Arithmetic Final 2

8/12/2019 Arithmetic Final 2

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ArithmeticAptitude

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Index

1. Basic Calculations2. Number Systems3. L.C.M & H.C.F4. Percentages5. Profit & Loss6. Averages7. Ratio & Proportions8. Time & Distance9. Time & Work 10.Mensurations 2D & 3D11. Interest12. Clocks13. Calendars14. Probability15. Partnership16. Mixture or Allegation17. Problem on Ages18. Problem on Trains19. Boats and Streams

20. Pipes & Cisterns

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BASIC CALCULATIONS

VBODMAS

The order of various operations in exercises involving brackets and functions must be performed strictlyaccording to the order of the letters of the word VBODMAS. Each letter of the word VBODMAS stands afollows:

V for Vinculum : - (bar)B for Bracket : [{( )}]O for Of : ofD for Division : M for Multiplication : xA for Addition : +S for Subtraction : -

Note: There are three brackets. 1. ( ) 2. { } 3. [ ]

They are removed strictly in the order ( ), { } and [ ].Solved Example:

1. Simplify:85

41

131131

5of 21

451

321

4

Sol: Given expression

=85

45

3113

16of

29

516

29

=85

3113

16of

29

516

29

=8

1911

316

of 29

516

29

=8

693

16of 29

516

29

=8

693

1629

516

29

=8

69241

516

29

=120

10351629

=120

105129 =

120105154 0

=120511

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Square Root And Cube Root

Square : A number multiplied by itself is known as the square of the given number.E.g. square of 3 is 3 x 3 = 9

Square Root : Square root of a given number is that number which when multiplied by itself is equal to thegiven number. It is denoted by the symbol .

E.g. square root of 16 is 4 because 42 = 4 x 4 = 16 Thus, 16 = 4.

Methods of finding the Square Root:

I. Prime Factorization Method: This method is used when the given number is a perfect square or whenevery prime factor of that number is repeated twice. Follow the steps as mentioned below.

1. First find the prime factors of the given number.2. Group the factors in pairs.

3. Take one number from each pair of factors and then multiply them together. This product is the squaof the given number.

E.g. Find the square root of 225.Sol: 225 = 5 x 5 x 3 x 3

So, 225 = 5 x 3 = 15.

II. Method of Division: This method is used when the number is large and the factors cannot be easilydetermined.

E.g. Find the square root of 180625.

So, the square root of 180625 i.e. 180625 is 425.

Explanation :

1. First separate the digits of the number into periods of two beginning from the right. The last period either single digit or a pair.

2. Find a number (here it is 4) whose square may be equal or less then the first period (here it is 18).3. Find the remainder (here it is 2) and bring down the next period (here it is 06).4. Double the quotient (here 4) and write to the left (here 8).5. The divisor of this stage will be equal to the above sum (here 8) with the quotient of this stage (here

suffixed to it (here 82).6. Repeat this process till all the periods get exhausted.7. The final quotient is equal to the square root of the given number (here it is 425).

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Square root of a Decimal: If the given number is having decimal, separate the digits of it into periods of twthe right and left starting from the decimal point and then proceed as followed in the example.

E.g. 1. Find the square root of 1.498176.

So, 1.498176 = 1.224

Note: The square root of a decimal cannot found exactly, if it has an odd number of decimal places.

Try with finding the square root of 0.1790136

Square Root of a Fraction:

Case 1: If the denominator is a perfect square, the square root is found by taking the square root of thenumerator and denominator separately.

E.g. Find the square root of .49

2601

Sol:49

2601 =49

2601 =77

5151 =

7

51= 7

7

2

Case 2: If the denominator is not a perfect square, the fraction is converted into decimal and then square robtained or the denominator is made perfect square by mult iplying and dividing a suitable number and thesquare root can be determined.

E.g. Find the square root of .8

461

Sol: 8

461 =28

246 1 =16

92 2 =4

3644.30 = 7.5911 (nearly)

Cube: Cube of a number is obtained by multiplying the number itself thrice.

E.g. 64 is the cube of 4 as 64 = 4 x 4 x 4.

Cube Root: The cube root of a number is that number which when raised to the third power produces the gnumber, that is the cube root of a numbera is the number whose cube isa .

The cube root ofa is written as3 a .

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Methods to find Cube Root:1. Method of Factorization:

a. First write the given number as product of prime factors. b. Take the product of prime numbers, choosing one out of three of each type. This product g

the cube root of the given number.E.g. Find the cube root of 9261.

Sol: 9261 = 3 x 3 x 3 x 7 x 7 x 7so, 7392613 = 21

2. Method to find Cube Roots of Exact Cubes consisting the numbers up to 6 Digits:

Before we discuss the actual method it is better to have an overview of the following table.

Sl. No If the cube ends in then Cube root ends in Example 1 1 1 12 2 8 83 3 7 274 4 4 645 5 5 1256 6 6 2167 7 3 3438 8 2 5129 9 9 72910 10 0 1000

The method of finding the cube root of a number up to 6 digits which is actually a cube of some numberconsisting of 2 digits can be well explained with the help of the following examples.

E.g. 1. Find the cube root of 19683.

Sol: First make groups of 3 digits from the right side.

19,683 : 19 lies between 23 and 33 , so left digit is 2.687 ends in 3, so right digit is 7. [See the table.]

Thus, the cube root of 19683 is 27.

E.g. 2. Find the cube root of 614125.

614 125 : 614 lies between 83 and 93 , so left digit is 8.125 ends in 5, so right digit is 5. [See the table.]

Thus, the cube root of 614125 is 85.

Exercise1. Let a and b be two integers such that a + b = 10. Then the greatest value of a x b is ___

1. 20 2. 100 3. 21 4. 24

2. If a factory A makes x cars in an hour and another factory B, makes y cars every half an hour, how many cars both factories make in 4 hours?1. 4x+4y 2. 4x+8y 3. 8x+4y 4. 4x_2y

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3. Which of the following is the same as 50+12?1. 10(5+3) 2. (60 5)+(100 2) 3. 25 12x2 4. 50 4 3

4. If x*y = xy-y

x then the value of 6*3

1 is _______

1. 16 2. 17 3. -16 4. -17

5. If 4 32x , then the value of x is _______1. 2 2. 3 3. 4 4. 5

6. If the 23x5x = 5x103, then the value of x is ________ 1. 4 2. 3 3. 2 4. 1

7. If 5x =251 , then the value of x is ________

1.21 2. -2 3. 2 4. -4

8. If 7x5 xy

41 = 17, then the value of (x, y) is ________

1. (6, 2) 2. (7, 2) 3. (9, 2) 4. (8, 2)

9.32343243 = ?

1.49 2.

23 3.

1811 4.

367

10.371 of 111

31 of ? = 0

1. 1 2. 3 3. 9 4. 12

11. Which of the following fraction is greater than21 but less than

23 ?

1.3919 2.

4723 3.

5729 4.

4933

12.2)0014.0(2)011.0(2)004.0(

2)014.0(2)11.0(2)01.0( = ?

1. 0.01 2. 0.1 3. 100 4. 10

13. The number 1027-1 is not divisible by ____

1. 9 2. 90 3. 11 4. 1014.

090.069.02)3.2(

027.033.2 =

1. 0 2. 2.6 3. 2.3 4. None of these

15. If 102y = 25 then what is the value of 10y ?

1. -5 2. 5 3.251 4.

251

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16. If ab = 0.25 then what should the value of

92

ba2ba2 ?

1. 1 2.94 3.

95 4. 2

17. Which number is equal to1.0

01.001.01.0

1. 1.01 2. 1.1 3. 10.1 4. 10.01

18. What is the value of [0.3+0.3-0.3-0.3 x (0.3 x 0.30)]1. 0.09 2. 0.27 3. 0.60 4. None of these

19. Find the number which is equal to (50)3 + (-30)3 + (-20)31. 3x50x30x(-20) 2. 30x50x3x20 3. 3x50x(-30) 4. 3x(-30)x(-20

Find the value of the following:

20. 111111x11 = _______ 1. 122221 2. 1222221 3. 222221 4. 12222221

21. 5776800x11 = ___________ 1. 65344800 2. 63544800 3. 62544800 4. 63545800

22. 12369x11 = ________ 1. 135069 2. 136059 3. 136069 4. 135059

23. 15.60x0.30 = ?1. 4.68 2. 0.458 3. 0.468 4. 0.0468

24.01.0?

193420 x7 =?

1.9

35 2.5

63 3.7

18 4. None of these

25. If 2276 155 = 79.2, the value of 122.76 15.5 is equal to1. 7.092 2. 7.92 3. 79.02 4. 79.2

26.2.06.3?28.17 = 200

1. 120 2. 1.20 3. 12 4. 0.12

27. 12 0.09 of 0.3x2 = ?1. 0.80 2. 8.0 3. 80 4. None of these

28.?20

5.0820 12

1. 8 2. 18 3. 2 4. None of these

29. If 5 = 2.24, the value of 48.052

53 will be __________

1. 0.168 2. 1.68 3. 16.8 4. 168

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30.00121.0

289.0 = ?

1.11

17 0 2.11 017 3.

1117.0 4.

1117

31. 5555+6666-9999-1111 = ?1. 1001 2. 1011 3. 1111 4. 1221

32. Which is greater among1915

,95 ,

98

and87 ,

1.95 2.

1915 3.

87 4.

98

33. (17)2+(23)2 = ?1. 718 2. 818 3. 988 4. 8283

34. 132-123 = ?1. 369 2. 396 3. 496 4. 469

35. 441

872

1371

321 = ?

1. 52811 2. 5

2813 3. 6

2811 4. 6

2815

36. What approximate value should come in place of the question mark (?) in the following equation?66

62 % of ? = 32.78x18.44

1. 900 2. 880 3. 920 4. 940

37. What should come in place of the question mark (?) in the following equation?85.147+34.912x6.2+? = 802.2931. 8230 2. 8500 3. 8410 4. 8600

38. What should come in place of question mark (?) in the following equation?5679+1438-2015 = ?1. 5192 2. 5012 3. 5102 4. 5002

39. Four of the five parts numbered (1), (2), (3), (4) and (5) are exactly equal. Which of the parts is not equal to tfour? The number of that part is the answer.1. 40% of 160+

31 of 240 2. 120% of 1200

3. 38x12-39x8 4. 62

1 of 140-2.5x306.4

40. In the following equation what value would come in place of question mark(?)?5798-? = 7385-46321. 3225 2. 2595 3. 2775 4. 3045

41. What should come in place of question mark (?) in the following equation?197x?+162 = 26201. 22 2. 12 3. 14 4. 16

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42. Which of the following numbers are completely divisible by seven?A. 195195 B. 181181 C. 120120 D. 8918911. only A & B 2. only B & C 3. only D & B 4. All are divisible

43. what should come in the place of the question mark (?) in the following equation

1710

125

209

2521 = ?

1.12 577 2. 11

109 3.

45011 9 4. 1

9029

44. What should come in the place of the question mark (?) in the following equation?

112?

?28

1. 70 2. 56 3. 48 4. 64

45. What should come in the place of the question mark (?) in the following equation?48 ? +32 ? = 3201. 16 2. 2 3. 4 4. 32


8149

2?)7(

1. 9 2. 2 3. 3 4. 4


2

22

135

2745 .

1. 81 2. 1 3. 243 4. 9

48. Simplify: 18 410 +32 (4+10 2-1) = ___________ 1. 5 2. 9 3. 8 4. 7

49. Solve: 4- 345126 .1. 5 2. 4 3. 6 4. 8

50. If x=4, y=3, then the value of x 2xy is _________

1.47 2. 1

21 3.

54 4.

45

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NUMBER SYSTEM

Natural Numbers (N): Counting numbers 1, 2, 3, . . . . . are called Natural numbers. They are also called PIntegers. N = {1, 2, 3, . . . . . . . .}Whole Numbers (W): All the natural numbers including 0 together constitute the set of Whole numbers.W = {0, 1, 2, 3, . . . . . . . .}Integers (I or Z): All the whole numbers including negative counting numbers together constitute the set

Integers.I or Z = {. . . . . . ., -3,-2,-1, 0, 1, 2, 3 . . . . . . . .}Rational Numbers (Q): Numbers which are in the form of

qp

, where p, q are integers andq 0, are called

Rational numbers.Q = {

qp , (q 0)/ p, q Z}

E.g. -3, 1, 3.2,13

, 227

, etc.

Note:1. Rational numbers are divided into two groups, namely integers and non-integers.2. Non-integer belonging to the set of rat ional numbers is cal led fract ion.

Fraction: A number expressed in the form ofpq

is also called fraction, where p is the numerator and q is the

denominator. Fraction is a part of an integer.

E.g. ,56

,72

,61 etc.

Proper Fraction: Fractions in which Numerator < Denominator are called Proper Fractions.E.g. ,

51 ,

73

,97 etc.

Improper Fraction: Fractions in which Numerator > Denominator are called Improper Fractions.E.g. 8/3, 7/5, 9/4, etc.Mixed Fraction: It has two parts. One is integer and the other is a fraction.E.g. 1 1/3, 2 3/5, 5 4/3, etc.

Note:1. All the mixed fractions can be converted into improper fractions.2. A rational number can be expressed in the decimal form.3. The decimal form of a rational number is either recurring or a terminating decimal.

E.g. 10/3 = 3.3333 (recurring)3/4 = 0.75 (terminating)

Irrational Numbers (Q) : A number which cannot be expressed in the form of rational number is called anIrrational number.

For an irrational number, the decimal part is non-recurring and non-terminating.E.g. 2 = 1.414. It is non-recurring and non-terminating.Even number: An integer divisible by 2 is called an Even number.

E.g. 2, 4, 6, 8,..Odd Number: An integer not divisible by 2 is called an Odd number.

E.g. 1, 3, 5, 7,.Prime Numbers: Numbers which are not divisible by any number other then 1 and itself are called Prime

E.g. 2, 3, 5, 7,.Composite Numbers: Except 1, the numbers which are not prime are called Composite numbers

E.g. 4, 6, 9, 12,Co-prime Numbers: Numbers which do not have any common factor other than 1 are called Co-prime num

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E.g. (4, 15), (9, 22), (12, 29),

Note:1. 1 is neither prime nor composite.2. 2 is an even prime number.3. Co-prime numbers can be prime or composite numbers.4. Any two prime numbers are always Co-prime numbers.5. Any two consecutive positive integers are always co -primes.Place Value of a Digit in a Nu meral : The value of where the digit is in the number, such as units, tens, hun

etc.Face Value: Face Value of a number is the number itself.Consider the number 12654:

Place Value of 4 = 4 ones = 4, Face Value of 4 = 4Place Value of 5 = 5 tens = 50, Face Value of 5 = 5Place Value of 6 = 6 hundreds = 600, Face Value of 6 = 6Place Value of 2 = 2 thousands = 2000, Face Value of 2 = 2Place Value of 1 = 1 ten thousands = 10,000, Face Value of 1 = 1

Perfect Numbers: If the sum of the factors of a given number is twice the number, the number is said to bPerfect number.

E.g. Factors of 6 = 1, 2, 3, 6 and 1 + 2 + 3 + 6 = 1228, 496, etc.are the other examples of perfect numbers.

MULTIPLICATION TIPS:

1. For multiplication of a given number by 9, 99, 999, etc., that is by 10n 1, the easy way is: Put as many zerothe right of the multiplicant as there are nines in the multip lier and from the result subtract the multiplget the answer.

E.g. Multiply 2893 by 99.

Sol: 2893 x 99 = 2893 (100 1) = 289300 2893 = 286407.2. For multiplication of a given number by 11, 101, 1001, etc., that is by 10n + 1, the easy way is: Placen zeros to

the right of the multiplicant and then add the multiplicant to the number so obtained.

E.g. Multiply 3782 by 11.

Sol: 3782 x 11 = 3782 (10 + 1) = 37820 + 3782 = 41602.

3. For multiplication of a given number by 15, 25, 35, etc. Double the multiplier and then multiply the m by this new number and finally divide the product by 2.

E.g. Multiply 5054 x 15 = (5054 x 30) = (151620) = 75810

4. For multiplication of a given number by 5, 25, 125, 625, etc., that is, by a number which is some powePlace as many zeros to the right of the multiplicant as is the power of 5 in the multiplier, then divide thso obtained by 2 raised to the same power as is the power of 5.

E.g. 2982 x 5 = 29820/2 = 149105739 x 25 = 573900/22 = 143475

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a) No. of factors of a given number: If .....r q p c baN then the number of factors of N = ( p + 1) (q + 1) (r + 1)., where a, b, c are pr ime factors of N and p , q , r ,. are positive integers.

E.g. Find the number of factors of 24.Sol: 24 = 13 32

The number of factors of 24 = (3 + 1) (1 + 1) = 8.

b) Sum of the factors of a given number: If .....r q p cba then the sum of the factors of N =

..................1c

1c1b

1b1a

1a 1r1q1p where a, b, c are prime factors of N and p , q , r ,. are positive

integers.

E.g. Find the sum of the factors of 24.Sol: 24 = 13 32

Sum of the factors of 24 = .6013

13

12

12 1113

c) No. of ways of expressing a given number as a product of two factors:

If .....r q p cba where a, b, c are pr ime factors of N and p, q, r,. are positive integers then thof ways in which N can be expressed as product of two factors = ).... .1r)(1q)(1p(

21 .

E.g. Find the number of ways of expressing 48 as a product of two factors.Sol: 48 = 14 32 No. of ways = 5)11)(14(

21

)1q)(1p(21 .

d) No. of ways of expressing a given number which is a perfect square as a product of two factors:

If .....cbaN rqp where a, b, c are prime factors of N and p, q, r,. are positive integers then the

of ways in which N can be expressed as product of two factors = 1)........1r)(1q)(1p(21 .E.g. Find the no. of ways of expressing 36 as a product of two factors.Sol: 36 = 22 32 No. of ways = 51)12)(12(

21

1)1q)(1p(21 .

TIPS ON SQUARES:

Condition Method Example To square anynumber ending

with 5.(a 5)2 = {a (a+1 )}25

(35)2 = {3(3+1)}25 =1225

To square anumber in which

every digit is one.

Count the number of digits in the given number and start writingnumbers in ascending order from one to this number and then in

descending order up to one.(11)2 = 121,(111)2=12321

To square anumber which isnearer to 10 x .

Use the formula:22222 y)yx)(yx(y)yx(x

(1004) =(1004 4)

(1004 + 4) + (4)2 =1000(1008) + 16 =

1008016

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DIVISION:

Dividend = (Divisor x Quotient) + Remainder

E.g.

TESTS OF DIVISIBILITY

Divisibilityby Rule

Example Explanation

2

Units digit of the number should be zero or

divisible by 2.4, 2, 102,

etc.

3Sum of the digits in the number should be

divisible by 3. 17821+7+8+2 = 18 which is divisible by

3 hence 1782 also divisible by 3.

4 Number formed by the last two digits should

be divisible by 4 or are both zero. 4784,300, etc.

4784 Since 84 is divisible by 4,4784 is also divisible by 4.

5 Units digit of the number should be 0 or 5. 120, 625,etc.

6 Should satisfy divisibility rules of 2 and 3. 4518

7

The unit digit of the given number is doubledand then it is subtracted from the numberobtained after omitting the unit digit. If the

result is divisible by 7, then the given numberis also divisible by 7.

448 448 44 8(2) = 44 16 = 28which is divisible by 7 and hence448 is also divisible by 7.

8 Number formed by the last three d igi ts should

be divisible by 8. or zeros 15761576 576 is divisible by 8 andhence 1576 is also divisible by 8.

9Sum of the digits in the number should be

divisible by 9. 13951395 1+3+9+5 = 18 is divisible

by 9 and hence 1395 is alsodivisible by 9.

10 Number should end in zero. 1000

11Sum of digits at odd places Sum of digits at

even places should be 0 or divisible by 11.38797

38797 Sum of digits at odd places = 3+7+7 = 17 Sum of digitsat odd places = 8+9 =17 and 17 17 =0, hence 38797 is divisible by

11.12 Should satisfy divisibility rules of 3 and 4. 156 156 is divisible by 2 and 3 hence156 is also divisible by 12.14 Should satisfy divisibility rules of 2 and 7. 322 322 is divisible by 2 and 7 hence322 is also divisible by 14.

Last two digits in the number should be 0 ordivisible by 25.

175 75 is divisible by 25 andhence 175 is also divisible by 25.

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25 175

125Last three digits in the number should be 0 or

divisible by 125. 22502250 250 is divisible by 125 andhence 2250 is also divisible by 125.

Steps to find whether a given number is prime number or not:

1. Find the least positive integer,a such thata 2 > given number.2. Test the divisibility of given number by every prime number that is less thana .3. The given number is a prime number only if it is not divisible by any of these primes.

E.g. Check whether 923 is a prime number or not?

1. 923 lies between 900 and 961 which are perfect squares having square roots 30 and 31 respectively.2. Prime numbers less than 31 are 2,3,5,7,11,13,17,19,23,29.3. 923 are not divisible by any of these numbers and hence it is a prime number.

a) To find the number in the unit place for odd numbers: When there is an odd digit in the unit place except 5,multiply the number by itself until you gets 1 in the unit place.

n)1(... = ( 1)n4)3(... = ( 1)n4)7(... = ( 1)n2)9(... = ( 1) where n = 1, 2, 3, . . .

b) To find the number in the units place for even numbers: When there is an even digit in the unit place, multipthe number by itself until you gets 6 in the unit place.

n4)2(... = ( 6)n2)4(... = ( 6)

n)6(... = ( 6)n4)8(... = ( 6) where n = 1, 2, 3, . . .

c) If there is 1, 5 or 6 in the units place of the given number: If there is 1, 5 or 6 in the unit place of the givennumber, then after any times of its multiplication, it will have the same digit in the unit place.

n)1(... = ( 1)n)5(... = ( 5)n)6(... = ( 6) where n = 1, 2, 3, . . .

Solved Examples1. On dividing 64652 by a certain number, the quotient is 101 and the remainder is 12. Find the divisor

Sol: Here, the required number is divisor.

Divisor =Quotient

Remainder-Dividend

= 640101

64640101

1264652

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2. A number when divided by 160 leaves a remainder 52 and the quotient is 15. Find the number.

Sol: Here, the required number is dividend.Dividend = (Divisor x Quotient) + Remainder

= (160 x 15) + 52= 2452

3. Find the least number of 5 digits which is exactly divisible by 642.

Sol: The least number of 5 digits is 10,000.Dividing this number by 642, the remainder is 370.So, the required number is 10,000 + (642 - 370) = 10272.

4. Find the greatest number of 5 digits which is exactly divisible by 642.

Sol: The greatest number of 5 digits is 99,999.Dividing this number by 642, the remainder is 489.So, the required number is 99,999 - 489 = 99510.

5. Find the number nearest to 14800 which is exactly divisible by 245.

Sol: The remainder on dividing 14800 by 245 is 100.So, the number required number = 14800 100 = 14700 which is exactly divisible by 245.

6. Find whether 577 is a prime number.

Sol: (24)2 = 576 < 577 and (25)2 = 625 > 577n = 25

Prime numbers less than 25 are 2, 3, 5, 7, 11, 13, 17, 19 and 23.Since, 577 is not divisible by any of these numbers, it is a prime number.

7. How many numbers up to 531 are divisible by 15?Sol: Divide 531 by 15.

531 = 35 x 15 + 6The quotient is the required number and here it is 35.So, there are 35 numbers up to 531 are divisible by 15.

8. How many numbers up to 200 are divisible by 5 and 7 together?

Sol: L.C.M. of 5 and 7 = 35.Divide 200 by 35.200 = 5 x 35 + 20The quotient is the required number and here it is 5.

So, there are 5 numbers up to 200 are divisible by 35.9. Find the number in the unit place in 59)729( .

Sol: 9729)1(...729)729()729( 5859 in the unit place.

10. Find the number in the unit place in 42)98( .Sol: 4)4(...)6(...)98()98()98( 210442 in the unit place.

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11. Find the number in the unit place in 36)636( .Sol: 6)6(...)636( 3636 in the unit place.

12. Convert 0.4444.. into a rational number.Sol: Let x = 0.4444..(1)

Since 1 digit (4) is repeating multiply equation 1 on both sides by110 .10 x = 4.4444.(2)Subtract Equation 1 from 2 on both sides

10 x = 4.4444.- x = 0.4444.----------------------

9 x = 4.0000.----------------------

.94

x4x9

13. Convert 5.626262.. into a rational number.Sol: Let x = 5.626262.. (1)

Since 2 digits (62) is repeating multiply equation 1 on both sides by210 .100 x = 562.6262.(2)Subtract Equation 1 from 2 on both sides

100 x = 562.6262.- x = 5.6262..---------------------------

99 x = 557.0000.---------------------------

99x = 557 x = 99557

Exercise

1. 98+1O. x11 (3x3)+6x7 14+7 3x3 = ________

1.9

89 2. 1 3.9

45 4.9

10 1

2. The number divisible by 99 is_________ 1. 3572404 2. 135792 3. 913464 4. 114345

3. The smallest number which when subtracted from 43079 makes it exactly divisible by 9 is1. 4 2. 5 3. 6 4. 7

4. The number 111, 111, 111, 111 is divisible by1. 9, 11 2. 5, 11 3. 5, 9 4. 3, 11

5. The largest natural number by which the product of three consecutive even natural numbers is always divisib1. 16 2. 24 3. 48 4. 96

6. The number of divisors of 120 individuality is1. 8 2. 12 3. 14 4. 16

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7. The greatest number which exactly divides 826 is1. 2 2. 7 3. 59 4. 413

8. The least number which is a perfect square as well as a factor of 936 is1. 36 2. 9 3. 4 4. 16

9. Which of the following is a perfect number?1. 28 2. 30 3. 14 4. 16

10. Which of the following numbers is divisible by 11?1. 21434799 2, 74325566 3. 85437657 4. 93825677

11. The value of 320.16x320.04 is __________ 1. 2. 2 2 3. 2 4. 4. 2

12. Find the sum of 1+2+6+12+..+90.1. 550 2. 440 3. 330 4. 388

13. How many numbers are there between 200 and 300 in which 0 occurs only once?1. 18 2. 885 3. 115 4. 1000

14. 115885115115885885115115115885885885 =

1. 770 2. 885 3. 115 4. 1000

15. The difference between the squares of two numbers is 256000 and the sum of the numbers is 1000. The num1. 628, 372 2. 600, 400 3. 640, 630 4. None

16. A number is as much greater than 21 as is less than 71. The number is1. 39 2. 46 3. 41 4. 49

17. The sum of three numbers is 102. If the ratio between first and second be 2:3 and that between second and th5:3, then the second number is _____ 1. 30 2. 45 3. 27 4. 48

18. The value of 1+

91

1

11

1 = _________

1.1929 2.

1910 3.

1029 4.

910

19. The value of 9+9x9-9 9 is ___________ 1. 17 2. 89 3. 9 4. None

20. On dividing a number by 999, the quotient is 377 and the remainder is 105. The number is1. 476727 2. 376538 3. 376728 4. 359738

21. 0.77777 is equal to1.

8377 2.

7799 3.

9977 4.

10 977

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22. 199+299+399+..9999 is divisible by1. 99 2. 98 3. 100 4. 101

23. If the fraction60

63 ,

153

136 ,

156

143 and16

17 are arranged in ascending order, then which is the third number?

1.15 614 3 2.

15 313 6 3.

1617 4.

6063

24. How many three-digit numbers, when divided by 15, leave the remainder 15?1. 49 2. 50 3. 60 4. 62

25. The value of (112+122+132++202) is1. 3854 2. 2485 3. 3485 4. 2870

26. A boy was asked to multiply a certain number by 25. He multiplied by 52 and got his answer more than the cone by 324. The number to be multiplied was1. 12 2. 15 3. 25 4. 52

27. 9999972................

752

532

312 is equal to

1.999

5 2.999

1001 3.3

1001 4.1001

3

28. A number consists of two digits whose sum is 15. If 9 is added to the number, then the digits change their plnumber is1. 69 2. 78 3. 87 4. 96

29. One of the two consecutive positive integers, the sum of whose squares is 761, is1. 15 2. 24 3. 20 4. 25

30. The denominator of a rational number is 3 more than its numerator. If the numerator is increased by 7 and thdenominator is decreased by 7, we obtain 2. The rational number is

1.41 2.

1815 3.

107 4.

118

31. Find the value of n1n

1nn

33

33 is

1.21 2.

34 3.

32 4.

43

32. A number N when divided by 120 gives a remainder 76. What is the remainder obtained when the same numdivided by 8?

1. 2 2. 3 3. 4 4. 5

33. What is the value of 1+

71

1

11

1 ?

1.1319 2.

1913 3.

1318 4.

67

34. The sum of two consecutive prime numbers is 30. The two numbers are1. 18, 12 2. 7, 23 3. 11, 19 4. 13, 17

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35. By how much is 11% of 22.2 less than 10% of 24.4?1. 0.2 2. 0.02 3. 0.002 4. 0.0002

36. The sum of two numbers is 29 and the difference of their squares is 145. The difference between the number1. 13 2. 5 3. 8 4. 11

37. The ratio between a two digit number and the sum of the digits of that number is 4:1. If the digit in the units3 more than the digit in the tenth place, what is the number?1. 24 2. 63 3. 36 4. Cant be determined

38. The difference between a two digit number and the number obtained by interchanging the digits is 27. What sum of the two digits of the number?1. 3 2. 6 3. 9 4. Cant be determined

39. 0.02% of a number is 20. What will be 20% of the number?1. 200 2. 4000 3. 20,000 4. 2000

40. A number whose fifth part increased by 5 is equal to its fourth part diminished by 5 is1. 160 2. 200 3. 180 4. 220

41. What will be the remainder when 2975 is divided by 30?1. 28 2. 27 3. 1 4. 29

42. If a number is divided by 527, the remainder 42. What will be the remainder if it is divided by 17?1. 8 2. 7 3. 16 4. 15

43. Two different numbers when divided by 47, leave remainders 13 and 23 respectively. If their sum is divided same number 47, what will be the remainder?1. 35 2. 36 3. 20 4. 25

44. The units digit in the product (2767)135x(576)298 is

1. 2 2. 6 3. 8 4. 345. Find the number of divisors of 240?

1. 20 2. 10 3. 24 4. 18

46. Find the sum of the divisors of 360?1. 1107 2. 1170 3. 1175 4. 1180

47. What is the total number of possible values of p so that the number 3p8p4709 is divisible by 9?1. 0 2. 1 3. 2 4. 3

48. A man travels31 of a distance by car,

31 of the remaining by bus and

31 of the still remaining by train and the

remaining 72 km by scooter. Find the distance he traveled?1. 216 2. 243 3. 336 4. 576

49. What fraction of a week is a second?1.

604801 2.

864001 3.

6048001 4. 604800

50. 54 of a number exceeds its 2/3 by 8. The number is

1. 30 2. 60 3. 75 4. 90

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51. The expression 1/1.2 + 1/2.3 + 1/3.4 + . + 1/n(n+1) for any natural number n, is _____ 1. Always greater than 1 2. Always less than 13. Always equal to 1 4. Not definite

52. A two digit number becomes five-sixth of itself when its digits are reversed. The difference of the two digits number is ______ 1. 45 2. 54 3. 56 4. 65

53. The sum of the squares of two numbers is 3341 and the difference of their squares is 891. The numbers are __1. 25, 46 2. 35, 46 3. 25, 36 4. None of these

54. Of the three numbers, the sum of the first two is 45, the sum of the second and third is 55 and the sum of the thrice the first is 90. The third number is _______ 1. 30 2. 27 3. 39 4. 52

55. Three numbers are in the ratio 3 : 4 : 5. The sum of the largest and the smallest equals the sum of the third anThe smallest number is _______ 1. 20 2. 27 3. 39 4. 52

56. Of the three numbers, the first is twice the second and is half of the third. If the average of these numbers be numbers in order are _________ 1. 48, 96, 24 2. 48, 24, 96 3. 96, 24, 48 4. 96, 48, 24

57. The number of prime factors in the expression (6)10 x (7)17 x (11)27.1. 54 2. 64 3. 71 4. 81

58. The number of prime factors in 2222 x 3333 x 5555 is _______ 1. 3 2. 1107 3. 1110 4. 1272

59. The total number of prime factors of the product (8)20 (15)24 (7)15 is _______ 1. 59 2. 98 3. 123 4. 138

60. 24 is divided into two parts such that 7 times the first part added to 5 times the second part makes 146. The fiis _________ 1. 11 2. 13 3. 16 4. 17

61. The sum of squares of the numbers is 80 and the square of their difference is 36. The product of the two num ______ 1. 22 2. 44 3. 58 4. 116

62. The product of two numbers is 120. The sum of their squares is 289. The sum of the two numbers is _______1. 20 2. 23 3. 169 4. None of these

63. A fraction becomes 4 when 1 is added to both the numerator and denominator and it becomes 7 when 1 is sub

from both the numerator and denominator. The numerator of the given fraction is ________ 1. 2 2. 3 3. 7 4. 15

64. Three numbers are in the ratio 5 : 4 : 3. The sum of the largest and the smallest equals the sum of the third anThe largest number is ________ 1. 60 2. 27 3. 90 4. 65

65. The sum of three numbers is 132. If the first number be twice the second and third number be one-third of ththen the second number is _______ 1. 32 2. 36 3. 48 4. 60

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66. If the unit digit in the product 75 ? x 49 x 867 x 943 be 1, then the value of ? is ______ 1. 1 2. 3 3. 7 4. 9

67. What is the value of the digit in the place of $ in 3875 $ 622, if the number is divisible by 9?1. 4 2. 3 3. 2 4. 1

68. What is the least value of A + B, if 407A234B is exactly divisible by 9?1. 3 2. 4 3. 5 4. 7

69. If a number is subtracted from the square of its one half, the result is 48. The square root of the number is __________ 1. 4 2. 5 3. 6 4. 8

70. Two numbers are such that the ratio between them is 3 : 5, but if each is increased by 10, the ratio between th becomes 5 : 7. The numbers are _________ 1. 3, 5 2. 7, 9 3. 13, 22 4. 15, 25

Answers

1) 4 16) 2 31) 3 46) 2 61) 12) 4 17) 2 32) 3 47) 2 62) 23) 2 18) 1 33) 1 48) 2 63) 44) 4 19) 2 34) 4 49) 3 64) 35) 3 20) 3 35) 2 50) 2 65) 26) 4 21) 3 36) 2 51) 2 66) 47) 3 22) 3 37) 3 52) 1 67) 28) 3 23) 1 38) 4 53) 2 68) 49) 1 24) 3 39) 3 54) 1 69) 1

10) 3 25) 2 40) 2 55) 3 70) 411) 3 26) 1 41) 4 56) 212) 3 27) 3 42) 1 57) 213) 1 28) 2 43) 2 58) 314) 4 29) 3 44) 3 59) 315) 1 30) 2 45) 1 60) 2

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L.C.M. AND H.C.F.

Common Multiple: A common multiple of two or more numbers is a number which is exactly divisible by them.

E.g. 32 is a common factor of 8 and 16Least Common Multiple (L.C.M): The least multiple among all the common multiples of given numbers is Least Common Multiple.

Methods of finding L.C.M.

1. Method of Prime Factorsa. Resolve each given number into prime factors. b. Take out all the factors with highest powers tha t occur in given numbers.c. Find the product of these factors. This product will be L.C.M.

E.g. Find the L.C.M. of 12, 14 and 20.

12 = 22 x 314 = 2 x 720 = 22 x 5

So, the L.C.M. = 22 x 3 x 5 x 7 = 420

2. Method of Division

E.g. Find the L.C.M. of 12, 15, 18 and 20.

So, the L.C.M. = 2 x 2 x 3 x 5 x 3 = 180

Common Factor: A common factor of two or more numbers is a number which divides each ofthem exactly.

E.g. 4 is a common factor of 8 and 12

Highest Common Factor (H.C.F): Highest common factor of two or more numbers is thegreatest number that divides each one of them exactly. It is also called Greatest Common Divisoror Greatest Common Measure.

Methods of finding H.C.F.

1. Method of Prime Factors

E.g. Find the H.C.F. of 50 and 70

Sol: 50 = 2 x 5 x 570 = 2 x 5 x 7

Common factors are 2 and 5.So, H.C.F. = 2 x 5 = 10

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www.BankExamsToday.com, best study material for bank exams CRT


7. Find the greatest number which on dividing 152, 277 and 427 leaves equal remainder.

Sol: Required number =H.C.F. of |(152 277)|, |(277 - 427)|, |(427 152)|= H.C.F. of 125, 275 and 150 = 25.

8. Find the greatest number of 4 digits which, when divided by 12, 18, 21 and 28 leaves 4 as a remainder case.

Sol: L.C.M. of 12, 18, 21 and 28 = 252.Greatest 4-digit number = 9999.The remainder when 9999 is divided by 252 = 171So, the required number = (9999 171) + 4 = 9832.

9. Find the greatest number of 4 digits which, when divided by 12, 15, 20 and 35 leaves no remainder.

Sol: L.C.M. of 12, 15, 20 and 35 = 420.The remainder when 9999 is divided by 420 = 339So, the required number = (9999 339) = 9660.

10. Find the least number of 4 digits which is divisible by 2, 4, 6 and 8.

Sol: L.C.M. of 2, 4, 6 and 8 is 24.The least number of 4 digits = 1000The remainder when 1000 divided by 24 = 16.So, the required number = 1000 + (24 16) = 1008.

11. Find the smallest number of 4 digits when divided by 12, 18, 21 and 28 leaves remainder 5 in each case

Sol: L.C.M. of 12, 18, 21 and 28 = 252The least number of 4 digits = 1000The remainder when 1000 divided by 252 = 244.So, the required number = 1000 + (252 244) + 5 = 1013.

12. Two numbers when divided by a certain divisor give remainders 16 and 12 respectively. When their sudivided by the same divisor, the remainder is 4. Find the divisor.

Sol: Required divisor = 16 +12 4 = 24.

13. A number on being divided by 10 and 11 successively leaves the remainders 5 and 7, respectively. Findremainder when the same number is divided by 110.

Sol: Required remainder = 10 x 7 + 5 = 75.

14. Find the least number which when divided by 8, 10 and 15 leaves the remainders 3, 5 and 10, respectivSol: Here, 8 3 = 10 5 = 15 10 = 5

L.C.M. of (8, 10, 15) = 120The required least number = 120 5 = 115.

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Exercise

1. Find the L.C.M. of 12, 25, 30 and 20.1. 240 2. 300 3. 320 4. 150

2. Find the L.C.M. of 2 x 3 x 52 x 7, 23 x 32 x 5 and 22 x 33 x 52.1. 23 x 53 x 32 2. 23 x 32 x 5 x 7 3. 23 x 32 x 5 x 7 4. 23 x 33 x 52 x 7

3. Find the L.C.M of 158

and152

,103

,54 .

1.53

4 2.54

3 3.54

4 4.53

3

4. Find the L.C.M. of 0.01, 0.1, 0.001 and 0.0001.1. 0.1 2. 0.01 3. 0.001 4. 1

5. Find the H.C.F. of 300, 450 and 525.1. 125 2. 50 3. 75 4. 25

6. Find the H.C.F. of 32 x 52 x 7 , 3 x 52 x 7, 33 x 72 x 52.1. 3 x 5 x 72 2. 3 x 5 x 7 3. 3 x 52 x 7 4. 32 x 52 x 72

7. Find the G.C.D. of 2512

and256

,209

,53 .

1.103 2.

10 03 3.

10003 4. 3

8. Find the G.C.D. of 0.01, 0.1, 0.001, 0.0001.1. 0.1 2. 0.01 3. 0.001 4. 0.0001

9. Find the G.C.D. of 0.2, 2.0, 0.02, 20.0.1. 0.2 2. 0.02 3. 0.002 4. 2

10. L.C.M. of two numbers is 192 and their H.C.F. is 16. If one of them is 48 then find the other.1. 32 2. 64 3. 48 4. 68

11. The product of two numbers is 1575 and their L.C.M. is 315. Find the numbers.1. 30 and 45 2. 35and 45 3. 45 and 65 4. 45 and 75

12. The L.C.M. of two numbers is 12 times their H.C.F. The sum of L.C.M. and H.C.F. is 65. If one of them is 2the other.1. 15 2. 18 3. 25 4. 16

13. The L.C.M. of two numbers is 4 times their H.C.F. The product of numbers is 900. Find their L.C.M.1. 64 2. 48 3. 45 4. 60

14. Two numbers are in the ratio 5 : 8 and their H.C.F. is 4. Find the numbers.1. 25 and 40 2. 20 and 32 3. 30 and 48 4. 15 and 24

15. Three numbers are in the ratio 3 : 6 : 8 and their L.C.M. is 120. Find the smaller number.1. 16 2. 18 3. 15 4. 12

16. Find the greatest number which will divide 1026, 1215 and 2349 exactly.1. 24 2. 27 3. 29 4. 37

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17. Find the greatest possible length which can be used to measure the length 6 m 76 cm, 4 m 81 cm and 7 m 67exactly.1. 13 cm 2. 15 cm 3. 23 cm 4. 17 cm

18. Find the greatest number that will divide 1657 and 2037 to leave remainders 6 and 5 respectively.1. 121 2. 123 3. 129 4. 127

19. Find the greatest number that will divide 520, 1140 and 1220 leaves the remainders 7, 6 and 5 respectively.1. 29 2. 23 3. 27 4. 21

20. Find the greatest number which divides 208, 1194 and 1449 leaving 4 as the remainder in each case.1. 27 2. 19 3. 17 4. 13

21. Find the greatest number which divides 284, 678 and 1618 leaves the same remainder in each case.1. 2 2. 3 3. 4 4. 6

22. Find the least number which is exactly divisible by 10, 12, 14, 16 and 18.1. 5010 2. 5030 3. 5040 4. 5020

23. Find the least number which when divided by 12, 16, 18 leaves 5 as remainder in each case.1. 143 2. 149 3. 139 4. 144

24. Find the smallest number which when decreased by 3 is exactly divisible by 9, 12, 15, 18 and 21.1. 1253 2. 1263 3. 1267 4. 1243

25. Which of the following when increased by 4 is exactly divisible by 12, 14, 16, 28 and 20?.1. 5034 2. 5006 3. 5026 4. 5036

26. Find the least number which when divided by 35, 45 and 55 leaves remainders 20, 30 and 40 respectively.1. 3450 2. 3250 3. 3420 4. 3410

27. Find the least number of five digits which is exactly divisible by 4, 6, 8, 12 and 16.1. 10025 2. 10024 3. 10032 4. 10034

28. Find the greatest number of five digits which is exactly divisible by 16, 24, 28 and 32.1. 99456 2. 99556 3. 99446 4. 99566

29. Find the greatest number less than 800 which is exactly divisible by 12, 15, 20 and 30.1. 740 2. 720 3. 780 4. 760

30. Find the greatest number between 600 and 700 which is exactly divisible by 20 and 301. 660 2. 640 3. 620 4. 480

31. Find the least number of four digits which when divided by 4, 5 and 6 leaving the remainder 2 in each case.

1. 1022 2. 1012 3. 1032 4. 100232. Find the greatest number of four digits which when divided by 10, 15 and 20 leaves remainder 3 in each case

1. 9983 2. 9663 3. 9963 4. 9960

33. Find the least number which when divided by 8, 12, 15 and 18 leaves a remainder 3 in each case and exactly by 11.1. 360 2. 361 3. 363 4. 383

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34. Find the least number which when divided by 15, 20, 24 and 30 leaves 1 as remainder but when divided by 1no remainder.1. 228 2. 361 3. 380 4. 399

35. Five bells toll at an interval of 6, 9, 12, 15 and 18 seconds respectively beginning together. After what intervtime will they toll again together?1. 1 min 2. 4 min 3. 2 min 4. 3 min

36. 4 traffic signals at four different places change at an interval of 5 sec, 10 sec, 15 sec and 25 sec respectively. change simultaneously at 10 : 20 A.M., at what time will they again in change simultaneously?1. 10 : 22: 30 A.M. 2. 10 : 24: 30 A.M. 3. 10 : 21: 30 A.M. 4. 10 : 24: 40 A.M.

37. 4 wheels moving 24, 32, 48 and 52 revolutions in a minute starting at a certain point on the circumferencedownwards. After what interval of time will they come together again in the same position?1. 15 sec 2. 30 min 3. 30 sec 4. 15 min

38. Four men start together to travel the same way around on a circular path of 36 km with speeds 2 km/h, 3 kmkm/h and 6 km/h respectively. When they meet at the same point again after?1. 24 hours 2. 36 hours 3. 12 hours 4. 18 hours

39. Find the least number of square tiles required to pave the floor of a room of 5 m 44 cm long and 3 m 74 cm b1. 178 2. 164 3. 176 4. 172

40. Find the least number of square tiles required to pave the floor of a room of 8 m 99 cm long and 6 m 67 cm b1. 713 2. 723 3. 717 4. 719

41. A trader has three kinds of liquids of first kind 117 lit, second kind 130 lit and the third kind 143 lit. Find thenumber of full casks of equal sizes which this can be stored without mixing them.1. 28 2. 24 3. 36 4. 30

42. The H.C.F. of two numbers is 11 and their product is 1452. How many pairs of such numbers possible?

1. 1 2. 2 3. 3 4. 443. The H.C.F. of two numbers is 15 and their sum is 195. How many pairs of such numbers possible?

1. 1 2. 3 3. 4 4. 6

44. The H.C.F. of two numbers is 23 and their sum is 184. How many such pairs of numbers are possible?1. 0 2. 2 3. 3 4. 4

45. Find the least perfect square which is exactly divisible by 20, 30, 40 and 60.1. 2500 2. 3600 3. 900 4. 1600

46. Find the least number by which 240 must be multiplied in order to produce a multiple of 300.1. 2 2. 4 3. 5 4. 6

47. 64 mango trees, 48 apple trees and 80 orange trees have to plant in rows such that each row contains the samnumber of trees of one variety only. Find the least number of rows in which the trees may be planted.1. 18 2. 12 3. 16 4. 24

48. Find the greatest number of five digits which can be divisible by to 15, 20, 24, 27, 32 and 36.1. 90039 2. 90139 3. 86400 4. 90019

49. Find the smallest number of five digits which when divided by 52, 56, 78 and 91 leaves remainders 28, 32, 5respectively.1. 10896 2. 10196 3. 10496 4. 10696

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Percentages

Understanding Percentages:

Percent : The term per cent means per hundred or for every hundred. The word is derived from the Latin wo percentum. Percentage : A fraction whose denominator is 100 is called a percentage.Rate per cent : The numerator of the fraction is called rate per cent.

E.g.10 0

5 and 5 percent means the same thing i.e. 5 parts out of every hundred parts.

Basic Formulae:

1. To convert any fractionn1 into a rate per cent, multiply it by100 and put % sign i.e. %100

n1

E.g. What percentage is equivalent to43 ?

Sol:43 x 100 = 25%

2. To convert a per cent into a fraction, drop the per cent sign and divide the number by 100.

E.g. What fraction is 831 %?

Sol: 831 % =

325 =

325 x

10 01 =

121

3. x% of a given number (N) =10 0

x x N

E.g. 75 % of 800 = ?

Sol: 75 % of 800 =10 075 x 800 = 600

4. If A is x % more than that of B, then B is less than that of A by 100x100

x %.

5. If A is x % less than that of B, then B is more than that of A by 100x100

x %.

6. If A is x % of C and B is y % of C, then A = y x x 100% of B.

7. If two numbers are respectively x % and y % more than a third number, then the first number is

100y100x100 % of the second and

the second number is 100x100y100 % of the first.

8. If two numbers are respectively x % and y % less than a third number, then the first number is 100y100x100

% of the second and

the second number is 100x100y100 % of the first.

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9. If the price of a commodity increases by N%, then the reduction in consumption so as not to increase th

expenditure is 100N100

N %.

10. If the price of a commodity decreases by N%, then the increase in consumption so as not to decrease th

expenditure is 100N100

N %.

11. If a number is changed (increased/decreased) successively by x % and y % then net % change is given

100xy

yx % which represents increase or decrease in value according as the sign is +ve or ve.

Note: If x and y indicates decrease in percentage, then put ve sign before x and y else +ve sign.

12. If the population of a town (or the length of a tree) is P and its annual increase is r%, then:

(i) Population (or length of tree) after n years = .100

r1P

n

(ii) Population (or length of tree) n years ago = .100

r1

Pn

13. If the population (or value of a machine in rupees) is P and annual decrease (or depreciation) is r%, the

(i) Population (or value of machine) after n years = .100

r1P

n

(ii) Population (or value of machine) n years ago = .

100r

1

Pn

14. If a number K is increased successively by x % followed by y % and z %, then the final value of K wi

K 100

x1

100y

1100

z1

15. In an examination, the minimum pass percentage is x%. If a student scores y marks and fails by z mathe maximum marks in the examination is

x)zy(100 .

16. In an examination a % and b % students respectively fail in two different subjects while c % studen both the subjects, then the percentage of students who pass in both the subjects will be (100 ( a + b -

Percentage Change:A change can be of two types an increase or a decrease.When a value is changed from initial value to a final value,

% change = (Difference between initial and final value/initial value) X 100Eg: If 20 changes to 40, what is the % increase?Soln: % increase = (40-20)/20 X 100 = 100%.Note:

1. If a value is doubled the percentage increase is 100.2. If a value is tripled, the percentage change is 200 and so on.

Percentage Difference:% Difference = (Difference between values/value compared with) X 100.

Eg: By what percent is 40 more than 30?Soln: % difference = (40-30)/30 X 100 = 33.33%

(Here 40 is compared with 30. So 30 is taken as denominator)

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Eg: By what % is 60 more than 30?Soln: % difference = (60-30)/30 X 100 = 100%.

(Here is 60 is compared with 30.)Hint: To calculate percentage difference the value that occurs after the word than in the question can directly be uthe denominator in the formula.

Important Points to Note:1. When any value increases by

a. 10%, it becomes 1.1 times of itself. (since 100+10 = 110% = 1.1) b. 20%, it becomes 1.2 times of itself.c. 36%, it becomes 1.36 times of itself.d. 4%, it becomes 1.04 times of itself.

Thus we can see the effects on the values due to various percentage increases.2. When any value decreases by

a. 10%, it becomes 0.9 times of itself. (Since 100-10 = 90% = 0.9) b. 20%, it becomes 0.8 times of itself c. 36%, it becomes 0.64 times of itself d. 4%, it becomes 0.96 times of itself.

Thus we can see the effects on a value due to various percentage decreases.Note:1. When a value is multiplied by a decimal more than 1 it will be increased and when multiplied by less than 1 it wdecreased.2. The percentage increase or decrease depends on the decimal multiplied.Eg: 0.7 => 30% decrease, 0.67 => 33% decrease, 0. 956 => 4.4% decrease and so on.Eg: When the actual value is x, find the value when it is 30% decreased.Soln: 30% decrease => 0.7 x.Eg: A value after an increase of 20% became 600. What is the value?Soln: 1.2x = 600 (since 20% increase)

x = 500.Eg: If 600 is decrease by 20%, what is the new value?Soln: new value = 0.8 X 600 = 480. (Since 20% decrease)

Thus depending on the decimal we can decide the % change and vice versa.Eg: When a value is increased by 20%, by what percent should it be reduced to get the actual value?Soln: (It is equivalent to 1.2 reduced to 1 and we can use % decrease formula)

% decrease = (1.2 1)/1.2 X 100 = 16.66%.3. When a value is subjected multiple changes, the overall effect of all the changes can be obtained by multip

the individual factors of the changes.Eg: The population of a town increased by 10%, 20% and then decreased by 30%. The new population is what % ooriginal?Soln: The overall effect = 1.1 X 1.2 X 0.7 (Since 10%, 20% increase and 30% decrease)

= 0.924 = 92.4%.Eg: Two successive discounts of 10% and 20% are equal to a single discount of ___Soln: Discount is same as decrease of price.

So, decrease = 0.9 X 0.8 = 0.72 => 28% decrease (Since only 72% is remaining).SOLVED EXAMPLES

1. If Srujana salary is 20% more than that of Deepa, then how much percent is Deepas salary less than thSrujana?

Sol: Here, x = 20.

Required answer = 10 0x100

x %

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= 10 020100

20 %

=6

100 % =64

16 % =32

16 %.

2. If Anitas income is 30% less than that of Saritha, then how much percent is Sarithas income more thaAnitha?

Sol: Here, x = 30.

Required answer = 100x100

x %

= 10030100

30 %

=7

30 0 % =76

42 %.

3. If A is 25% of C and B is 30% of C, then what percentage of A is B?

Sol: A = 25% of C and B = 30% of C

AB =

2530

B =2530 x 100% of A

= 120% of A

4. Two numbers are respectively 25% and 50% more than a third number. What percent is the first of the

Sol: Here, x = 25 and y = 50

So, First number = 100y10 0 x10 0 % of the second

= 10 05010025100 % of the second

=6

50 0 % of the second

= 62

83 % =31

83 % of the second.5. Two numbers are respectively 20% and 32% less than a third number. What percent is the second of the

Sol: Here, x = 20 and y = 32

So, Second number = 100x100

y100 % of the first

= 1002010032100 % of the first

= 85% of the first.

6. If the price of a commodity increases by 50%, find how much percent its consumption be reduced so asincrease the expenditure.

Sol: Reduction in consumption = 100N100

N %

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= 10 050100

50 %

=3

10 0 % =

3

133 %.

7. If the price of a commodity decreases by 50%, find how much percent its consumption be increased so decrease the expenditure.

Sol: Increase in consumption = 100N100

N

= 10050100

50 %

= 100%.

8. If the salary of Mr. Shashi is first increased by 18% and thereafter decreased by 15%, what is the net chhis salary?

Sol: Here, x = 18 and y = -15

So, the net % change in the salary =10 0xy

yx %

=100

)15)(18(1518 %

=100

)15)(18(1518 %

= 0.3%.Since the sign is +ve, there is an increase in the salary of person by 0.3%.

9. The population of a town is decreased by 20% and 40% in two successive years. What percent populatidecreased after two years?

Sol: Here, x = - 20 and y = - 40

So, the net % change in population =10 0xy

yx %

=100

)40)(20(4020 %

=100800

60 %

= - 52%.

Since the sign is -ve, there is decrease in population after two years by 52%.

10. If the side of a square is increased by 10%, its area increased by k%. Find the value of k.

Sol: Area of square = side x side

So, net % change in area =10 0xy

yx %

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=100

)10)(10(1010 % [Take x, y = 10]

= 21%Hence, the area is increased by 21%.

Here, k = 21.11. The radius of a circle is increased by 4%. Find the percentage increase in its area.

Sol: Area of circle = x radius x radius

So, net % change in area =10 0xy

yx %

=100

)4)(4(44 % [Take x, y = 4]

=10016

8 % =254

8 %.

12. The tax on a commodity is diminished by 12% and its consumption increases by 10%. Find the effect o

revenue.Sol: Revenue = tax x consumption

So, net % change in revenue =100xy

yx %

=100

)10)(12(1012 % [Take x = - 12, y = 10]

=1001202 % = - 3.2%.

So, the revenue decreases by 3.2%.13. The population of a town increases by 4% annually. If its present population is 12500, what will it be i

time?

Sol: Here, P = 12500, r = 4 and n = 2.

Population after n years = .100

r1P

n

Population after 2 years =2

10 04

112500

=2

251

112500

=2

2526

12500

=2526

2526

12500

= 13520.

14. The population of a town increases by 10% annually. If its present population is 12100, what it was 2 yago?

Sol: Here, P = 12100, r = 10 and n = 2.

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Population n years ago =n

100r

1

P

Population 2 years ago = 210010

1

12100

=2

1011

12100 =

1011

101112100

= 10101111

12100 = 10000.

15. The population of a town is 100000. It increases by 5% during the first year. During the second year, itdecreases by 10% and increases by 15% during the third year. What is the population after 3 years?

Sol: Here, P = 100000, x = 5, y = - 10 and z = 15

So, Population after 3 years = P 100x1 100y1 100z1

= 10000010 0

51

10 010

110015

1

= 108675100100100

11590105100000

16. In an examination, a student must get 50% marks to pass. If a student, who gets 110 marks, fails by 50 find the maximum marks.

Sol: It is clear, Minimum Marks = 110 + 50 = 160Let the maximum marks = xSo, 50% of x = 160

x = 32017. In an examination, 40% students failed in Mathematics and 51% failed in Science. If 16% failed in bot

subjects, find the percentage of those who passed in both the subjects.

Sol: Here, a = 40, b = 51 and c = 16Percentage of students passing both the subjects

= (100 ( a + b - c ))%= (100 (40+5116))%= 25%.

Percentages Fractions Conversions:The following is a table showing the conversions of percentages and decimals into fractions:

Percentage Decimal Fraction

10% 0.1 1/1012.5% 0.125 1/816.66% 0.1666 1/620% 0.2 1/525% 0.25 1/430% 0.3 3/1033.33% 0.3333 1/340% 0.4 2/550% 0.5 1/260% 0.6 3/562.5% 0.625 5/866.66% 0.6666 2/3

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70% 0.7 7/1075% 0.75 3/480% 0.8 4/583.33% 0.8333 5/690% 0.9 9/10100% 1.0 1

Similarly we can go for converting decimals more than 1 from the knowledge of the above cited conversions as folWe know that 12.5% = 0.125 = 1/8Then, 1.125 = [8(1)+1]/8 = 9/8 (i.e., the denominator will add to numerator once, denominator remaining the sameAlso, 2.125 = [8(2)+1]/8 = 17/8 (here the denominator is added to numerator twice)3.125 = [8(3)+1]/8 = 25/8 and so on.Thus we can derive the fractions for decimals more than 1 by using those les than 1.We will see how use of fractions will reduce the time for calculations:Eg: What is 62.5% of 320?Soln: Value = (5/8) X 320 (since 62.5% = 5/8)

= 200.Part-1

Exercise1. What percentage is 150 ml of 3 litres?

1. 2% 2. 5% 3. 8% 4. 50%

2. If 40% of a number added to 1800, it gives the number itself. Find the number?1. 3000 2. 257 3. 1500 4. 4500

3. The difference between 12 % and 37% of a number is 750. Find the number?1. 1500 2. 4500 3. 3000 4. 257

4. A number exceeds its 75% by 50. Then, find the number?1. 175 2. 500 3. 750 4. 200

5. If 30% of a number added to 441, it gives the number itself. Find the number?1. 350 2. 630 3. 580 4. 250

6. The price of an article X is 10% less than that of article Y. By how much percent of the price article Y is morthat of the price of X?

1. 9111 % 2. 11

91 % 3. 90% 4. 10%

7. In a library 50% of the books in Hindi, 40% of the remaining is English, and 30% of the remaining is Teluguremaining 18,900 books are Marathi. Find the total number of books?1. 3,15,000 2. 45,000 3. 63,000 4. 90,000

8. The cost of 1 kg of sweets increases every year by 10%, After two years the cost of 1 kg of sweets will incre1. 10% 2. 20% 3. 21% 4. 25%

9. The marks obtained by two candidates A and B in arithmetic are 6.25 and 5.75 respectively. Marks percentagexceeds that of B by1. 7.8% 2. 8% 3. 8.7% 4. 8.8

10. Gopal spends 30% of his monthly income on food articles, 40% of the remaining on conveyance and clothessaves 50% of the remaining. If his monthly salary is Rs.18,400, how much money does he save every month?1. Rs.3864 2. Rs.4264 3. Rs.3624 4. Rs.5888

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11. A man spends 25% of his income on house rent, 45% of this income on food and 40% of the balance on convIf he is left with Rs.540 his income is1. Rs.4500 2. Rs.3000 3. Rs.1350 4. Rs.900

12. In an examination 450 candidates were boys and 550 were girls. If 32% of the boys and 38% of the girls pasexamination, the percentage of failed candidates is1. 35.3% 2. 62% 3. 64.7% 4. 68%

13. After having spent 25% of his money on machinery, 30% on raw material and 10% on staff, a person is left wRs.70,000. The amount spent on raw material is1. Rs.64000 2. Rs.60000 3. Rs.48000 4. Rs.40000

14. The price of kerosene increases by 25%. By how much percentage a family must reduce consumption of keras not to increase the monthly expenditure on kerosene?1. 15% 2. 20% 3. 25% 4. 30%

15. Fifty percent students in a class are N.C.C cadets and 40 percent of the rest are scouts. Find the percentage ototal who are neither N.C.C. nor Scouts.1. 18% 2. 15% 3. 25% 4. 30%

16. A monthly return railway ticket costs 25 percent more than a single ticket. A weeks extension can be had foformer by paying 5 percent of the monthly tickets cost. If the money paid for the monthly ticket (with extenRs.84, the price of the single ticket is1. Rs.48 2. Rs.80 3. Rs.64 4. Rs.72

17. Mahesh gets a certain sum as pocket money every month. He spends 20 percent on purchasing books and 25 of the remainder on stationary, 10 percent of what is left is given as charity. The rest he uses for purchasing setc, and this works out to be Rs.108. His monthly pocket money is1. Rs.200 2. Rs.250 3. Rs.350 4. Rs.500

18. The population of a city increases at the rate of 10% annually. Its present population is 90.51 lakhs. The pop

years ago was nearly ________ 1. 60 lakhs 2. 68 lakhs 3. 71 lakhs 4. 72.8 lakhs

19. 8% of the people eligible to vote are between 18 and 21. In an election, 85% of those eligible to vote, who w between 18 and 21, actually voted. In that election, the number of persons between 18 and 21, who actually vwas what percent of those eligible to vote?1. 4.2 2. 6.4 3. 6.8 4. 8

20. When the price of wheat is increased by 20%, a family has to reduce the consumption by 4 kg keeping theexpenditure fixed at Rs.80. The increased price per kg is1. Rs.20 3. Rs.5 3. Rs.4 4. Rs.10

21. In an examination every candidate took History or Geography of both, 64.8% took History and 60.2% of can

took Geography. If the total number of candidates is 2000, the candidates who took both are1. 500 2. 400 3. 375 4. None of these

22. In a school of 400 students, 40% of girls, 20% of the boys and 30% of the students failed in the exam. The %students passed in the exam is:1. 75% 2. 76% 3. 45% 4. None

23. In an election, the winning candidate received 10400 votes, which represented 52% of the electorate. The oncandidate secured votes from 23% of the electorate, calculate the number of people who did not cast their vot1. 3500 2. 4000 3. 4800 4. 5000

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24. In a competitive examination, a candidate obtained 45% marks and failed to get selected by 120 marks. Anocandidate obtained 58% marks and failed to get selected by 16 marks. Find the maximum marks and the percof marks for selection?1. 600, 60% 2. 750, 55% 3. 800, 65% 4. 800, 60%

25. Three persons A, B and C whose salaries together amount to Rs.2560, spend 90, 85 and 80 percent in their sarespectively. If their savings are 5:6:14, find their respective salary?1. 800, 640, 1120 2. 840, 600, 1120 3. 780, 760, 1020 4. 800, 760, 1000

26. In an examination a candidate attempts 90% of the total questions at of these 80% his answers are correct. Equestion carries 2 marks for correct answer and negative (-1) mark for wrong answer. If the marks secured bycandidate is 189 what is the total number of questions?]1. 250 2. 150 3. 100 4. 350

27. The sales of a company, reduced to 20%. After how much percentage increase the sales of the company be inoriginal?1. 20% 2. 25% 3. 10% 4. 30%

28. In a town 60% of people read Hindu, 50% of read the Times of India, 30% read both. Find the percentage of people who read neither?1. 80% 2. 20% 3. 10% 4. 40%

29. The population of the town 20,000. If the number of males are increased by 40% then at the end of the year t population will be 23,200. Find the number of females, males in the town at the beginning?1. 12000, 8000 2. 18000, 12000 3. 10000, 10000 4. None

30. In an election where there are two candidates only contested. One candidate who gets 70% of the votes is elemajority of 2000 votes. What is total number of votes?1. 5000 2. 6000 3. 3000 4. None

31. A candidate has to obtain 33% of the total marks to pass the examination. Ram got 25% of the total marks an

by 40 marks. What are the maximum marks?1. 400 2. 750 3. 500 4. 5000

32. In an examination A gets 28% of the total marks and fails by 24 marks. While B gets 31% of the total marks48 marks more than the pass mark. Find the maximum marks and pass %?1. 3000, 29% 2. 2400, 29% 3. 2500, 29% 4. 3000, 30%

33. In an examination there were 5000 candidates out of which 3000 were boys. If 60% of the boys and 20% of passed. Find the percentage of failed candidates?1. 60% 2. 76% 3. 56% 4. 62%

34. If the price of sugar decrease by 25%. Find by how much percent must a householder increase his consumptithat article so as not to reduce his expenditure?

1. 3313 % 2. 1119 % 3. 3100 % 4. None35. If the price of wheat decreased by 20%, find by how much percent must a householder increase his consump

that article so as not to reduce his expenditure?

1. 1123

% 2. 1323

% 3. 1523

% 4. 1623

%

36. The population of a town is 1,00,000 now. If it increases at a rate of 10% p.a. find the population after 2 yea1. 112000 2. 121000 3. 120100 4. None

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37. If the population of a town is 5000 now and it increases at 10% first year 20% in the second year then its popafter 2 years?1. 6600 2. 5500 3. 6000 4. 5000

38. Two numbers are 20% and 40% of the third number. How much % of the first number in the second number1. 20% 2. 200% 3. 10% 4. 100%

39. A reduction of 20% in the price of oranges would enable a person to get 2 dozen more for Rs.50. Find the re price per dozen.1. Rs.3 2. Rs.5 3. Rs.4 4. Rs.4.5

40. The tax on a commodity is diminished by 15% and its consumption increased by 10%. What is the effect on revenue derived from it?1. 5.5% increase 2. 5.5% decrease 3. 6.5% decrease 4. 6.5% increase

Answers

1) 2 11) 2 21) 1 31) 32) 1 12) 3 22) 4 32) 23) 3 13) 2 23) 4 33) 34) 4 14) 2 24) 4 34) 15) 2 15) 4 25) 1 35) 46) 2 16) 3 26) 2 36) 27) 4 17) 1 27) 2 37) 18) 3 18) 2 28) 2 38) 29) 3 19) 3 29) 1 39) 2

10) 1 20) 3 30) 1 40) 3

Exercise:1. If 20% of 40% of a = 25% of a% of b, then what is b?

a. 8/5 b. 16/25 c. 8/25 d. None

2. By what % is 200 more than 50?a. 100 b. 200 c. 300 d. None

3. A value changes from 30 to 80. What is the percentage change?a. 125 b. 166.66 c. 156 d. None

4. The population of a city is increased by 30% and thus became 78000. What is the original population?a. 76000 b. 64200 c. 60000 d. None

5. In a theatre, the number of seats is increased by 20% and the price per ticket is increased by 10% but the public decreased by 30%. What is the net effect on the economy of the theatre?

a.10% rise b. 7% fall c. 7% rise d. None

6. A saves 20% of his income. His income is increased by 20% and so he increased his expenditure by 30%. What percentage change in his savings?

a. 20% fall b. 4% fall c. 20% rise d. 4% rise

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22. If 240 lt of oil is poured into a tank, it is still 20% empty. How much more oil is to be poured to fill the tank?a. 300 lt b. 60 lt c. 120 lt d. None

23. A and B were hired for the same salary. A got two 40% hikes whereas B got a 90% hike. What is the percentagdifference in the hikes they got?

a. 16% b. 6% c. 10% d. 8%

24. The population of a town doubled every 5 years from 1960 to 1975. What is the percentage increase in populatthis period?

a. 800 b. 400 c. 700 d. 600

25. In a test of 80 questions, Jyothsna answered 75% of the first 60 questions correctly. What % of the remaining qshe has to answer correctly so that she can secure an overall percentage of 80 in the test?

a. 80% b. 90% c. 85% D. 95%

Answers

1. 1/5 X 2/5 X a = X a X b => b = 8/252. % difference = (200-50)/50 X 100 = 300 %3. % increase = (80-30)/30 X 100 = 166.66 %4. 1.3 x = 78000 => x = 60000.5. Net effect = 1.2 X 1.1 X 0.7

= 0.924 => 7.6% decrease.6. Let I be the income.

Expenditure = 0.8I Savings = 0.2I => 20% New income = 1.2I (since 20% rise) New expenditure = (0.8I) X 1.3 (Since 30% rise)

= 1.04ISo, new savings = 1.2I 1.04I = 0.16I => 16%(So income decreased form 20% to 16%)% decrease = (20-16)/20 X 100 = 20%.

7. It is equivalent to 1.25 decreased to 1.% decrease = (1.25-1)/1.25 X 100 = 20%

8. % change in area = 1.2 X 1.2 (since area = side X side)= 1.44 => 44%.

9. It is equivalent to 1.25 decreased to 1. So 20% decrease.10. Valid Votes:

A got 56% => B got 44%Difference = 12% = 48000So, 100% = 400000. These are valid votes.But valid votes are only 80% of total votes.So, 80% of total votes = 400000 => total votes = 500000

11. Total money = Rs. 300.Saving of the lady = 10% of 300 = 30/-With 30/- she bought 5 kg sugar => each kg costs Rs. 6/-12. In 20lt, salt = 20% => 4 lt.

New volume = 18 lt (2 lt evaporated)So, new % = 4/18 X 100 = 22.22%

13. % correction = (58-46.4)/58 X 100 = 20%14. Three successive decreases of 20%, 20% and 10% => 0.8 X 0.8 X 0.9 = 0.576

Again 10% decrease => 0.576 0.1 = 0.476.So, 0.476 x = 9520 => x = 20000.

15. Total discount = 0.9 X 0.8 X 0.7 = 0.504 of actual price.So, price = 0.504 X 10000 = 5040.

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16. In 10 lt, alcohol is 10% = 1 lt.Let x lt alcohol is added.So, (1+x)/(10+x) = 20% = 1/5 => x = 1.25 lt.

17. A = 2B and B = 3C (ince 200% more)A = 6C => 500 % more.

18. 50% of max marks 40% of max marks = 25 max marks = 250

Pass marks = 40% of max + 10 => 100 + 10 = 110.19. A = 1.2 B => B = A/1.2 => 0.8333A => 16.66%.

(OR) Decrease from 1.2 to 1 => 16.66%.20. 1.1 X 1.1 X x = 12100 => x = 10000.21. Salt = 15% of x = 0.15x (x = volume of solution)

Now, 0.15x/(x-30) = 20% = 1/5 (since 30 lt evaporated)x = 120 lt

22. 20% empty => 80 % full = 240 lt => 20% = 60 lt23. A => 1.4 X 1.4 = 1.96

B => 1.9 => 6% difference.24. From 1960 to 1975, in 15 years population doubled every 5 yrs => three times

So, 2 X 2 X 2 = 8 times => 700% more.25. [(75% X 60) + (x% X 20)] / 80 = 80% => x = 95. (since required is 80%)

(OR) 60 out of 80 is 3/4. So, (3/4 X 75) + (1/4 X x) = 80 => x =95.

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Profit and Loss

Introduction:

1. In any business transaction, it is common to have either profit or loss.2. But the aim of any business is to earn profit.3. The most commonly used term involving sale and purchase of any business are Cost Price and Selling Pric

What is Profit?When a person does a business transaction and gets more than what he had invested, then he is said to hav

The profit he gets will be equal to the additional money he gets other than his investment.So profit can be understood as the extra money one gets other than what he had invested.

Eg: A person bought an article for Rs. 100 and sold it for Rs. 120. Then he got Rs. 20 extra and so his profit is

What is Loss?When a person gets an amount less than what he had invested, then he is said to have a loss. The loss will b

to the deficit he got than the investment.Eg: A person bought an article at Rs. 100 and sold it for Rs. 90. Then he got a deficit of Rs. 10 and so his loss is

Cost Price (CP):The money that the trader puts in his business is called Cost Price. The price at which the articles are b

called Cost Price.In other words, Cost Price is nothing but the investment in the business.

Selling Price (SP):The price at which the articles are sold is called the Selling Price. The money that the trader gets from the

is called Selling Price.In other words, Selling Price is nothing but the returns from a business.

Marked/Market/List Price (MP):

The price that a trader marks or lists his articles to is called the Marked Price.This is the only price known to the customer.

Discount:The waiver of cost from the Marked Price that the trader allows a customer is called Discount.

Note:1. Profit or loss percentage is to be applied always to the Cost Price only.2. Discount percentage is to be applied always to the Marked Price only.

Relationship Among CP, SP and MP:A trader adds his profit to the investment and sells it at that increased price.Also he allows a discount on Marked Price and sells at the discounted price.So, we can say that,

o SP = CP + Profit. (CP applied with profit is SP)o SP = MP Discount. (MP applied with discount is SP)

Understanding Profit and Loss:So, by now we came to know that if CP is increased and sold it would result in profit and vice versa.Also whatever increase is applied to CP, that increase itself is the profit.For Rs. 10 profit, CP is to be increased by RS. 10 and the increased price becomes SP.For 10% profit, CP is to be increased by 10% and it is the SP.(From previous chapter we know that any value increased by 10% becomes 1.1 times.)So, for 10% profit, CP increased by 10% => 1.1CP = SP.

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o SP = 1.1CP => SP/CP = 1.1 => 10% profito SP = 1.07CP => SP/CP = 1.07 => 7% profito SP = 1.545CP => SP/CP = 1.545 => 54.5% profit and so on.

Similarly,o SP = 0.9CP => SP/CP = 0.9 => 10% loss (Since 10% decrease)o SP = 0.76CP => SP/CP = 0.76 => 24% loss and so on.

So, to calculate profit % or loss %, it is enough for us to find the ratio of SP to CP.Note:

1. If SP/CP > 1, it indicates profit.2. If SP/CP < 1, it indicates loss.

Multiple Profits or losses:A trader may sometimes have multiple profits or losses simultaneously. This is equivalent to having m

changes and so all individual changes are to be multiplied to get the overall effect.

Formulae:

1. Gain % =C.P.

100Gain

2. Loss % =C.P.

100Loss

3. When S.P. and gain% are given, then

S.P.Gain%100

100 C.P.

4. When C.P. and gain% are given, then

C.P.100

Gain%100 S.P.

5. When S.P. and loss% are given, then

S.P.Loss%100100

C.P.

6. When C.P. and loss% are given, then

C.P.100

Loss%100 S.P.

7. If a man buys a items for Rs.b andsells c items for Rs.d, then the

Gain or loss % =ad bc 100bc

Note: a) In case of gain percent, the result will be positive. b) In case of loss percent, the result will be negative.

8. If the C.P. of x articles = S.P. of y articles, then

Gain or loss % = %100y

yx

Note: a) If x > y, it is % gain. b) If x < y, it is % loss.

9. The Cost Price of an article is C.P. If it is sold at S.P1., then gain% or loss% is x and if it is sold at a price S.P2.,gain% or loss% is y then

y100S.P

x100S.P 21 (Or)

yxS.PS.P

100C.P 21

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Note: a) If x or y is negative it indicates a loss. b) If x or y is positive it indicates a gain.

10. A sells

Arithmetic Final 2

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