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81. If the numerator of a fraction is increased by $\frac{1}{4}$ and the denominator is decreased byy $\frac{1}{3}$ , the new fraction obtained is $\frac{33}{64}$ . What was the original fraction ?

A. $\frac{3}{7}$

B. $\frac{5}{7}$

C. $\frac{7}{9}$

D. Cannot be determined

Discuss

Solution:

Let the fraction be

\frac{x}{y}

Then,

\begin{aligned} \Leftrightarrow \frac{x + \frac{1}{4}}{y - \frac{1}{3}} = \frac{33}{64} \\ \Leftrightarrow \frac{3 (4 x + 1)}{4 (3 y - 1)} = \frac{33}{64} \\ \Leftrightarrow \frac{4 x + 1}{3 y - 1} = \frac{33}{64} \times \frac{4}{3} \\ \Leftrightarrow \frac{4 x + 1}{3 y - 1} = \frac{11}{16} \\ \Leftrightarrow 16 (4 x + 1) = 11 (3 y - 1) \\ \Leftrightarrow 64 x + 16 = 33 y - 11 \\ \Leftrightarrow 64 x - 33 y = - 27 \end{aligned}

Which cannot be solved to find

\frac{x}{y}

Hence, the original fraction cannot be determined from the given data.

82. If doubling a number and adding 20 to the result gives the same answer as multiplying the number by 8 and taking away 4 from the product, the number is :

A. 2

B. 3

C. 4

D. 6

Discuss

Solution:

Let the number be x
Then,
⇔ 2x + 20 = 8x - 4
⇔ 6x = 24
⇔ x = 4

83. Out of six consecutive natural numbers if the sum of first three is 27, what is the sum of the other three ?

A. 24

B. 25

C. 35

D. 36

Discuss

Solution:

Let the six numbers be, x , (x + 1), (x + 2), (x + 3), (x + 4) and (x + 5)
Then,
⇔ x + (x + 1) + (x + 2) = 27
⇔ 3x + 3 = 27
Required sum :
= (x + 3) + (x + 4) + (x + 5)
= 3x + 12
= (3x + 3) + 9
= 27 + 9
= 36

84. The difference between a two-digit number and the number obtained by interchanging the two digits is 63. Which is the smaller of the two numbers ?

A. 29

B. 70

C. 92

D. Cannot be determined

Discuss

Solution:

Let the ten's digit be x and unit's digit be y
Then,

\begin{aligned} \Leftrightarrow (10 x + y) - (10 y + x) = 63 \\ \Leftrightarrow 9 (x - y) = 63 \\ \Leftrightarrow x - y = 7 \end{aligned}

85. If the numerator of a fraction is increased by 200% and the denominator is increased by 300%, the resultant fraction is $\frac{15}{26}$ . What was the original fraction ?

A. $\frac{8}{11}$

B. $\frac{10}{11}$

C. $\frac{9}{13}$

D. $\frac{10}{13}$

Discuss

Solution:

Let the fraction be

\frac{x}{y}

Then,

\begin{aligned} \Leftrightarrow \frac{x + 200 % of x}{y + 300 % of y} = \frac{15}{26} \\ \Leftrightarrow \frac{3 x}{4 y} = \frac{15}{26} \\ \Leftrightarrow \frac{x}{y} = \frac{15}{26} \times \frac{4}{3} \\ \Leftrightarrow \frac{x}{y} = \frac{10}{13} \end{aligned}

86. If 50 is subtracted from two-third of number, the result is equal to sum of 40 and one-fourth of that number. What is the number ?

A. 174

B. 216

C. 246

D. 336

Discuss

Solution:

Let the number be x
Then,

\begin{aligned} \Leftrightarrow \frac{2}{3} x - 50 = \frac{1}{4} x + 40 \\ \Leftrightarrow \frac{2}{3} x - \frac{1}{4} x = 90 \\ \Leftrightarrow \frac{5 x}{12} = 90 \\ \Leftrightarrow x = (\frac{90 \times 12}{5}) \\ \Leftrightarrow x = 216 \end{aligned}

87. The sum of two numbers is 25 and their difference is 13. Find their product :

A. 104

B. 114

C. 315

D. 325

Discuss

Solution:

Let the numbers be x and y
Then,
x + y = 25 and x - y = 13

\begin{aligned} \Leftrightarrow 4 x y = {(x + y)}^{2} - {(x - y)}^{2} \\ \Leftrightarrow 4 x y = {(25)}^{2} - {(13)}^{2} \\ \Leftrightarrow 4 x y = 625 - 169 \\ \Leftrightarrow 4 x y = 456 \\ \Leftrightarrow x y = 114 \end{aligned}

88. The sum of seven consecutive numbers is 175. What is the difference between twice the largest number and thrice the smallest number ?

A. 7

B. 8

C. 10

D. 12

Discuss

Solution:

Let the seven numbers be x, (x + 1), (x + 2), (x + 3), (x + 4), (x + 5) and (x + 6)
Then,
⇔ x + (x + 1) + (x + 2) + (x + 3) + (x + 4) + (x + 5) + (x + 6) = 175
⇔ 7x + 21 = 175
⇔ 7x = 154
⇔ x = 22
Required difference :
= 2(x + 6) - 3x
= 12 - x
= 12 - 22
= -10

89. The number obtained by interchanging the two digits of a two-digit number is lesser than the original number by 54. If the sum of the two digit of the number is 12, then what is the original number ?

A. 28

B. 39

C. 82

D. Cannot be determined

Discuss

Solution:

Let ten's digit = x
Then, unit's digit = (12 - x)

∴ [10 x + (12 - x)] -

[10 (12 - x) + x]

= 54

\begin{aligned} \Leftrightarrow 18 x - 108 = 54 \\ \Leftrightarrow 18 x = 162 \\ \Leftrightarrow x = 9 \end{aligned}

So, ten's digit = 9 and unit's digit = 3
Hence, original number = 93

90. A fraction is such that if the double of the numerator and the triple pf the denominator is changed by +10 percent and -30 percent respectively, then we get 11 percent of $\frac{16}{21}$ . Find the fraction :

A. $\frac{2}{25}$

B. $\frac{3}{25}$

C. $\frac{4}{25}$

D. None of these

Discuss

Solution:

Let the fraction be

\frac{x}{y}

Then,

\begin{aligned} \Leftrightarrow \frac{110 % of 2 x}{70 % of 3 y} = 11 % of \frac{16}{21} \\ \Leftrightarrow \frac{22 x}{21 y} = \frac{11}{100} \times \frac{16}{21} \\ \Leftrightarrow \frac{x}{y} = (\frac{11}{100} \times \frac{16}{21} \times \frac{21}{22}) \\ \Leftrightarrow \frac{x}{y} = \frac{2}{25} \end{aligned}

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