Numbers & Simple Equations

8/4/2019 Numbers & Simple Equations

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NUMBERS & SIMPLE

EQUATIONS

- Prepared by KAPIL DATTA


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First, lets getthe basicsright.

What are the two basic categories

of nos.?


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TYPES OF NUMBERS

RATIONALNOS.: can be expressed inthe form of p/q. Eg: 2, 5, 1050, , etc.

IRRATIONALNOS.: cannot be written inp/q form. Eg: 5, 35, (9)1/3, e,.


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RATIONALNUMBERS

INTEGERS: Positive, Negative & Neutral.

NATURALNOS.- EVEN (divisible by 2)

(2n), ODD (not divisible by 2) (2n1) WHOLENOS.- 0 + Natural Nos.

PRIMES- nos. which have only two factors.

Is 1 a prime no.?


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DIVISIBILITYRULES:

2: last digit to be 0, 2, 4, 6, 8. 3: sum of digits to be divisible by 3. 4: last 2 digits to be divisible by 4 or to

be 00. 5: last digit to be 0 or 5.

6: to be divisible by both 2 & 3. 8: last 3-digits to be divisible by 8 or to

be 000.


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DIVISIBILITYRULES:

9: sum of digits to be divisible by 9. 11: the difference between the sum of

the alternate digits to be either divisibleby 11 or to be 0. 12: to be divisible by both 3 & 4. 15: to be divisible by both 3 & 5. 16: last 4-digits to be divisible by 16 or to

be 0000.


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DIVISIBILITYRULES:

18: even no. divisible by 9.

21: divisible by both 3 & 7.

22: even no. divisible by 11.

24: divisible by both 3 & 8.

25: last 2-digits to be 25, 50, 75 or 00.

32: last 4-digits to be divisible by 32 or to

be 0000.


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How many primes are there

from 1 to 100?

There are total 25 primes from 1 to 100,viz. 15 from 1 to 50 and 10 from 50 to

100.1 to 50: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29,

31, 37, 41, 43, 47.

50 to 100: 53, 59, 61, 67, 71, 73, 79, 83,89, 97.


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Do yourememberSquares & Cubes?

1, 4, 9, 16, 25, 36, 49, 64, 81, 100

121, 144, 169, 196, 225, 256, 289, 324, 341,

400441, 484, 529, 576, 625, 676, 729, 784, 841,

900

961, 1024, 1089, 1156, 1225, 1296, 1369,1444, 1521, 1600

1, 8, 27, 64, 125, 216, 343, 512, 729, 1000,1331, 1728, 2197, 2744, 3375, 4096


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ASSIGNMENT:

List down all the prime nos. from 100 to200.

Hint: There are 21 prime nos. from 100to 200.

Memorize squares for nos. from 1 to 40

and cubes for nos. from 1 to 16.


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CO-PRIMESORRELATIVEPRIMES:

These are pairs of those nos. whichhave only 1 as a common factor.

Examples: (5,7), (16,27), (8,15)

(15,18) is not a pair of Co-primes as

these two nos. have 3 as their HCF.


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VBODMAS

_______

VINICULUM BRACKETS [ { ( ) } ] OF DIVISION MULTIPLICATION xADDITION + SUBTRACTION -


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________

(25 25 10) + 5of5 125 x 25 = ?

= (25 15) + 25 125 x 25

= 10 + 1/5 x 25

= 10 + 5

= 15


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CYCLICALPATTERN:

For2: 2, 4, 8 and 6. For3: 3, 9, 7 and 1.

For7: 7, 9, 3 and 1. For8: 8, 4, 2 and 6. For4: 4 and 6.

For9: 9 and 1. For5: 5 For6: 6


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Find the digitin the units place

of(2347)343.

Units digit of 2347 is 7, whose power

when raised follows a cyclical pattern of4 different digits occurring in units place.

343 = 85 x 4 + 3

The third no. in the cyclical pattern of 7 is3, which is the units digit of the given no.


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LCM- Lowest Common Multiple

FACTORISATIONMETHOD

Find the LCM of 27 & 48.

27 = 33

48 = 24 X 31

The highest powerof prime factor 2 is 4

and that of 3 is 3.Hence, LCM = 24 x 33 = 432.


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HCF-

Highest C

omm

on Fact

or

FACTORISATIONMETHOD

Find the HCF of 36 & 48.

36 = 22 x 32

48 = 24 x 31

2 & 3 are common prime factors. Thelowest powerof 2 is 2 and that of 3 is 1.

HCF = 22 x 31 = 12.


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Sum of first n natural nos.,their

squares and cubes:

n = n(n+1)

2

n2= n(n + 1)(2n + 1)

6

n3 = [n(n + 1)]2

4


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(5+10+15++55)-(3+6+9++33)=?

= 511 - 311

=2

11= 2 x 11 x 12

2

= 132


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If7p6p is divisible by 11,whatisthe

maxm. no.ofvaluesthat p could take?

For 7p6p has to be divisible by 11, then7+6=p+p, since p has to be an integer.

2p=13 i.e., p is not an integer. (7+6) 2p = 1113 2p = 11

p=

1For only one value of p, 7p6p is divisible

by 11.


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Iftwo positive nos.have theirHCF as23

and product as 10580,then how many

possible valuesofLCMcan they have?

Let the two nos. be 23a and 23b.

232ab = 10580

ab = 20

:. Possible pairs are (1,20), (2,10), (4,5)

Only (1,20) and (4,5) have HCF as 23.

(Why not (2,10)??)

:. There are 2 possible values ofLCM.


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4 bellstoll at an interval of4, 6,8 and 10 min.

resp.Ifthey toll togetherat2am,when will

they to

ll to

gether forthe firsttime after2am?

The LCM of 4, 6, 8 and 10 = 120min.

120 min. = 2 hours

The bells will toll together at 2 + 2 =

4am


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How many nos. from 1 to5000 (both

inclusive) are bothsquares and

cubes?

For a number to be both a square and acube, it can be expressed as n6.

16= 1.26= 64.

36= 729.

46= 4096.Hence, there are four such nos.


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SIMPLEEQUATIONS:

Linear equations have one or morevariables, the maximum power of each

variable is one. One variable: 3x + 7 = 22 Two variables: 2x - 3y = 28 & x y = 26

2 eqns required to solve for the 2 variables. Three variables: 5x + 2y z = 6;

2x 4y + 3z = 3 & 3x 5y + 4z = 53 eqns required to solve for the 3 variables.


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In a fraction,ifthe denom.isincreased by

1,it becomes .Ifthe num.isincreased

by 9,it becomes 1.Find the denominator.

Let the given fraction =x/y

If the denom. is increased by 1, it becomes .

=> x/(y+1) = => 2x y 1 = 0 .eqn(1)

If the num. is increased by 9, it becomes 1.

(x+9)/y=

1=>

x y + 9=

0 .eqn(2)Solving eqns (1) & (2), we get y = 19.


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Rams age is 1/5thhis fathers age.The fathers

age after10 yrswill be twice Ritas age after10

yrs.If3 yrs agoRitas age was 12 yrs, find Rams

present age.

Let Rams age = x.Then fathers age = 5x.

Ritas age 3 yrs ago=12=>Ritas present age=15

After 10 yrs,Ritas age=25

fathers age=2(25)=50:. Fathers present age = 50 10 = 40Now 5x = 40 => x = 8yrs Rams present age


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A fathers age is 6 yrs more than 3timeshis

sons age.If after4 yrs, fathers age would be

12 yrs more than 2timesthe sons age, find

the present age ofthe father.

Let the sons age be = x yrs.

Then the fathers age=

3x +6After 4 yrs, (3x + 6 + 4) 2(x + 4) = 12

X = 10

:. Fathers age will be = 3x + 6 =36yrs.


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Sum ofthree nos.is432.First no.is

timesthe second and the third no.ishalf

ofthe 2nd no.Find the smallestofthe nos.

Let the second no. = x.

:. 3x/4 + x + x/2 = 432

:. 3x + 4x + 2x = 432 => x = 192

4

The smallest no. is x/2.192 =96.

2


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The sum ofthe digitsof a 2-digit no.is 10.If 18

issubtracted from the no.,the place valuesof

the digits getinterchanged.Find the number.

Let the number be xy.

10x = y 18 = 10y + x

x y = 2 ...eqn (1)Also given that x + y = 12 .eqn (2)

:. (1) + (2) gives 2x=

12 or x=

6.y = 4

The required no. is 64.


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Five burgers, 6 pizzas & 7cokescostRs.178.

Six burgers,4 pizzas & 2cokescostRs.124.

Find the costofone burger, 1 pizza & 1 coke.

Given 5b + 6p + 7c = 178 .(1)

6b + 4p + 2c = 124 .(2)2 x eqn(1) + eqn(2)

16b + 16p + 16c = 480

b + p + c =Rs.30


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In 5hrsAcan walk 4 kms more than whatB

can walk in 8hrs.In 7hrsBcan walk 2 kms

more than whatAcan walk in 3hrs. WhatisAs

speed?

5A = 4 + 8B .(1)

7B = 2 + 3A .(2)Solving (1) & (2), we get

B = 2 and A = 4

:. As speed is 4km/hr.


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Rs.1420 is distributed among 7 men, 11 women & 10

boys.Eachwoman getsthrice the amountthat a boy gets

& the share ofone man equalsthe sum ofthe sharesof a

woman and a boy.Find the sum ofthe sharesreceived by

the women and the boys puttogether.

1 woman = 3 boys1Man = 1 woman + 1 boy = 4 boys

7 men : 11 women : 10 boys (7 x 4) boys : (11 x 3) boys : 10 boys 28 : 33 : 10. Hence, each part of the ratio is

1420 71 = 20Share of 11 women & 10 boys = (33+10) x 20=Rs.860


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THANK YOU


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Numbers & Simple Equations