Question 1

How many 1/8's are there in 37 1/ 2?

Accepted Answer

Number of 1/8 s = (75/2) / (1/8) = (75/2) x 8 = 300

Question 2

3/48 is what part of 1/12?

Accepted Answer

Let N of 1/12 = 3/48. Then, N = 3/48 x 12 = 3/4

Question 3

A boy was asked to multiply a given number by (8/17). Instead he divided the given number by (8/17) and got the result 225 more then what he should have got if he had multiplied the number by ( 8/17). The given number was?

Accepted Answer

∵ N x 17/8 - N x 8/17 =225 ⇒ 225 x N / 136 = 225 ∴ N = (225 x 136) / 225 = 136

Question 4

In an examination, a student was asked to find (3/14) of a certain number, by mistake he found (3/4) of it. His answer was 150 more then the correct answer. The given number is?

Accepted Answer

∵ 3N/4 - 3N/14 =150 ⇒ 15N/28 = 150 ∴ N = (150 x 28) /15 = 280

Question 5

If we multiply a fraction by itself and divided the product by its reciprocal, the fraction thus obtained is 18 26/ 27. The original fraction is?

Accepted Answer

∴ N x N x 1/N = 1826/27 ⇒ N3 = 512/27 ⇒ N3= (8/3)3 ∴ N = 8/3 = 22/3

Question 6

The smallest fraction which should be subtracted from the sum of 1 3/ 4, 2 1/ 2, 5 7/ 12, 3 1/ 3 and 2 1/ 4 to make the result a whole number is?

Accepted Answer

∴ 7/4 + 5/2 + 67/12 + 10 + 9/4 = (21 + 30 + 67 + 40 + 27/12) = 185/12 This is nearly greater than 15. Let required fraction be x. Then, 185/12 - x = 15 ∴ x = (185/12) - 15 = 5/12

Question 7

Gopal was asked to find 7/9 of a fraction. But he made a mistake of dividing the given fraction by 7/9 and got an answer which exceeded the correct answer by 8/21. The correct answer is?

Accepted Answer

∵ Let the fraction = N ∵ (9 x N)/7 - (7 x N)/9 = 8/21 ⇒ (32 x N)/63 = 8/21 ⇒ N = (8/21) x (63/32) = 3/4 ∴ Correct answer = N x 7/9 = 7/9 x 3/4 = 7/12

Question 8

The highest score in an innings was 3/11 of the total and the next highest was 3/11 of the remainder. If the scores differed by 9, then the total score is?

Accepted Answer

Let total score be N. Then highest score = 3N/11 Remainder = (N - 3N/11) = 8N/11 Next highest score = 3/11 of 8N/11 = 24N/121 Now, ∵ 3N/11 - 24N/121 = 9 ⇒ 9N/121 = 9 ∴N = 121

Question 9

In a family, the father took 1/4 of the cake and he had 3 times as much as others had. The total number of family members is?

Accepted Answer

Let there be N members, other than father . Father's share = 1/4, other's share = 3/4. Each of other's share = 3/4N ∵ 3 x 3/4N = 1/4 ∴ N = 9 Hence, the total number of members = N + 1 = 10

Question 10

Ravi earns twice as much in January as in each of the other months. What part of his annual earnings he earns in that month?

Accepted Answer

Suppose Ravi earns Rs. N in each of the 11 months. Then earning in January = Rs. 2 x N. ∴ Total annual income = (11 x N + 2 x N) = Rs. 13 x N Part of total earning in January = (2 x N) / (13 x N) = 2/13

Question 11

√ 132 + 28 / 4 - (3)3 + 107 = (?) 2

Accepted Answer

(?)2 = √132 + 28 / 4 - (3)3 + 107 = √169 + 7 - 27 + 107 = √256 = √16 = 4

Question 12

If a 2 + b 2 = 234 and ab = 108, find the value of a + b / a - b .?

Accepted Answer

We know that, (a + b)2 = a2 + b2 + 2ab = 234 + 2 x 108 = 450 (a - b)2 = a2 + b2 - 2ab = 234 - 2 x 108 = 18 ∴ (a + b)2/(a - b)2 = 450/18 = 25 ⇒ [(a + b)/(a - b)]2 = 25 ∴ (a + b)/(a - b) = √25 = 5

Question 13

If a 2 + 1 = a, then the value of a 12 + a 6 + 1 is?

Accepted Answer

a2 + 1 = a ⇒ a + 1/a = 1 On squaring both sides, we get a2 + 1/a2 + 2 = 1 On cubing both sides, we get (a2 + 1/a2)3 = (-1)3 ⇒ a6 + 1/a6 + 3a2 x 1/a2 (a2 + 1/a2) = -1 ⇒ a6 + 1/a6 + 3 x (-1) = -1 Now, a6 + 1/a6 + 1 = 3 As, a12 + a6 + 1 can be written as a6 + 1/a6 + 1 ∴ a12 + a6 + 1 = 3

Question 14

If a, b and c are non-zero, a + 1/b = 1 and b + 1/c = 1, then the value of abc is?

Accepted Answer

∵ a + 1/b = 1 ab + 1 = b ...(i) Also, b + 1/c = 1 ⇒ b = 1 - 1/c ...(ii) From Eqs. (i) and (ii), we get ab + a = 1 - 1/c ⇒ ab = -1/c ⇒ abc = -1

Question 15

If a + b + c = 0, then the value of [(a + b)/c + (b + c)/a + (c + a)/b] x [a/(b + c) + b/(c + a) + c/(a + b)] is?

Accepted Answer

Since, a + b + c = 0 Then, a + b = -c ....(i) a + c = -b .........(ii) b + c = -a ....(iii) Now, [(a + b)/c + (b + c)/a + (c + a)/b] x [a/(b + c) + b/(c + a) + c/(a + b)] Now, putting the value of a + b , b + c and c + a from Eqs. (i), (ii) and (iii) , we get [(-c)/c + (-a)/a + (-b)/b] x [a/-a + b/-b + c/-c] [(-1) + (-1) + (-1)] [(-1) + (-1) + (-1)] = (-3) x (-3) = 9

Question 16

If a + b + c = 14 and a 2 + b 2 + c 2 = 96, then (ab + bc + ca) =?

Accepted Answer

We know that, (a + b + c)2 = (a2 + b2 + c2) + 2(ab + bc + ca ) ⇒ 196 = 96 + 2(ab + bc + ca) ⇒ 2(ab + bc + ca) = 196 - 96 = 100 ∴ (ab + bc + ca) = 100/2 = 50

Question 17

[5 - [3/4 + {2 1/ 2 - (1/2 + 1/6 - 1/7)}]] / 2 =?

Accepted Answer

? = [5 -[3/4 + {21/2 - (1/2 + 1/6 - 1/7)}]]/2 = [5 - [3/4 + {5/2 -(1/2 + 7 - 6/42)}]]/2 = [5 - [3/4 + {5/2 - (1/2 + 1/42)}]]/2 = [5 - [3/4 + {5/2 - (21 + 1 /42)}]]/2 = [5 - [3/4 + {5/2 - 22/42}]]/2 = [5 - [3/4 + {105 - 22/42}]]/2 = [5 - [3/4 + 83/42]] / 2 = [5 - [63 + 166/84]]/2 = (191/84)/2 =191/168 = 123/168

Question 18

[0.5 x 0.5 x 0.5 + 0.2 x 0.2 x 0.2 + 0.3 x 0.3 x 0.3 - 3 x 0.5 x 0.3 x 0.2 ] / [0.5 x 0.5 + 0.2 x 0.2 + 0.3 x 0.3 - 0.5 x 0.2 - 0.2 x 0.3- 0.5 x 0.3] =?

Accepted Answer

Given expression [a3 + b3 + c3 - 3abc] / [a2 + b2 + c2 - ab - bc - ca] = a + b + c = 0.5 + 0.2 + 0.3 where, a = 0.5, b = 0.2, c = 0.3

Question 19

Solve [(999 + 588) 2 - (999 - 588) 2] / (999 x 588)

Accepted Answer

Given expression = [(a + b )2 - (a - b )2] / ab = 4ab/ab = 4 where, a = 999, b = 588

Question 20

Solve [(238 + 131) 2 + (238 - 131) 2] / (238 x 238 + 131 x 131)

Accepted Answer

Given expression = [(a + b)2 + (a - b)2] / [a2 + b2] = 2(a2 + b2) / (a2 + b2) = 2 where, a = 238, b = 131

Simplification Questions