UPCAT Review Math Chapter 10 of 16.pdf

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SECTION 10: PRE-REVIEW TEST

Directions: You have 30 minutes to answer 20 items. Box your final answer.

1. Arthur is 36 years old. He is 3 years older than thrice his son’s age. Find

the age of his son.

2. If 5 Ɵ 7 =

. Evaluate

Ɵ

.

3. Patrick is twice as old as his friend, James. James is 8 years older than Rica. In 8 years, Patrick will be three times as old as Rica. How old is James now?

4. Fifty liters of a punch that contains 20% water is mixed with 100 liters of another punch. The resulting fruit punch is 50% water. How many percent of water is in the 100 liters of punch.

5. Charles left home and drove at the rate of 60 mph for 3 hours. He stopped for lunch then drove for another 2 hours at the same rate to reach his destination. How many miles did Charles drive in total?

6. Mark wrote three consecutive even integers on a piece of paper. Three times the smallest number is greater by 114 than the sum of the other two. Find the three numbers.




7. Dindo and Eduard wanted to be far apart. They left from the same place with Dindo going to the east at 30 kph while Eduard going to the south at 40 kph. In how many hours will they be 225 km apart?

8. A snack company is mixing corn and peanuts to make a mixture worth P8 per kilo. Peanuts cost P9.50 per kilo and they use 3 kilos per batch. If corn cost P6 per kilo, how many kilos of corn do they need per batch?

9. Two printers would finish a task in 24 minutes. Printer 1 alone would finish the task in 60 minutes. How many pages does the task contain if printer 2 prints 5 pages a minute more than printer 1?

10. Five carpenters can complete constructing a house in 3 weeks. If there are fifteen carpenters how many days they can finish the same job?

11. Marco’s father is 6 times older than Marco and Marco is twice as old as his sister Danica. In three years time, the sum of their ages will be 69. How old is Marco now?

12. In a lab experiment, Fred needs to increase the concentration of a 10% acid solution to 30%. If the original solution is 30 liters, how many liters of a 60% acid solution does he need to add the original solution?

13. Three years from now Grace will be 42 years old. In 12 years, the sum of the ages of Grace and Lovely will be 125. How old is Lovely right now?




14. The product of two odd consecutive integers 575. Find the two integers.

15. A van travelled the SLEX in 4 hours while a second van in 3 hours. If the second van is 35 kph faster than the first, how fast is the slower van?

16. Flora is 7 years more than thrice the age of her daughter. The age of daughter is 10. Find the age of mother and find the difference between their ages.

17. If a pole 10 meters high casts a shadow 12 meters long, how long a shadow would a pole 14 meters high?

18. What is the two consecutive even integers whose sum is 138?

19. 8 years from now Moly will be 22 years old. The current sum of the ages of Mary and Britanni is 43. How old is Britanni right now?

20. Adrian and Jonathan who live 18 miles apart start at noon to walk toward each other at rates of 4 mph and 5 mph respectively. In how many hours will they meet?




SECTION 10:

WORD PROBLEMS

Now that you know how to write number sentences from the last chapter, we need to

build on this to help you deal with number problems. This chapter will provide you

different kinds of number problems and give you helpful tips on how to understand the

relationships contained in the problems and solve the problem.

A. AGE PROBLEMS

Age problems are generally easiest to solve as it only requires writing the correct

number sentence and solving for the age being asked.

Examples:

Mariah is twice the age of Nick now. What is Mariah’s age if Nick is 21?

m=2n, where n=21

m=2(21)

m=42

Mariah is 42 years old.

Britney is 5 times the age of her oldest son. Five years from now, Britney will be

thrice the age of her son. What is her son’s age now?

Equation (1): b = 5s

Equation (2): b+5=3(s+5), substitute Eq.1 into Eq.2

5s+5 = 3s+15

5s-3s = 10

2s = 10

s = 5

Britney’s son is 5 years old.

Ten years ago, Mary was twice the age of her sister, Carol. If Carol is 20 years old

now, how old is Mary now?

m-10 = 2(c-10), where c=20

m-10 =2(20-10)

m-10 =2(10)




m -10 =20

m = 30

Mary is 30 years old now.

B. NUMBER PROBLEMS

Number problems, like age problems, are simple applications of number sentence writing skills. The usual thing that they require is knowledge of spaces between numbers (between consecutive numbers = 1, between even numbers = 2, etc.).

Here is a table of how to easily represent variables in number problems:

… in the word problem Representation in the number sentence

A FEW TIPS!

Sum of two consecutive numbers

x + (x+1)

Sum of three consecutive numbers

x + (x+1) + (x+2) OR

(x-1) + x + (x+1)

Better to use (x-1)… because the sum cancels out the constant and leaves you with 3x = … (this leaves less room for manipulation mistakes) just remember that this way, you get x as the middle number in the sequence NOT the first number in the sequence

Sum of two even/odd numbers

x + (x+2)

Product of two consecutive numbers

x(x+1)

Product of two consecutive numbers

x(x+2) OR

(x-1) (x+1)

I prefer to use (x-1) (x+1). This way x = the even number in between the two odd numbers. This is easier than using x(x+2) because you will result to 2x – 1 = something. Then you just add 1 to both sides and get the square root of both sides rather than doing factoring.




Examples:

The sum of two consecutive integers is 45. Find the numbers.

n + (n + 1) = 45 2n + 1 = 45 2n = 44 n = 22; so n+1 =23

The numbers are 22 and 23.

The sum of the first and third of three consecutive odd integers is 67 less than three times the second integer. Find the three integers. x + (x + 4) = 3(x + 2) – 67 2x + 4 = 3x + 6 – 67 67 + 4 – 6 = x 65 = x x=65, x+2=67, x+4=69

The three numbers are 65, 67 and 69.

What positive consecutive odd numbers have a product of 35?

(x)(x + 2) = 35 x2 + 2x = 35 x2 + 2x – 35 = 0 (x + 7)(x – 5) = 0

x= 5 (rejecting -7 since the problem asks for positive odd) x+2=7 The consecutive odd numbers are 5 and 7.

Let’s try the tip discussed above about using (x-1) and (x+1) instead of (x) and (x + 2).

(x-1)(x + 1) = 35 x2 - 1 = 35 x2 = 36 x = 6 (-6 is rejected)

x -1 = 5, x + 1 = 7

The numbers are 5 and 7.




When the smallest of three consecutive odd integers is added to four times the largest, it produces a result 55 more than four times the middle integer. Find the numbers. x + 4 (x + 4) = 55 + 4 (x + 2) x + 4 x + 16 = 55 + 4x + 8 x = 47 x + 2 = 49 x + 4 = 51

The numbers are 47, 49 and 51.

Exercise 2

(Directions: You have 8 minutes to answer 5 items

1. Find the two consecutive odd numbers whose product is 575.

2. Mr. Blanco is 27 years older than Renz and 10 years from now

he will be twice as old as Renz. How old is each now?

3. Michael is 27 years old. The sum of the ages of Michael and Roldan is 65. How old is Roldan?

4. Seven years ago, Claire’s age was half of the age she will be in 10 years. How old is she now?

5. The product of two even consecutive integers is 840. Find the two integers.




C. RATIO AND PROPORTION There are only two types: inverse proportion and direct proportion.

Direct proportion – the larger x gets, the larger y gets.

Example:

Shadows – the longer the object casting the shadow, the longer its shadow.

A 10-meter flagpole casts a shadow of 5 meters long. A 20-meter shadow will

cast a shadow 10 meters long.

Ratio of flagpoles = Ratio of shadows

f1:f2 = s1:s2

10: 20 = 5:10

Indirect proportion – the larger x gets, the smaller y gets.

Example:

Work – the more workers working on the same job, the shorter the time it takes

to finish the job.

Ten construction workers finish a house in 5 days. If 20 construction workers

build exactly the same house, it will take only 2.5 days.

Ratio of workers = Reciprocal of the Ratio of days (thus the name inverse

proportion)

w1:w2 = d2:d1

10: 20 = 2.5:5

Examples:

A flagpole 15 meters in height casts a shadow 3 meters long, how many meters

is the shadow of a 5 meter lamp post?

Number sentence:

This is a direct proportion so:




15x = 5(3)

15x = 15

x =

The shadow of the lamp post is 1 meter.

Five men can finish a printing job in three days. How many days can 7 men finish

the same job?

Number sentence:

This is an inverse proportion so:

3(5) = 7x

15 = 7x

x =

days

It will take 7 men

days to finish the printing job.

D. DISTANCE PROBLEMS

Distance problems use the distance formula (Distance = rate x time) given

different scenarios. Usually, exams ask for the time at which two moving objects

will meet, the time at which two accelerating objects will have the same speed,

the total distance travelled, and the difference of the distance travelled by two

objects moving at different speeds or started moving at different times. In

essence, this is just asking you to imagine a complex scenario using the simple

distance formula.

D.1 Different speeds, same starting time, opposite directions

Example:

Two bikes started from the same starting point, at 9am, traveling in opposite directions at 20 and 30 kph respectively. At what time will they be 150 km apart?




D r t

Man 20t 20 t

Carriage 30t 30 t

Let t= number of hours after 9 am that the bikes will be 150 km apart

After t hours the distances D1 and D2 traveled by the two bikes are D1 = 20t and D2 = 30t

After t hours the distance D separating the two bikes is given by D = D1 + D2 = 20t + 30t = 50t

Distance D will be equal to 150 km when D = 50t = 150 t = 3 hours. 9 am + 3 hours = 12 nn

The bikes will be 150 km apart by 12 noon.

D.2 Different speeds, different starting time, same direction

Example:

At 7am, a man began to run from point A at 10 kph. At 9am, a carriage started running from the same point at 20 kph in the same direction as the man. At what time will the carriage pass the man?

Let t=time elapsed from 7am that the carriage will pass the man

D r t

Man 10t 10 t

Carriage 30(t-2) 30 t-2

After t hours the distances D1 traveled by the man is D1 = 10t

The carriage starts at 9am and therefore would have travelled two hours less than the man when it passes him. After (t - 2) hours, distance D2 traveled by car B is given by D2 = 20(t-2)




When the carriage passes the man, they are at the same distance from point A so D1 = D2 which is 10t = 30 (t-2) 10t = 30t – 60 60 = 20t 3 = t

The carriage passes the man at 7am + 3 hours = 10am

D.3 Different speeds, same starting time, towards each other

Example:

Two lovebirds traveling to meet in the middle left from two nests that are 720 meters apart, at the same time. The strong-winged female flies at 45 meters per minute while the male flies at 15 meters per minute .How many minutes after they left their nests will they meet?

D r t

Female 45t 45 t

Male 16t 15 t

After t minutes, the two birds would have traveled distances D1 and D2 given by D1 = 45t and D2 = 15t

After t minutes, the total distance D traveled by the two birds is given by D = D1 + D2 = 45t + 15t = 60t

At total distance D, equal to 720 meters, the two birds will meet each other. 60t = 720 t = 12 minutes

It will take the two lovebirds 12 minutes to meet.

D.4 Different speeds, different time, one object

Example:

A bird left its nest for its hunting ground and flew at the rate of 15 kph for 2 hours. It stopped to rest and flew for another 4 hours at the rate of 25 kph to reach its hunting ground. How many miles was the nest from the hunting ground?




Total distance D traveled by the bird is D = 15(2) + 25(4) = 130 km

The nest was 130 km apart from the hunting ground.

Rina and Chris wanted to be far apart. They left from the same point in a right angle, Chris traveling east at 30 kph and Rina travelling south at 40 kph. In how many hours will they be 225 km apart?

D r t

Chris 30t 30 t

Rina 40t 40 t

30t East

.

40t 225

South

The two cars are traveling in directions that are at right angle. Let x and y be the distances traveled by the two cars in t hours. Hence x = 30t and y =40t

Since the two directions are at right angle, the Pythagorean Theorem can be used to find distance D (225) between the two cars as follows:




√( ) ( )

√

The two cars will 225 km apart in 4.5 hours

D.5 Different rate, same distance

Example:

Sophie traveled from Quezon City to Manila in 3 hours. At a rate that was 30 kph faster than Sophie’s, Andres traveled the same distance in 2 hours. Find the distance between the two cities.

D r t

Sophie 3x x 3

Andres 2(x+30) x+30 2

3x = 2(x+30) 3x = 2x +60 x = 60

D = 3x D = 3(60) D = 180

The distance between the two cities is 180.

D.6 Distance problems against and in the direction of a current.




The only complication of a distance problem with and against a current is that we add or subtract the current’s speed to the given “on-calm-water”, “ground” speed, etc.

Against the current, we subtract the current speed. Biking without wind, a plane flies at 100 kph. Wind velocity is at 20 kph. So when the plane is flying against the wind, it flies slower at the speed of:

= Ground speed – Wind speed = 100 - 20

Flying in the same direction of the wind, the plan rides the current thus making it fly faster at the speed of:

= Ground speed + Wind speed = 100 + 20

Example:

Mariah’s luxury yacht travels through a river end-to-end for 3 hours with a current of 5 kph and then returns against the current in 8 hours. What is the yacht’s speed without the river current? How long is the river?

Distance upstream = distance upstream

6(y + 5) = 8(y – 5) 6y + 30 = 8y – 40 70 = 2b b = 35

D = 6(35 + 5) D = 6(40) D = 240

The yacht’s speed without the river current is 35 kph. The river is 240 km long.

D r t

downstream d = 6(y + 5) y + 5 6

upstream d = 8(y – 5) y – 5 8




Airport A and B are 120 km apart. Plane 1 and 2 are identical planes flying at the same speed. Plane 1 left airport A and plane 2 left airport B at the same time and passed each other 2 hours later. A wind current blows from airport A to B at 10 kph. Find the ground speed of the two planes.

D r t

Plane going against wind

d = 2(p - 10) p - 10 2

Plane going the same direction as wind

d = 2(p + 10) p + 10 2

2(p – 10) + 2(p + 10) = 120 4p = 120 p = 30

The ground speed of the planes is 30 kph.

Exercise 3

(Directions: You have 5 minutes to answer 5 items

1. A motorboat can go 15 miles downstream in 30 minutes. It takes 40 minutes for this boat to go back. Find the speed of the current in miles per hour.

2. The ground speed of a boat is 30 mph. It takes the same time for the boat to travel 5 miles upstream as it does to travel 10 miles downstream. What is the speed of the current?

3. Two cars left the same place and travelled in different directions. If the first car travelled at 60 mph and the second car travelled at 80 mph, in how many hours will they be 420miles apart?

4. Janet takes 2 hours to wash 4 kilos of dirty clothes, and Glenda takes 3 hours to wash 5 kilos of dirty clothes. How long will they take, working together, to wash 15 kilos of dirty clothes?

5. If 10 men can finish a job in nine days, how many men would it take to finish the job in 15 days?




E. WORK PROBLEMS

Work problems can also be treated like distance problems with one slight bit of nuance, D becomes the amount of work (which is usually = 1) r becomes the part of the work done per unit of time.

Example:

In a flowershop, it takes 3 hours for Kirsten make a flower arrangement. Julia can do the same arrangement in 2 hours. If they work together, how long will it take Kirsten and Julia to make an arrangement?

Kirsten’s rate of work: 1/3 Julia’s rate of work: 1/2

Work done by Kirsten alone: (1/3)t Work done by Linda alone: (1/2)t

Working together: (1/3)t + (1/2)t = 1 6[(1/3)t + (1/2)t] = 6 2t + 3t = 6 t = 6/5

Working together, they will finish an arrangement in 6/5 hours.

A faucet fills a Jacuzzi in 30 minutes The drain can empty it in 45 minutes. How long will it take to fill if both the faucet and the drain are left open?

= Work done per minute by faucet alone: 1/30

= Work done per minute by drain alone: -1/45 (drain works against the faucet) = Total work done per minute together: (1/30) – (1/45) = 1/90

With both faucet and drain open, the Jacuzzi will be full in 90 minutes.

In a factory, Donna takes 1 hour to make 50 bracelets, and Criselle takes 3 hours to make 60 bracelets. If they work together to make 350 bracelets, how long will

it take?




Here, the given is outputs per unit of time, not what part of the job is completed. So,

Donna can do 50 bracelets per hour and Criselle can do 20 bracelets per hour (60/3). Together, they can make 50 + 20 = 70 bracelets an hour.

It will take them 5 hours to make 350 bracelets.

Two workers are painting a house. Alone, the master carpenter can complete the job in 2 hours, but his assistant takes 10 hours. They worked together for the first hour, then master carpenter went home sick. How long will it take the assistant to finish painting the rest of the house?

After the first hour, the part of the job left unfinished

=

Since the remaining work will be done by the assistant, to get t, which is

divide the part of the job for him to do by his rate, 1/10:

= 4 hours

The assistant will finish the rest of the job in 4 hours.

F. MIXTURE PROBLEMS

Problems involving mixtures are more complex problems involving fractions. It is made

easy by understanding the whole and the part and whether you are adding to the

dividend or the divisor.

Imagine a solution made of salt and water. Usually, the given is a solution of x% salt

then you either add water or salt to create a y% salt solution. In other instances, the

given are products which you will sell at together (like mixed nuts made of peanuts and

cashews, or different kinds of chocolate put in a bag, etc.).




Examples:

A fruit factory makes a mixture of mangoes and guava for a pie company. For

each batch, it adds 3.5 kilos of guavas costing P4.50 per kilo to mangoes costing

P6.75 per kilo. The resulting mixture costs P5.875 per kilo. How many kilos of

mangoes does the factory need for each batch?

3.5(4.5) + 6.75x = 5.875(3.5+x)

15.75 + 6.75x = 20.5625 + 5.875x

0.875x = 4.8125

x = 5.5 kilos

The factory needs 5.5 kilos of mangoes for each batch.

How many liters of a 70% salt solution must be added to 30 liters of a 20% salt solution to produce a 50% salt solution?

liters of the solution

% alcohol

liters of alcohol

Added: 70% sol’n x 0.7 0.7x

Old: 20% sol’n 30 0.2 (0.2)(30) = 6

New: 50% sol’n 50 + x 0.5 0.5(50 + x)

New = Old + Added

0.5 =

15 + 0.5 x = 6 + 0.7x 9 = 0.2x x = 45

45 liters of the 70% solution needs to be added.

Maria is making fresh tortillas. How many kilos of flour that cost P30 per kilo must she mix with 6 kilos of corn that costs P55 per kilo to make a tortilla mix that costs P35?

kilos Price/kilo Cost

Flour x 30 30x

Corn 6 55 6(55)=330

Tortilla mix 6 + x 35 35(x+6)




New = Old + Added 35(x+6) = 330 + 30x 35x + 210 = 330 + 30x 5x = 120 x = 24

Maria needs 24 kilos of flour.

G. SYMBOLS-AS-OPERATIONS PROBLEMS

Entrance exams here and abroad like throwing in a bit of a surprise like using weird symbols that signify some mathematical relationship between numbers. There really is not much to do but to stay confident because the symbol problems are easy after some worked examples.

Examples:

Evaluate 9 7 when 6 1 = ( ( ) )

( ).

( ( ) )

( )

( )

Let the operation be defined by d =5c2 – 2d, for all integers c and d.

Evaluate 5 4.

( ) ( )




TIP:

For symbol problem like the two above, you just have to substitute variables into the equation and evaluate using the PEMDAS rule.

Exercise 4

(Directions: You have 8 minutes to answer 10 items. Box your final answer.)

1. If 𝑀 𝐼 𝐶

=

𝑀 ∙ I ∙C

, what is

9

?

2. A building casts a shadow 25 meters while a 12 meter post cast 15 meters. How long is a building?

3. Sam walks 15 kilometers at an average of 2.5 kph and returns on a bicycle at an average rate of 7.5 kph. How long does the entire trip take him?

4. It takes 3 hours for a boat to travel 30 miles downstream in 2 hours. Find the speed of the boat and the current.

5. John added four pints of pure alcohol to a 30% solution in water. If the new concentration is 40% how many pints of the original solution were there?




SECTION 10: POST-REVIEW TEST

Directions: You have 30 minutes to answer 20 items. Box your final answer.

1. If x ♫ y♫ z = ( )( )

, what is -10 ♫ 2 ♫ 5?

2. Luckie and Roy are painting a tutorial center. Alone, Luckie can complete the job in 2 hours, but Roy takes 10 hours. They worked together for the first hour, Luckie left. How long will take Roy to finish painting the rest of the tutorial center?

3. Kevin is four times as old as his friend Ron. Ron is 6 years older than Jen. In 7 years, Kevin will be twice as old as Jen. How old is Ron now?

4. Sixty liters of fruit juice that contains 15% water is mixed with 120 liters of another fruit juice. The resulting fruit juice is 40% water. Find the percent of water in the 120 liters of fruit juice.

5. Paulo travelled 480 kilometers in 8 hours. What is his average speed?

6. Lance wrote three consecutive odd integers. Four times the smallest number is greater by 64 than the sum of the other two. Find the three numbers.

7. Reggie and Charles left Mandaluyong and travelled in opposite directions. Reggie’s speed average is 12 km per hour less than Charles’ speed. After 3 hours, they are 396 km apart. What is the average speed of each car?




8. Tricia, Lawrence and Nash share the cost of their projects. Tricia pays

of the

cost, Lawrence pays

and Nash pays the rest. What fraction of the cost does

Nash pay?

9. Twelve boys can finish painting a fence in 8 days. If there are fifteen boys, how many days can they finish the same job?

10. Edu is 45 years old. He is 5 years older than five times his son’s age. Find the age of his son.

11. Jimson’s father is 7 times older than Jimson and Jimson is thrice as old as his sister, Jane. In five years time, the sum of their ages will be 81. How old is Jimson now?

12. In an experiment, Jim needs to increase the concentration of a 15% acid solution to 45%. If the original solution is 45 liters, how many liters of a 90% acid solution does he need to add the original solution?

13. Tessie will be 32 years old five years from now. In 6 years, the sum of the ages of Tessie and Cely will be. How old is Cely right now?

14. The product of three negative even integers is -504. What are the integers?




15. Two buses travelled the same road. The trip took the first bus 6 hours while the second bus, 5 hours. If the second bus is 25 kph faster than the first, how fast is each bus?

16. Jamela is 5 years more than twice the age of her sister. The age of her sister is 4. Find the age of Jamela.

17. A tree casts a shadow 18 meters long while a 6 -meter pole casts a shadow 8 meters long. How tall is a tree?

18. The sum of three consecutive even integers is 72. What is the smallest number?

19. 12 years from now Nancy will be 33 years old. The actual sum of the ages of Nancy and Maricel is 56. How old is Maricel right now?

20. Andy and Rey, who live 18 miles apart, start at 1: 00 to walk toward each other at same rate of 3 mph. In how many hours will they meet?

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UPCAT Review Math Chapter 10 of 16.pdf