INDEX
TOPIC NAME
Page No.
1. Basic Calculations
2-9
2. Number System
10-21
3. L.C.M. & H.C.F
22-29
4. Percentages
30-38
5. Average
39-48
6. Ratio and Proportion
49-59
7. Partnership
50-65
8. Mixtures (or) Alligations
66-71
9. Profit and Loss
72-81
10. Problems on Ages
82-86
11. Time and Work
87-94
12. Pipes and Cisterns
95-101
13. Time and Distance
102-111
14. Problems on Trains
112-118
15. Boats and Streams
119-124
16. Simple Interest
125-133
17. Compound Interest
134-142
18. Clocks
143-148
19. Calendars
149-153
20. Mensuration - 2D
154-164
21. Mensuration - 3D
165-170
Key to Assignments
171-173
1. Basic Calculations
VBODMAS
The order of various operations in exercises involving brackets and functions must be performed strictly according to the order of the letters of the word VBODMAS. Each letter of the word VBODMAS stands as follows:
V for Vinculum : - (bar)
B for Bracket : [{( )}] O for Of : of D for Division : ÷ M for Multiplication : x A 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: 8 5 4 1 1 3 1 1 3 1 5 of 2 1 4 5 1 3 2 1 4Sol: Given expression
= 8 5 4 5 3 1 1 3 1 6 of 2 9 5 1 6 2 9 = 8 5 3 1 1 3 1 6 of 2 9 5 1 6 2 9 = 8 1 9 1 1 3 1 6 of 2 9 5 1 6 2 9 = 8 6 9 3 1 6 of 2 9 5 1 6 2 9 =
8
69
3
16
2
9
5
16
2
9
= 8 6 9 2 4 1 5 1 6 2 9 = 1 2 0 1 0 3 5 1 6 2 9 = 1 2 0 1 0 5 1 2 9 = 1 2 0 1 0 5 1 5 4 0 = 1 2 0 5 1 1 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 the given number. It is denoted by the symbol .
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 when every prime factor of that number is repeated twice. Follow the ste ps 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 square root of 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 easily determined.
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 may be 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 th is stage (here 2) 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).
Square root of a Decimal: If the given number is having decimal, separate the digits of it
into periods of two to the 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 the numerator and denominator separately.
E.g. Find the square root of . 4 9 2 6 0 1 Sol: 4 9 2 6 0 1 = 4 9 2 6 0 1 = 7 7 5 1 5 1 = 7 5 1 = 7 7 2
Case 2: If the denominator is not a perfect square, the fraction is converted into decimal and
then square root is obtained or the denominator is made perfect square by multiplying and dividing a suitable number and then its square root can be determined.
E.g. Find the square root of . 8 4 6 1 Sol: 8 4 6 1 = 2 8 2 4 6 1 = 1 6 9 2 2 = 4 3 6 4 4 . 3 0 = 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 given number, that is the cube root of a number a is the number whose cube is a.
The cube root of a is written as 3
a
.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 gives 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 7
so, 3926137 = 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 1 2 2 8 8 3 3 7 27 4 4 4 64 5 5 5 125 6 6 6 216 7 7 3 343
8 8 2 512
9 9 9 729
10 10 0 1000
The method of finding the cube root of a number up to 6 digits which is actually a cube of some number consisting 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.
Brainstorming
1. 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 will both factories make in 4 hours?
1. 4x+4y 2. 4x+8y 3. 8x+4y 4. 4x_2y
3. Which of the following is the same as 50+12?
1. 10(5+3) 2. (605)+(1002) 3. 2512x2 4. 5043
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 x 32, 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 =
2 5
1 , then the value of x is ________
1. 2 1 2. -2 3. 2 4. -4 8. If 7 x 5xy 4
1 = 17, then the value of (x, y) is ________
1. (6, 2) 2. (7, 2) 3. (9, 2) 4. (8, 2) 9. 3 2 3 4 3 2 4 3 = ? 1. 4 9 2. 2 3 3. 1 8 1 1 4. 3 6 7
10. 3 7 1 of 111 3 1 of ? = 0 1. 1 2. 3 3. 9 4. 12
11. Which of the following fraction is greater than 2 1
but less than 2 3 ? 1. 3 9 1 9 2. 4 7 2 3 3. 5 7 2 9 4. 4 9 3 3 12. 2 ) 0 0 1 4 . 0 ( 2 ) 0 1 1 . 0 ( 2 ) 0 0 4 . 0 ( 2 ) 0 1 4 . 0 ( 2 ) 1 1 . 0 ( 2 ) 0 1 . 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. 10 14. 0 9 0 . 0 6 9 . 0 2 ) 3 . 2 ( 0 2 7 . 0 3 3 . 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. 2 5 1 4. 2 5 1 16. If a b
= 0.25 then what should the value of
9 2 b a 2 b a 2 ? 1. 1 2. 9 4 3. 9 5 4. 2
17. Which number is equal to
1 . 0 0 1 . 0 0 1 . 0 1 . 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)3
1. 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. 0 1 . 0 ? 1 9 3 4 2 0 x7 =? 1. 9 3 5 2. 5 6 3 3. 7 1 8 4. None of these
25. If 2276155 = 79.2, the value of 122.7615.5 is equal to
26. 2 . 0 6 . 3 ? 2 8 . 1 7 = 200 1. 120 2. 1.20 3. 12 4. 0.12 27. 120.09 of 0.3x2 = ? 1. 0.80 2. 8.0 3. 80 4. None of these 28. ? 2 0 5 . 0 8 2 0 12 1. 8 2. 18 3. 2 4. None of these 29. If 5 = 2.24, the value of 4 8 . 0 5 2 5 3 will be __________ 1. 0.168 2. 1.68 3. 16.8 4. 168 30. 0 0 1 2 1 . 0 2 8 9 . 0 = ? 1. 1 1 1 7 0 2. 1 1 0 1 7 3. 1 1 1 7 . 0 4. 1 1 1 7 31. 5555+6666-9999-1111 = ? 1. 1001 2. 1011 3. 1111 4. 1221
32. Which is greater among 1 9 1 5 , 9 5 , 9 8 and 8 7 , 1. 9 5 2. 1 9 1 5 3. 8 7 4. 9 8 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. 4 4 1 8 7 2 1 3 7 1 3 2 1 = ? 1. 5 2 8 1 1 2. 5 2 8 1 3 3. 6 2 8 1 1 4. 6 2 8 1 5
36. What approximate value should come in place of the question mark (?) in the following equation?
66 6
2% 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+? = 802.293
1. 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 the other four? The number of that part is the answer.
1. 40% of 160+ 3 1 of 240 2. 120% of 1200 3. 38x12-39x8 4. 6 2 1 of 140-2.5x306.4 40. In the following equation what value would come in place of question mark(?)?
5798-? = 7385-4632
1. 3225 2. 2595 3. 2775 4. 3045
41. What should come in place of question mark (?) in the following equation? 197x?+162 = 2620
1. 22 2. 12 3. 14 4. 16
42. Which of the following numbers are completely divisible by seven?
A. 195195 B. 181181 C. 120120 D. 891891
1. 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
1 7 1 0 1 2 5 2 0 9 2 5 2 1 = ? 1. 1 2 5 7 7 2. 11 1 0 9 3. 4 5 0 1 1 9 4. 1 9 0 2 9
44. What should come in the place of the question mark (?) in the following equation? 1 1 2 ? ? 2 8 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 ? = 320
1. 16 2. 2 3. 4 4. 32
46. What should come in the place of the question mark (?) in the following equation? 8 1 4 9 2 ?) 7 ( 1. 9 2. 2 3. 3 4. 4
47. What should come in the place of the question mark (?) in the following equation?
2 2 2 1 3 5 2 7 4 5 . 1. 81 2. 1 3. 243 4. 9 48. Simplify: 18104 +32(4+102-1) = ___________ 1. 5 2. 9 3. 8 4. 7 49. Solve: 4-
6
12
543
. 1. 5 2. 4 3. 6 4. 850. If x=4, y=3, then the value of x
yx2
is _________ 1. 4 7 2. 1 2 1 3. 5 4 4. 4 52. NUMBER SYSTEM
Natural Numbers (N): Counting numbers 1, 2, 3, . . . are called Natural numbers. They are also
called Positive Integers. 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 of Integers.
I or Z = {. . . ., -3,-2,-1, 0, 1, 2, 3 . . . .}
Rational Numbers (Q): Numbers which are in the form of
q p
, where p, q are integers and q ≠ 0, are
called Rational numbers. Q = { q p , (q
0)/ p, q
Z} E.g. -3, 1, 3.2, 1 3, 22 7 , etc. Note:1. Rational numbers are divided into two groups, namely integers and non-integers. 2. Non-integer belonging to the set of rational numbers is called fraction.
Fraction: A number expressed in the form of p
qis also called fraction, where „p‟ is the numerator and „q‟ is the denominator. Fraction is a part of an integer.
E.g. , 5 6 , 7 2 , 6 1 etc.
Proper Fraction: Fractions in which Numerator < Denominator are called Proper Fractions. E.g. , 5 1 , 7 3 , 9 7 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) 1) 4 16) 1 31) 3 46) 3 2) 2 17) 3 32) 4 47) 1 3) 2 18) 2 33) 2 48) 4 4) 3 19) 2 34) 4 49) 3 5) 2 20) 2 35) 3 50) 3 6) 1 21) 2 36) 1 7) 2 22) 2 37) 2 8) 3 23) 1 38) 3 9) 2 24) 4 39) 2 10) 2 25) 2 40) 4 11) 4 26) 4 41) 2 12) 1 27) 4 42) 4 13) 3 28) 2 43) 4 14) 4 29) 2 44) 2 15) 2 30) 1 45) 1
3/4 = 0.75 (terminating)
Irrational Numbers (Q’): A number which cannot be expressed in the form of rational number is
called an Irrational 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 numbers.
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
numbers.
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 Numeral: The value of where the digit is in the number, such as units,
tens, hundreds, 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 = 4 Place Value of 5 = 5 tens = 50, Face Value of 5 = 5 Place Value of 6 = 6 hundreds = 600, Face Value of 6 = 6 Place Value of 2 = 2 thousands = 2000, Face Value of 2 = 2 Place 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 be a Perfect number.
E.g. Factors of 6 = 1, 2, 3, 6 and 1 + 2 + 3 + 6 = 12
28, 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 zeros to the right of the multiplicant as there are nines in the multiplier and from the result subtract the multiplicant and get 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: Place n 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 multiplicant by this new number and finally divide the product by 2.
4. For multiplication of a given number by 5, 25, 125, 625, etc., that is, by a number which is some power of 5. Place as many zeros to the right of the multiplicant as is the power of 5 in the multiplier, then divide the number so obtained by 2 raised to the same power as is the power of 5.
E.g. 2982 x 5 = 29820/2 = 14910
5739 x 25 = 573900/22 = 143475
a) No. of factors of a given number: If
N
a
p
b
q
c
r...
then the number of factors of N = (p + 1) (q + 1) (r + 1)…………., where a, b, c are prime factors of N and p, q, r,………. are positive integers.E.g. Find the number of factors of 24. Sol:24 = 3 1
3
2
The number of factors of 24 = (3 + 1) (1 + 1) = 8.
b) Sum of the factors of a given number: If
N
a
p
b
q
c
r...
then the sum of the factors of N = ... ... 1 c 1 c 1 b 1 b 1 a 1 ap 1 q 1 r 1 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 = 2331
Sum of the factors of 24 = 6 0. 1 3 1 3 1 2 1 23 1 1 1 c) No. of ways of expressing a given number as a product of two factors:
If
N
a
p
b
q
c
r...
where a, b, c are prime factors of N and p, q, r,………. are positive integers then the number of ways in which N can be expressed as product of two factors =
(p 1)(q 1)(r 1)...
2 1
.
E.g. Find the number of ways of expressing 48 as a product of two factors. Sol:48 = 4 1
3
2
No. of ways =
(4 1)(1 1)
5 2 1 ) 1 q )( 1 p ( 2 1 .d) No. of ways of expressing a given number which is a perfect square as a product of two factors:
If Napbqcr... where a, b, c are prime factors of N and p, q, r,………. are positive integers then the number of ways in which N can be expressed as product of two factors =
(p 1)(q 1)(r 1)... 1
21 .
E.g. Find the no. of ways of expressing 36 as a product of two factors. Sol:36 = 2 2
3
2
No. of ways =
(2 1)(2 1) 1
5 2 1 1 ) 1 q )( 1 p ( 2 1 . TIPS ON SQUARES:Condition Method Example
To square any number ending with 5. (a5)2 = {a(a+1)}25 (35)2 = {3(3+1)}25 = 1225 To square a number in which every digit is one.
Count the number of digits in the given number and start writing numbers in ascending order from one to this number and then in descending order up to one.
(11)2 = 121,
(111)2=12321
number which is nearer to 10 x. 2 2 2 2 2 (x y ) y (x y)(x y) y x (1004 – 4) (1004 + 4) + (4)2 = 1000(1008) + 16 = 1008016 DIVISION:
Dividend = (Divisor x Quotient) + Remainder
E.g.
TESTS OF DIVISIBILITY Divisibility
by… Rule Example Explanation
2 Unit‟s digit of the number should be zero or divisible by 2. 4, 2, 102, etc.
3 Sum of the digits in the number should be divisible by 3. 1782
1+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. 300, etc. 4784,
4784 Since 84 is divisible by 4, 4784 is also divisible by
4.
5
Unit‟s 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 doubled and then it is subtracted
from the number obtained after omitting the unit digit. If the result
is divisible by 7, then the given number is also divisible by 7.
448 448 44 – 8(2) = 44 – 16 = 28 which is divisible by 7 and hence 448 is also divisible by
7.
8
Number formed by the last three digits should be divisible by 8. or
zero‟s 1576
1576 576 is divisible by 8 and hence 1576 is also
divisible by 8.
9
Sum of the digits in the number
should be divisible by 9. 1395
1395 1+3+9+5 = 18 is divisible by 9 and hence 1395 is also divisible by 9.
10 Number should end in zero. 1000
11
Sum 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 digits at 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 hence 156 is also divisible by
12. 14 Should satisfy divisibility rules of 2 and 7. 322
322 is divisible by 2 and 7 hence 322 is also divisible by
25
Last two digits in the number should be 0 or divisible by 25.
175
175 75 is divisible by 25 and hence 175 is also
divisible by 25.
125 should be 0 or divisible by 125. Last three digits in the number
2250
2250 250 is divisible by 125 and hence 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 that a2 > given number.
2. Test the divisibility of given number by every prime number that is less than a. 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) n 4)
3
(...
= (… 1) n 4)
7
(...
= (… 1) n 2)
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, multiply the number by itself until you gets 6 in the unit place.
4n
)
2
(...
= (… 6) 2n)
4
(...
= (… 6) n)
6
(...
= (… 6) 4n)
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 given number, 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 Examples
1. 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 = Q uotient Remainder -Dividend = 6 4 0 1 0 1 6 4 6 4 0 1 0 1 1 2 6 4 6 5 2
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 > 577
n = 25Prime 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 + 6
The 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 + 20
The 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 (729)59.
Sol:(729)59 (729)58729(...1)7299 in the unit place.
10. Find the number in the unit place in (9 8)42.
Sol:(98)42 (98)410(98)2 (...6)(...4)4in the unit place.
11. Find the number in the unit place in (636)36.
Sol:
(
636
)
36
(...
6
)
36
6
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 by 1
10 x = 4.4444…….(2)
Subtract Equation 1 from 2 on both sides 10 x = 4.4444……. - x = 0.4444……. --- 9 x = 4.0000……. --- . 9 4 x 4 x 9
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 by 2
10
. 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 =99
557
Brainstorming
1. O .1+8÷9x11(3x3)+6x714+73x3 = ________ 1. 9 8 9 2. 1 3. 9 4 5 4. 9 1 0 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 is
1. 4 2. 5 3. 6 4. 7
4. The number 111, 111, 111, 111 is divisible by
1. 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 divisible is
1. 16 2. 24 3. 48 4. 96
6. The number of divisors of 120 individuality is
1. 8 2. 12 3. 14 4. 16
7. The greatest number which exactly divides 826 is
1. 2 2. 7 3. 59 4. 413
8. The least number which is a perfect square as well as a factor of 936 is
1. 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 __________
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. 115 885 115 115 885 885 115 115 115 885 885 885 = 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 numbers are
1. 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 is
1. 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 third be 5:3, then the second number is _____
1. 30 2. 45 3. 27 4. 48 18. The value of 1+ 9 1 1 1 1 1 = _________ 1. 1 9 2 9 2. 1 9 1 0 3. 1 0 2 9 4. 9 1 0 19. The value of 9+9x9-99 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 is
1. 476727 2. 376538 3. 376728 4. 359738 21. 0.77777……… is equal to 1. 8 3 7 7 2. 7 7 9 9 3. 9 9 7 7 4. 1 0 9 7 7 22. 199+299+399+………..9999 is divisible by 1. 99 2. 98 3. 100 4. 101 23. If the fraction 6 0 6 3 , 1 5 3 1 3 6 , 1 5 6 1 4 3 and 1 6 1 7
are arranged in ascending order, then which is the third number? 1. 1 5 6 1 4 3 2. 1 5 3 1 3 6 3. 1 6 1 7 4. 6 0 6 3
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) is
1. 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 correct one by 324. The number to be multiplied was
1. 12 2. 15 3. 25 4. 52 27. 9 9 9 9 9 7 2 ... ... 7 5 2 5 3 2 3 1 2 is equal to
1. 9 9 9 5 2. 9 9 9 1 0 0 1 3. 3 1 0 0 1 4. 1 0 0 1 3
28. A number consists of two digits whose sum is 15. If 9 is added to the number, then the digits change their places. The number is
1. 69 2. 78 3. 87 4. 96
29. One of the two consecutive positive integers, the sum of whose squares is 761, is
1. 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 the denominator is decreased by 7, we obtain 2. The rational number is 1. 4 1 2. 1 8 1 5 3. 1 0 7 4. 1 1 8
31. Find the value of nn 1 n n1 3 3 3 3 is 1. 2 1 2. 3 4 3. 3 2 4. 4 3
32. A number N when divided by 120 gives a remainder 76. What is the remainder obtained when the same number is divided by 8?
1. 2 2. 3 3. 4 4. 5
33. What is the value of 1+
7 1 1 1 1 1 ? 1. 1 3 1 9 2. 1 9 1 3 3. 1 3 1 8 4. 6 7 34. The sum of two consecutive prime numbers is 30. The two numbers are
1. 18, 12 2. 7, 23 3. 11, 19 4. 13, 17
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 number is
1. 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 units place is 3 more than the digit in the tenth place, what is the number?
1. 24 2. 63 3. 36 4. Can‟t be determined
38. The difference between a two digit number and the number obtained by interchanging the digits is 27. What is the sum of the two digits of the number?
1. 3 2. 6 3. 9 4. Can‟t 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 is
1. 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?
43. Two different numbers when divided by 47, leave remainders 13 and 23 respectively. If their sum is divided by the 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. 3
45. 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 travels 3
1 of a distance by car, 3
1 of the remaining by bus and 3
1 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. 6 0 4 8 0 1 2. 8 6 4 0 0 1 3. 6 0 4 8 0 0 1 4. 604800 50. 5 4
of a number exceeds its 2/3 by 8. The number is
1. 30 2. 60 3. 75 4. 90
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 1
3. 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 is 1. The 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 third and 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 and 52. The 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 56, the 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 _______
60. 24 is divided into two parts such that 7 times the first part added to 5 times the second part makes 146. The first part is _________
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 numbers is ______
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 subtracted 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 and 72. The 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 the first, then the second number is _______
1. 32 2. 36 3. 48 4. 60
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 them becomes 5 : 7. The numbers are _________
1. 3, 5 2. 7, 9 3. 13, 22 4. 15, 25 1) 4 16) 2 31) 3 46) 2 61) 1 2) 4 17) 2 32) 3 47) 2 62) 2 3) 2 18) 1 33) 1 48) 2 63) 4 4) 4 19) 2 34) 4 49) 3 64) 3 5) 3 20) 3 35) 2 50) 2 65) 2 6) 4 21) 3 36) 2 51) 2 66) 4 7) 3 22) 3 37) 3 52) 1 67) 2 8) 3 23) 1 38) 4 53) 2 68) 4 9) 1 24) 3 39) 3 54) 1 69) 1 10) 3 25) 2 40) 2 55) 3 70) 4 11) 3 26) 1 41) 4 56) 2 12) 3 27) 3 42) 1 57) 2 13) 1 28) 2 43) 2 58) 3 14) 4 29) 3 44) 3 59) 3 15) 1 30) 2 45) 1 60) 2
3. 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 each one of them.
E.g. 32 is a common factor of 8 and 16
Least Common Multiple (L.C.M): The least multiple among all the common multiples of given
numbers is called Least Common Multiple.
Methods of finding L.C.M. 1. Method of Prime Factors
a. Resolve each given number into prime factors.
b. Take out all the factors with highest powers that 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 3
14 = 2 x 7 20 = 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 of them 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 the greatest number that divides each one of them exactly. It is also called Greatest Common Divisor or 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 5
70 = 2 x 5 x 7 Common factors are 2 and 5.
So, H.C.F. = 2 x 5 = 10
2. Method of Division
E.g. 1. Find the H.C.F. of 3332, 3724. Sol:
So, the H.C.F. of 3332, 3724 is 196.
E.g. 2. Find the H.C.F. of 10, 15 and 23. Step 1: First find the H.C.F. of 10 and 15. It is 5
Step 2: Then find the H.C.F. of this 5 and 23. It is 1. So, the H.C.F. of 10, 15 and 23 is 1.
Note:
1. L.C.M. and H.C.F. of fractions
2. Product of two numbers = L.C.M. of two numbers x H.C.F. of two numbers.
3. To find the greatest number that will exactly divide a, b and c, simply find the H.C.F. of a, b and c.
4. To find the greatest number that will divide a, b and c leaving remainders x, y and z respectively, find the H.C.F. of (a – x), (b – y) and (c – z).
5. To find the least number which is exactly divisible by a, b and c, simply find the L.C.M. of a, b and c.
6. To find the least number when divided by a, b and c leaving remainders x, y and z respectively, find the (L.C.M. of a, b and c) – k, where k = (a – x) = (b – y) = (c – z). 7. To find the least number which when divided by a, b and c leaves the same remainder r
in each case, find (L.C.M. of a, b and c) + r.
8. Two numbers when divided by a certain divisor give remainders
r
1 andr
2. When their sum is divided by the same divisor, the remainder isr
3. Then the divisor is given by3 2
1 r r
r .
9. A number on being divided by d and 1 d successively leaves the remainders 2 r and 1 r , 2 respectively. If the number is divided by d x1 d , then the remainder = (2 d x1 r +2 r ). 1
10. To find the greatest number that will divide x, y and z leaving the same remainder r in each case:
Case 1: When the value of remainder r is given
Required number = H.C.F. of (x – r), (y – r) and (z – r).
Case 2: When the value of remainder is not given
Required number = H.C.F. of |(x – y)|, |(y – z)| and |(z – x)|. 11. To find the n-digit greatest number which when divided by x, y and z: a. Leaves no remainder i.e. exactly divisible
Step 1: Find the L.C.M. of x, y and z. Let it be L.
Step 2: Divide the n -digit greatest number by this L. Let the remainder be R. Step 3: Required number = (n-digit greatest number – R).
b. Leaves remainder k in each case:
Required number = (n-digit greatest number – R) + k.
12. To find the n-digit smallest number which when divided by x, y and z: a) Leaves no remainder i.e. exactly divisible
Step 1: Find the L.C.M. of x, y and z. Let it be L.
Step 2: Divide the n -digit smallest number by this L. Le the remainder be R. Step 3: Required number = n-digit smallest number + (L – R).
b) Leaves remainder k in each case:
Required number = n-digit smallest number + (L – R) + k.
Solved Examples
1. Find the greatest number that will exactly divide 200 and 310.
Sol: The required number = H.C.F. of 200 and 310 = 10.
2. Find the greatest number that will divide 148, 246 and 623 leaving remainders 4, 6 and 11 respectively.
Sol: The required number = H.C.F. of (148 – 4), (246 – 6), and (623 – 11)
= H.C.F. of 144, 240, 612 = 12.
3. Find the smallest number that is exactly divisible by 45, 63 and 50.
Sol: Required number = L.C.M. of 45, 63 and 50 = 3150.
4. Find the least number which when divided by 36, 48 and 64 leaves the remainders 25, 37 and 53 respectively.
Sol: (36 – 25) = (48 – 37) = (64 – 53. = 11
Required number = (L.C.M. of 36, 48 and 64) – 11 = 576 – 11 = 565
5. Find the least number which when divided by 12, 16 and 18, will leave the remainders 7 in each case.
Sol: Required number = (L.C.M. of 12, 16 and 18) + 7
= 144 + 7 = 151
6. Find the greatest number which will divide 772 and 2778 so as to leave the remainder 5 in each case.
Sol: Required number = H.C.F. of (772 – 5) and (2778 – 5)
= H.C.F. of 767 and 2773 = 59.
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 in each case.
Sol: L.C.M. of 12, 18, 21 and 28 = 252.
The remainder when 9999 is divided by 252 = 171 So, 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 = 339 So, 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 = 1000
The 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 = 252
The least number of 4 digits = 1000
The 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 sum is divided 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. Find the remainder 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, respectively.
Sol: Here, 8 – 3 = 10 – 5 = 15 – 10 = 5
L.C.M. of (8, 10, 15) = 120
The required least number = 120 – 5 = 115.Brainstorming
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 1 5 8 and 1 5 2 , 1 0 3 , 5 4 . 1. 5 3 4 2. 5 4 3 3. 5 4 4 4. 5 3 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.
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 2 5 1 2 and 2 5 6 , 2 0 9 , 5 3 . 1. 1 0 3 2. 1 0 0 3 3. 1 0 0 0 3 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. 35 and 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 20. Find the 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
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 67 cm exactly.
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.
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 30
1. 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. 1002
32. 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 divisible by 11.
1. 360 2. 361 3. 363 4. 383
34. Find the least number which when divided by 15, 20, 24 and 30 leaves 1 as remainder but when divided by 19 leaves no 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 interval of time 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. If they 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
circumference downwards. 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 km/h, 4 km/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 broad.
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 broad.
1. 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 the least number 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. 4
43. 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 same number 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, 54 and 67 respectively.
1. 10896 2. 10196 3. 10496 4. 10696
50. When finding the H.C.F. of two numbers by division method, the quotients obtained are 1, 2 and 5 respectively and the last divisor is 34. Find the numbers.
1. 374 and 544 2. 324 and 646 3. 314 and 624 4. 616 and 348
51. When finding the H.C.F. of two numbers by division method, the quotients obtained are 2, 2, 4 and 3 respectively and their H.C.F. is 23. Find the smallest number which is less than 100.
1. 686 2. 667 3. 647 4. 657
52. Four bells first begin to toll together with an interval of 5, 10, 15 and 20 seconds. How many times do they toll together in an hour?
1. 60 2. 59 3. 61 4. 58
53. 3 men start together to walk along a circular track at the same rate. The length of their tracks is 56 cm, 64 cm and 80 cm. How far will they be in step again?
1. 21 m 20 cm 2. 22 m 40 cm 3. 20 m 20 cm 4. 21 m 40 cm 54. Find the least number of soldiers in a regiment such that they stand in rows of 10, 15 and 20
and form a perfect square.
1. 400 2. 1600 3. 900 4. 2500
55. A trader has three kinds of sugar of 77 kg, 147 kg and 252 kg. Find the least number of bags of equal size required to pack them without mixing.
56. The sum of two numbers is 721 and their H.C.F. is 103. Find the numbers.
1. 319, 103 2. 309, 412 3. 103, 512 4. 520, 201
57. The L.C.M. and H.C.F. of two numbers are 693 and 11 respectively. One of them is 77. Find the other.
1. 91 2. 98 3. 97 4. 99
58. Find the least number of boys so that they can be arranged in the groups of 20 or 25 or 40.
1. 200 2. 250 3. 240 4. 216
59. Find the least perfect square which is divisible by 12, 15 and 20.
1. 225 2. 900 3. 1600 4. 2500
60. The product of two numbers is 768 and their H.C.F. is 8. What are the numbers? 1. 31 and 28 2. 32 and 24 3. 33 and 26 4. 35 and 24
1) 2 11) 2 21) 1 31) 1 41) 4 51) 2 2) 4 12) 1 22) 3 32) 3 42) 2 52) 3 3) 3 13) 4 23) 2 33) 3 43) 4 53) 2 4) 1 14) 2 24) 2 34) 2 44) 2 54) 3 5) 3 15) 3 25) 4 35) 4 45) 2 55) 2 6) 3 16) 2 26) 1 36) 1 46) 3 56) 2 7) 2 17) 1 27) 3 37) 1 47) 3 57) 4 8) 4 18) 4 28) 1 38) 2 48) 3 58) 1 9) 2 19) 3 29) 3 39) 3 49) 1 59) 2 10) 2 20) 3 30) 1 40) 1 50) 1 60) 2
4. PERCENTAGE
Percent: The term per cent means per hundred or for every hundred. The word is derived from
the Latin word per centum.
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.
1 0 0 5
and 5 percent means the same thing i.e. 5 parts out of every hundred parts.
Basic Formulae:
1. To convert any fraction n 1
into a rate per cent, multiply it by100 and put % sign i.e. ×1 0 0% n
1
E.g. What percentage is equivalent to
4 3? Sol: 4 3 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 8
3 1 %? Sol: 8 3 1% = 3 2 5 = 3 2 5 x 1 0 0 1 = 1 2 1 3. x% of a given number (N) = 1 0 0 x x N
E.g. 75 % of 800 = ? Sol: 75 % of 800 =
1 0 0
7 5 x 800 = 600
4. If A is x % more than that of B, then B is less than that of A by
x 1 0 0 1 0 0 x %.
5. If A is x % less than that of B, then B is more than that of A by
x 1 0 0 1 0 0 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 1 0 0 y 1 0 0 x 1 0 0
% of the second and
the second number is
1 0 0 x 1 0 0 y 1 0 0 % of the first.
8. If two numbers are respectively x % and y % less than a third number, then the first
number is 1 0 0 y 1 0 0 x 1 0 0
% of the second and
the second number is
1 0 0 x 1 0 0 y 1 0 0 % of the first.
9. If the price of a commodity increases by N%, then the reduction in consumption so as not to increase the expenditure is
N 1 0 0 1 0 0 N %.
10. If the price of a commodity decreases by N%, then the increase in con sumption so as not to decrease the expenditure is
N 1 0 0 1 0 0 N %.
11. If a number is changed (increased/decreased) successively by x % and y % then net %
change is given by 1 0 0 xy y
x % 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 = .
1 0 0 r 1 P n
(ii) Population (or length of tree) n years ago = . 1 0 0 r 1 P n
13. If the population (or value of a machine in rupees) is P and annual decrease (or depreciation) is r%, then
(i) Population (or value of machine) after n years = . 1 0 0 r 1 P n
(ii) Population (or value of machine) n years ago = . 1 0 0 r 1 P n
14. If a number K is increased successively by x % followed by y % and z %, then the final value of K will be K 1 0 0 x 1 1 0 0 y 1 1 0 0 z 1
15. In an examination, the minimum pass percentage is x%. If a student scores y marks and fails by z marks, then the maximum marks in the examination is
x ) z y ( 1 0 0 .
16. In an examination a % and b % students respectively fail in two different subjects while c % students fail in both the subjects, then the percentage of students who pass in both the subjects will be (100 – ( a + b - c ))%.
SOLVED EXAMPLES
1. If Srujana salary is 20% more than that of Deepa, then how much percent is Deepa‟s salary less than that of Srujana?
Sol: Here, x = 20. Required answer = x 1 0 0 1 0 0 x % = 2 0 1 0 0 1 0 0 2 0 % = 6 1 0 0 % = 6 4 1 6 % = 3 2 1 6 %.
2. If Anita‟s income is 30% less than that of Saritha, then how much percent is Saritha‟s income more than that of Anitha?
Sol: Here, x = 30. Required answer = x 1 0 0 1 0 0 x % = ×1 0 0 3 0 1 0 0 3 0 % = 7 3 0 0 % = 7 6 4 2 %.
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 A B = 2 5 3 0 B = 2 5 3 0 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 second?
Sol: Here, x = 25 and y = 50
So, First number =
1 0 0 y 1 0 0 x 1 0 0 % of the second = 1 0 0 5 0 1 0 0 2 5 1 0 0 % of the second = 6 5 0 0% of the second = 6 2 8 3 % = 3 1 8 3 % of the second.
5. Two numbers are respectively 20% and 32% less than a third number. What percent is the second of the first?
So, Second number = 1 0 0 x 1 0 0 y 1 0 0 % of the first = 1 0 0 2 0 1 0 0 3 2 1 0 0 % 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 as not increase the expenditure.
Sol: Reduction in consumption =
N 1 0 0 1 0 0 N % = 5 0 1 0 0 1 0 0 5 0 % = 3 1 0 0 % =
3
1
33
%.7. If the price of a commodity decreases by 50%, find how much percent its consumption be increased so as not decrease the expenditure.
Sol: Increase in consumption =
N 1 0 0 1 0 0 N = 5 0 1 0 0 1 0 0 5 0 % = 100%.
8. If the salary of Mr. Shashi is first increased by 18% and thereafter decreased by 15%, what is the net change in his salary?
Sol: Here, x = 18 and y = -15
So, the net % change in the salary =
1 0 0 xy y x % = 1 0 0 ) 1 5 )( 1 8 ( 1 5 1 8 % = 1 0 0 ) 1 5 )( 1 8 ( 1 5 1 8 % = 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 population is decreased after two years?
Sol: Here, x = - 20 and y = - 40
So, the net % change in population =
1 0 0 xy y x % = 1 0 0 ) 4 0 )( 2 0 ( 4 0 2 0 % = 1 0 0 8 0 0 6 0 % = - 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.
So, net % change in area = 1 0 0 xy y x % = 1 0 0 ) 1 0 )( 1 0 ( 1 0 1 0 % [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 =
1 0 0 xy y x % = 1 0 0 ) 4 )( 4 ( 4 4 % [Take x, y = 4] = 1 0 0 1 6 8 % = 2 5 4 8 %.
12. The tax on a commodity is diminished by 12% and its consumption increases by 10 %. Find the effect on revenue.
Sol: Revenue = tax x consumption
So, net % change in revenue =
1 0 0 xy y x % = 1 0 0 ) 1 0 )( 1 2 ( 1 0 1 2 % [Take x = - 12, y = 10] = 1 0 0 1 2 0 2 % = - 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 in 2 years time?
Sol: Here, P = 12500, r = 4 and n = 2.
Population after n years = . 1 0 0 r 1 P n
Population after 2 years = 2
1 0 0 4 1 1 2 5 0 0 = 2 2 5 1 1 1 2 5 0 0 = 2 2 5 2 6 1 2 5 0 0 = 2 5 2 6 2 5 2 6 1 2 5 0 0 = 13520.
14. The population of a town increases by 10% annually. If its present population is 12100, what it was 2 years ago?
Sol: Here, P = 12100, r = 10 and n = 2.
Population n years ago =
n 1 0 0 r 1 P
