PRESENTED BY: Er. Manit Choudhary

For any queries, contact E-mail id: manit.choudhary91@gmail.com

Q: Which one of the following fraction is the greatest? 3/4, 4/5 and 5/6 Ans: 5/6 Q: Which one of the following fraction is the greatest? 1/8, 4/9 and 7/10 Ans: 7/10 Short trick: When the numerator and the denominator of the fractions increased

by a constant value, the last fraction is the greatest. Fraction like: (x+a)/ (x+b) Where: a = increment in numerator b = increment in denominator This condition for when a=b or a>b

Q: Which one of the following is the greatest? 1/8, 2/12, 3/16 and 4/20 Ans: 4/20 Q: Which one of the following is the greatest? 2/7, 4/15 and 6/23 Ans: 2/7 Short trick: In the case of a First fraction then the last value is greatest 2. Increase in num/ Increase in den. < First fraction then the last value is least 3. Increase in num/ Increase in den. = First fraction then all the values are equal

Q: Which is greater: 5/8 or 9/14 Ans: 9/14 Q: Which is greater: 4/15 or 6/23 Ans: 4/15 Q: Arrange following in ascending order 3/7, 4/5, 7/9, 1/2 and 3/5 Ans: 3/7, 1/2, 3/5, 7/9, 4/5 Short trick: The fraction whose numerator after cross

multiplication gives the greater value is greater.

Some Rules on Counting Numbers 1. Sum of all the first n natural numbers =

n(n+1)/2 2. Sum of first n odd numbers = n2

3. Sum of first n even numbers = n(n+1) 4. Sum of squares of first n natural numbers =

n(n+1)(2n+1)/6 5. Sum of cubes of first n natural numbers =

(n(n+1)/2)2

6. The difference between the squares of two consecutive numbers is always an odd number.

Power and Index: If a number p is multiplied by itself n times,

the product is called nth power of p and is written as pn. In pn, p is called the base and n is called the index of the power.

Q: What is the number in the unit place in

(729)59 ? Ans: 9 Q: Find the number in unit place in (623)36, (122)22, (98)42 Ans: 1, 4 and 4

Short tricks: Rule I: For odd numbers (1)n = (1) (3)4n = (1) (7)4n = (1) where n = 1,2,3 Rule II: For even numbers (2)4n = (6) (4)2n = (6) (6)n = (6) (8)4n = (6) where n = 1,2,3 Rule III: (1)n = (1) (5)n = (5) (6)n = (6) where n = 1,2,3 (9)n = (9) where n = 1,3,5 (9)2n = (1) where n = 1,2,3

Q: The quotient arising from the division of 24446 by a certain number is 79 and the remainder is 35; what is the divisor?

(a) 307 (b) 309 (c) 312 (d) 315 Ans: (b) Q: What least number must be added to 8961 to make it

exactly divisible by 84? (a) 27 (b) 25 (c) 29 (d) 32 Ans: (a) Q: What least number must be subtracted to 8961 to make it

exactly divisible by 84? (a) 59 (b) 58 (c) 57 (d) 56 Ans: (c) Q: Find the least number of 5 digit which is exactly divisible

by 89. (a) 9943 (b) 10043 (c) 10570 (d) 10057 Ans: (d)

Q: A number when divided by 899 gives a remainder 63. What remainder will be obtained by dividing the same number by 29?

(a) 3 (b) 4 (c) 5 (d) 6 Ans: (c) Q: A number when divided by 899 gives a remainder 62. What

remainder will be obtained by dividing the same number by 31?

(a) 0 (b) 3 (c) 5 (d) 7 Ans: (a) Q: A number when divided by 12 gives a remainder 7. What

remainder will be obtained by dividing the same number by 7? (a) 6 (b) 3 (c) 0 (d) Que. wrong Ans: (d) Q: A boy multiplied 423 by a certain number and obtained 65589

as his answer. If both the fives are wrong, but the other figures are right, find the correct answer.

(a) 64089 (b) 60489 (c) 60849 (d) 60948 Ans: (b)

Formula: Divisor x Quotient = Dividend Remainder Q: By what number less than 1000 must 43259

be multiplied so that the last three digits to the right of the product may be 437?

(a) 643 (b) 634 (c) 753 (d) 743 Ans: (d)

Q: The sum of two numbers is 14 and their difference is 10. Find the product of the two numbers.

(a) 140 (b) 70 (c) 44 (d) 24 Ans: (d) Q: The sum of two numbers is 30 and their difference

is 6. Find the difference of their squares. (a) 160 (b) 180 (c) 190 (d) 200 Ans: (b) Q: The product of two terms is 39 and their difference

is 28. Find the difference of their reciprocals. (a) 28/39 (b) 39/28 (c) 7/39 (d) 13/28 Ans: (a)

Short trick: Product of the numbers: Product= (Sum2 Difference2)/4 OR Product= (Sum + Difference) (Sum Difference)/4 OR X=(Sum + Difference)/2 Y=(Sum Difference)/2 Difference of their Squares: =Sum Difference Difference or Sum of their reciprocals: =Difference or Sum/ Product

Q: If two-fifth of one-half of number is 8, find the number.

(a) 6.4 (b) 20 (c) 40 (d) 44 Ans: (c) Q: If one-fifth of one-third of one-half of

number is 15, find the number. (a) 420 (b) 450 (c) 470 (d) 480 Ans: (b) Short trick: Number = x (b/a) (d/c) (f/e)

Q: The sum of the digits of a two digit number is 9. If 63 subtracted from the number then digits are reversed. Find the number.

(a) 63 (b) 36 (c) 72 (d) 81 Ans: (d) Q: The sum of the digits of a two digit number is 8. If

the digits are reversed, the number is decreased by 54. Find the number.

(a) 71 (b) 62 (c) 53 (d) 80 Ans: (a) Short trick: Number = (11xy)/2 Where, x=sum of digits y=decrement or increment (+ve) sign for when y is decreased and (ve) sign for

when y is increased.

Q: If 40% of a number is 360, what will be 15% of 15% of that number?

(a) 20.2 (b) 20.4 (c) 20.25 (d) 20.3 Ans: (c) Q: The ratio of the sum and the difference of

two numbers is 7:1. Find the ratio of those two numbers..

(a) 7:3 (b) 3:7 (c) 3:4 (d) 4:3 Ans: (d)

Q: The difference between a two digit number and the number obtained by interchanging the digits is 27. What is the difference of the two digits of the number?

(a) 2 (b) 3 (c) 4 (d) 5 Ans: (b) Q: If the places of last two digits of a three-digit number are

interchanged, a new number greater than the original number by 54 is obtained. What is the difference between the last two digits of that number?

(a) 6 (b) 7 (c) 8 (d) 9 Ans: (a) Short trick: Difference of two digits = (Diff. in original & Interchanged number)/9

Q: The digit at the units place of a 2-digit number is increased by 50%. And the digit at the tens place of the same number is increased by 100%. Now, we find that the new number is 33 more than the original number. Find the original number.

(a) 66 (b) 33 (c) 36 (d) 69 Ans: (c) Q: It is given that 232+1 is exactly divisible by a certain number.

Which one of the following is also divisible by the same number?

(a) 296+1 (b) 216-1 (c) 216+1 (d) 7233 Ans: (a) Q: 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 tens place, what is the number?

(a) 36 (b) 47 (c) 58 (d) 69 Ans: (a)

Q: A number on being divided by 5 and 7 successively leaves the remainders 2 and 4 respectively. Find the remainder when the same number is divided by 57=35

(a) 20 (b) 21 (c) 22 (d) 23 Ans: (c) Short trick: The required number is = d1r2 +r1 Where, d1 = The first divisor r1 = the first remainder r2 = the second remainder

Q: Find the number of zeroes at the end of the products:

(1) 12 18 15 40 25 16 55 105 (2) 5 10 15 20 25 30 35 40 40 45 Ans: 6 and 10 Short trick: 1) If there is any zero at the end of any multiplicand 2) If 5 or multiple of 5 are multiplied by any even

number. To generalise the above two statements in (5)n (2)m n zeros when n

Q: Find the number of different divisors of 50, besides unity and the number itself.

Ans: 4 Q: The number of divisors of 40, except unity, is Ans: 7 Quick method: To find the number of different divisor 50 = 2 5 5 = 21 52 Total divisors = (1+1)(2+1) = 6 Excluding 1 and 50 = 6-2=4 Q: Find the different divisors of 37800, excluding unity. Ans: 95

Q: How many numbers up to 100 are divisible by 6? Ans: 16 Q: How many numbers up to 200 are divisible by 4 and 3 together? Ans: 16 Short trick: To find the number of number divisible by a certain integer The quotient obtained is the required number of numbers. Like in above question 100 = 16 6 + 4 Here quotient is 16 Q: How many numbers between 100 and 300 are divisible by 7? Ans: 28

Following points to be remembered: 1. The face value of a digit is the value of that

digit, may be at any place. For example, the face value of 4 in the numeral 547063 is 4, while its place value is 40000.

2. The square of a natural number never ends in 2, 3, 7 and 8.

3. The difference of square of two consecutive even integer is always divisible by 4.

4. The difference of square of two consecutive odd integer is always divisible by 4 and 8.

5. A given number is divisible by 2, if the unit digit in the number is any one of 0, 2, 4, 6, 8.

6. A given number is divisible by 3, if the sum of the digits of the given number is divisible by 3.

7. A given number is divisible by 4, if the number formed by last two digit, is divisible by 4.

8. A given number is divisible by 5, if the unit digit in the number is 0 or 5.

9. A given number is divisible by 8, if the number formed by the last 3 digits of the given number, is divisible by 8.

10. A given number is divisible by 9, if the sum of the digits of the number is divisible by 9.

11. A given number is divisible by 10, if the unit digit of the number is 0.

12. A given number is divisible by 11, if the difference of the sum of the digits in odd places and the sum of its digits in even places, is either 0 or a number divisible by 11.

Some other short tricks: 1. If a/b of a number is x then the number is x(b/a). 2. Any number is x more or less from the a/b of that number then the

number is (xb)/(b-a). 3. The sum of a two-digit number and the number obtained by

interchanging the digits is always multiple of 11 then the sum of digits is (sum of two numbers/11).

4. If x is added in any number then new number is y times the original number. So the original number is x/(y 1).

5. If x is added or subtracted from a/b part of a number then the number is equal to c/d part of that number. So the number is xbd/ (ad bc).

6. If a% of a number is added in the second number and increment in the second number is b% then the ratio of first and second number is b:a.

7. If any number is divided by a but by mistake multiplied by a instead of division and got an answer which exceeded by x then the number is xa/(a2 1).

8. If any number is multiplied by a but by mistake divide by b instead of multiplication and got an answer which decrease by x then the number is xab/(ab 1).

Q:1 What least number must be added to 1056, so that the sum is completely divisible by 23 ?

(a) 2 (b) 3 (c) 18 (d) 21 Q:2 The largest 4 digit number exactly divisible

by 88 is: (a) 9944 (b) 9768 (c) 9988 (d) 8888 Q:3 What is the unit digit in

{(6374)1793 x(625)317 x (341)491}? (a) 0 (b) 2 (c) 3 (d) 5

Q:4 The difference of two numbers is 1365. On dividing the larger number by the smaller, we get 6 as quotient and the 15 as remainder. What is the smaller number ?

(a) 240 (b) 270 (c) 295 (d) 360 Q:5 (753 x 753 + 247 x 247 - 753 x 247) (753 x 753 x 753 + 247 x 247 x 247) (a) 1/1000 (b) 1/506 (c) 253/500 (d) None Q:6 On dividing a number by 56, we get 29 as remainder.

On dividing the same number by 8, what will be the remainder ?

(a) 4 (b) 5 (c) 6 (d) 7

Q:7 How many natural numbers are there between 23 and 100 which are exactly divisible by 6 ?

(a) 8 (b) 11 (c) 12 (d) 13 Q:8 (963 + 476)2 + (963 - 476)2 (963 x 963 + 476 x 476) (a) 1449 (b) 497 (c) 2 (d) 4 Q:9 In dividing a number by 585, a student employed the

method of short division. He divided the number successively by 5, 9 and 13 (factors 585) and got the remainders 4, 8, 12 respectively. If he had divided the number by 585, the remainder would have been

(a) 24 (b) 144 (c) 292 (d) 584

Q:10 In a division sum, the divisor is 10 times the quotient and 5 times the remainder. If the remainder is 46, what is the dividend ?

(a) 4236 (b) 4306 (c) 4336 (d) 5336 Q:11 The sum of the two numbers is 12 and their

product is 35. What is the sum of the reciprocals of these numbers ?

(a) 12/35 (b) 1/35 (c) 35/8 (d) 7/32 Q:12 What is the unit digit in(795 - 358)? (a) 0 (b) 4 (c) 6 (d) 7

Q:13 If 60% of 3/5 of a number is 36, then the number is:

(a) 80 (b) 100 (c) 75 (d) 90 Q:14 The difference between the place value and

the face value of 6 in the numeral 856973 is: (a) 973 (b) 6973 (c) 5994 (d) None Q:15 A number was divided successively in order

by 4, 5 and 6. The remainders were respectively 2, 3 and 4. The number is:

(a) 214 (b) 476 (c) 954 (d) 1908

Q:16 The difference between the place values of two sevens in the numeral 69758472 is:

(a) 0 (b) 6993 (c) 699930 (d) None Q:17 A certain number of two-digits is three times the sum

of its digits and if 45 be added to it, the digits will be reversed. The number is:

(a) 32 (b) 72 (c) 27 (d) 23 Q:18 If 3 is added to the denominator of a fraction, it

becomes 1/3 and if 4 be added to its numerator, it becomes 3/4. The fraction is:

(a) 4/9 (b) 3/20 (c) 7/24 (d) 5/12

Q:19 The difference between squares of two numbers is 256000 and sum of the numbers is 1000. The numbers are:

(a) 628, 372 (b) 600, 400 (c) 640, 630 (d) None Q:20 There are two numbers such that the sum of twice

the first and thrice the second is 18, while the sum of thrice the first and twice the second is 17. The larger of the two is:

(a) 4 (b) 6 (c) 8 (d) 12 Q: 21 Which one of the following can't be the square of

natural number ? (a) 32761 (b) 81225 (c) 42437 (d) 20164

Q:22 Of the two numbers, 4 times the smaller one is less than 3 times the larger one by 5. But the sum of the numbers is larger than 6 times their difference by 6. The larger number is:

(a) 43 (b) 53 (c) 59 (d) 63 Q:23 2/3 of a number is 20 less than the original number. The

number is: (a) 40 (b) 60 (c) 80 (d) 90 Q:24 What is the value of M and N respectively? If M39048458N is

divisible by 8 and 11; Where M and N are single digit integers?

(a) 7, 8 (b) 8, 6 (c) 6, 4 (d) 5, 4 Q:25 If the number 42573 * is exactly divisible by 72, then the

minimum value of * is: (a) 4 (b) 5 (c) 6 (d) 7

Answers: 1. A 11. A 21. C 2. A 12. B 22. C 3. A 13. B 23. B 4. B 14. C 24. C 5. A 15. A 25. C 6. B 16. C 7. D 17. C 8. C 18. D 9. D 19. A 10. D 20. A