32. There are 50 socks in Daniela’s drawer. There are 25 black socks, 10 blue socks, 10 orange
socks and 5 striped (a rayas) socks.
a) Daniela picks a sock without looking. Write the sample space and find the probability of
each simple event. [Sol: S={Bk,Bl,Or,St}, P(Bk)=50%, P(Bl)=20%, P(Or)=20%, P(St)=10%]
b) She picks two socks without looking. What is the probability of picking the same colour?
[Hint: tackle it using a pruned tree]. [P(Same colour)=24.5%+3.7%+3.7%+0.82%=32.72%].
And the probability of picking different colours? [P(Different colours)=100-32.72=67.28%].
5 4
105
14. Cuerpos Geométricos y Volúmenes
- Poliedro: cuerpo geométrico cerrado con caras planas. Arista: cada lado de las caras.
Poliedro convexo: si el segmento que une dos puntos cualquiera está dentro del poliedro.
Poliedro cóncavo: si alguno de los segmentos sale fuera del poliedro.
Fórmula de Euler: en todo poliedro convexo (y todo cuerpo del “tipo” de la esfera)
Poliedro regular (sólidos platónicos): si todas las caras son polígonos regulares iguales y en
cada vértice concurre el mismo número de caras. Hay cinco.
Prisma: tiene dos caras iguales y paralelas “bases”, y las otras son paralelogramos.
Cuando la base es un paralelogramo, se llama paralelepípedo. Si además sus
ángulos son rectos, se llama ortoedro (como un rectángulo, con volumen).
Pirámide: una cara es un polígono y el resto triángulos que concurren en un punto.
Cuerpo de revolución: el obtenido haciendo girar una curva plana (generatriz) alrededor de
una recta (eje). Ejemplos: cilindro, cono y esfera.
Principio de Cavalieri: si dos cuerpos de la misma altura tienen secciones de
áreas iguales, entonces tienen el mismo volumen.
- Ejemplos: Volumen del cilindro: V=Abase·h. Volumen del cono:
Abase·h.
Volumen: V=
Abase·h.
Volumen: V=Abase·h.
Vértices – Aristas + Caras = 2.
Tetraedro Hexaedro (cubo) Octaedro Dodecaedro Icosaedro
106
GEOMETRIC BODIES AND VOLUMES
- Polyhedron: a solid figure bounded by polygonal faces. Edges: sides of the faces.
Convex polyhedron: if any line segment joining two points is inside the polyhedron.
Concave polyhedron: if there is a line segment joining two points outside the polyhedron.
Euler’s formula: in any convex polyhedron (or geometric body of the “type” of the sphere)
Regular polyhedra (platonic solids): when all its faces are equal regular polygons and on
every vertex meet the same number of faces. There are five.
Prism: has two equal parallel faces “bases”, and the others are parallelograms.
When the base is a parallelogram, it is called parallelepiped. Moreover if it is
right-angled, it is called cuboid (like a rectangle, with volume).
Pyramid: one face is a polygon and the others triangles that meet at one point
Solid of revolution: it is obtained by rotating a plane curve (generatrix) around a straight
line (axis). Examples: cylinder, cone y sphere.
Cavalieri’s principle: if two bodies have the same height and their sections
have the same areas, then they have the same volume.
- Examples: Volume of the cylinder: V=Abase·h. Volume of the cone:
Abase·h.
Volume: V=
Abase·h.
Volume: V=Abase·h.
Vertices – Edges + Faces = 2.
Tetrahedron Hexahedron (cube) Octahedron Dodecahedron Icosahedron
107
GEOMETRIC BODIES AND VOLUMES - PRACTICE
1. Find the volume of each prism
a) b) c) d) e)
Sol: V=48in3 V=30m
3 V=1152ft
3 V=693cm
3 V=120m
3
f) g) h) i) j)
Sol: V=24m3 V=120cm3 V=308cm
3 V=614.3ft
3 V=105cm3
2. Find the volume of these figures
a) b) c) d) e)
Sol: V=37.7cm3 V=1,539.4m
3 V=12 cm
3 V=4320cm
3 V=169.6ft
3
f) g) h) i) j)
Sol: V=64cm3 V=164.4ft
3 V=334.5in
3 V=75.4cm
3 V=235.6dm
3.
3. Find the volume, breaking the figure down.
[Sol: V=144+84=228in3]
Sol: V= 3,456+1,728=5,184in3 V=130,560+5,184=135,744in3.
3cm 3cm
4cm
6cm 4cm
8cm
6cm
8cm
9dm
10dm
7.3ft
5.5in
30cm
8cm 11cm
13ft 10.5ft 9
108
4. Find the volume of these figures (you may need to break them down)
Sol: V=664in3 V=960ft
3 V=508.8cm
3 V=972.1in
3.
Sol: V=1570-392.5=1177.5ft3 V=195m
3 V=171in
3 V=188.5yd
3.
Sol: V=230.8cm3 V=1,314.1in
3 V=1,835.9cm
3.
Sol: V=492.4m3 V=309.3in
3 V=272.7cm
3.
Sol: V=1,176cm3 V=140ft
3 V=1,815.8ft
3.
10cm 6cm 2cm
4cm
5cm
109
5. Count the number of vertices, edges and faces. Check it using Euler’s formula.
a) Dodecahedron b) Icosahedron
c) Truncated Tetrahedron d) Cuboctahedron
e) Truncated Cube f) Triangular Cupola
g) Great Rhombicuboctahedron h) Rhombicuboctahedron
i) Sphenocorona j) Pentagonal Antiprism
k) Rhombic Dodecahedron l) The hole causes Euler’s formula be equal to 0!
Solutions:
a) F = 12, E = 30, V = 20. b) F = 20, E = 30, V = 12. c) F = 8, E = 18, V = 12.
e) F = 14, E = 24, V = 12. f) F = 14, E = 36, V = 24. g) F = 8, E = 15, V = 9.
g) F = 26, E = 72, V = 48. h) F = 26, E = 48, V = 24. i) F = 14, E = 22, V = 10.
j) F = 12, E = 20, V = 10. k) F = 12, E = 24, V = 14. l) F = 16, E = 32, V = 16.
110
GEOMETRIC BODIES AND VOLUMES – WORD PROBLEMS
1. A swimming pool company makes a 15ft diameter circular pool that is available in 3ft,
3.5ft and 4ft depths. Fin the approximate number of water litres needed to fill each pool
2. Find the total volume of the barn (granero) pictured on the right.
3. A candy company has decided to change the shape of its box. The
old box was a rectangular prism with l=12cm, w=6cm and h=2cm.
Instead they will make a triangular prism in which the base of the
triangle is 6cm, and the height of the triangle is 12cm.
The height of the prism is 2cm. How does this new design change the volume of the box?
4. This figure has a volume of 360in3. How long is the distance
marked x?
5. The contents of a full baking-pan (molde para hornear) with
dimensions 8in. by 8 in. by 2 in. are poured (verter) into a
cylindrical container with a diameter of 5in. and a height of 8in. Will the cylindrical
container hold all the contents of the pan?
6. A cylinder-shaped bottle of aspirin is packaged inside a box that is a square-based prism.
The bottle has radius 2cm and height 6.6cm. The box has length 4.5cm, width 4.5cm , and
height 7cm. How much empty space is left inside the box once
the bottle has been put in it?
7. This building has a volume of 51.12m3. What is the length of
the longer side?
8. The lid (tapa) of a water bottle is pictured in the diagram.
a) Find the volume of the lid.
b) Find the volume of the bottle and its capacity (in litres).
9. The inside of a large tunnel in a children’s play area is to be painted. The tunnel is m long
and 1m tall. It is open at each end.
a) What is the area to be painted.
b) Cans of paint each cover 5m2. How many cans do they need to buy?
10. Find the volume and weight of this cement block. The holes are
square on top, 12cm on each edge. Cement weighs 1700kg per cm3.
111
11. Water is pouring into a conical tank at the rate of 1.8 m3 per minute.
Find to the nearest minute the number of minutes it will take to fill
the tank.
12. A scoop of ice cream (bola de helado) with diameter 6 cm is placed in
an ice-cream cone with diameter 5 cm. And height 12 cm. Is the cone big enough to hold
all the ice cream if it melts (derretirse)?
13. A concrete (cemento) bridge arch is shaped like the drawing at
right. The arch is in the shape of a semicircle. Find the volume of
concrete in the bridge.
14. When an elastic toy ball is fully inflated, it has a diameter of 7 inches. If some of the air is
removed, the diameter is only 5 inches. What is the difference in the
volume between the two sizes?
15. A child’s toy is fully filled with a heavy liquid in the hemisphere and lighter
liquids in the cone and cylinder so that the toy will always “right itself”
(stand up straight) as it is shown in the picture. What is the total volume
contained of the toy? (Radius of cone is 6 inches)
16. Estimate the amount of glass (cristal) in the paperweight (pisapapeles).
112
1º ESO Mathematics Trial Exam IES Extremadura
Bilingual section
Surname ______________________________________________ Mark
Name __________________________________
PROBABILITY (4 pts.)
1. (2.5 pts.). Find the sample space and the probability of simple events::
a. (0.5 pt.) Draw a card from a 40-cards-deck and write if it is a face card or not.
b. (0.5 pts.) Pick one marble at random from a bag containing five blue marbles, three
green marbles and two red marbles.
c. (1.5 pts.) You draw a card from each deck, and find their sum.
2. (1.5 pts) There are 14 socks in Juan’s drawer. There are 8 black socks, 4 white socks and 2
ankle socks (tobilleros). He picks two without looking. What is the probability of picking the
same type of socks?.
GEOMETRIC BODIES AND VOLUMES (6 pts.)
3. (1.5 pts) A farmer has left you his silo, a 20m high cylinder with a 4m diameter, and he has
told you that it's 65 percent full of feed.
a) What is the volume, in cubic metres of the silo? V=
b) What is the volume, in cubic metres of the feed inside the silo? V=
4. (3 pts.) Find the volume of figure (a). Write each solid’s name:
5. (1.5 pt.) Count the number of vertices, edges and faces of the Truncated Octahedron shown
in figure (b). Check it using Euler’s formula.
a) b)
8cm 5cm
14cm
8cm
8cm
4cm
2 3 4
5 6
5
Vertices
Edges:
Faces:
Euler’s Formula: Volume:
