Ratio - Aptitude MCQ Questions and Solutions with Explanations

y α (x + 3)
⇒ y = k (x + 3) for some constant k.
When y = 8, x = 1, y = k (x + 3)
⇒ 8 = k (1 + 3)
⇒ k = 2
When x = 2, y = 2(x + 3) = 2(2 + 3) = 2 × 5 = 10

$$\eqalign{ & {\text{y }}\alpha {\text{ }}\left( {{\text{x}} + \frac{1}{{\text{x}}}} \right) \cr & \Rightarrow {\text{y}} = {\text{kx}} + \frac{{\text{m}}}{{\text{x}}}{\text{,}} \cr} $$
Where k and m are constants,
Then,
$$\eqalign{ & = 4{\text{k}} + \frac{{\text{m}}}{4} = 6......({\text{i}}) \cr & and \cr & = 3{\text{k}} + \frac{{\text{m}}}{3} = 10......(ii) \cr} $$
Multiplying (i) by 3 and (ii) by 4, we get :
$$\eqalign{ & = 12{\text{k}} + \frac{{3{\text{m}}}}{4} = 18....({\text{iii}}) \cr & and \cr & = 12{\text{k}} + \frac{{4{\text{m}}}}{3} = \frac{{40}}{3}....({\text{iv}}) \cr} $$
Subtracting (iv) from (iii), we get :
$$\eqalign{ & = \frac{{3{\text{m}}}}{4} - \frac{{4{\text{m}}}}{3} = 18 - \frac{{40}}{3} \cr & \Rightarrow - \frac{{7{\text{m}}}}{{12}} = \frac{{14}}{3} \cr & \Rightarrow {\text{m}} = - 8. \cr & {\text{Putting m}} = - {\text{8 in (i),}} \cr & {\text{we get}}: \cr & 4{\text{k}} + \frac{{\left( { - 8} \right)}}{4} = 6 \cr & \Rightarrow 4{\text{k}} = 8 \cr & \Rightarrow {\text{k}} = 2 \cr & \therefore {\text{y}} = 2{\text{x}} - \frac{8}{{\text{x}}} \cr} $$

Let age of P = 5x
and age of Q = 6x
According to the question,
(5x + 4) + (6x + 4) = 52
11x = 44
x = 4
Present age of P = 5x + 4 = 5 × 4 + 4 = 24
Present age of Q = 6x + 4 = 6 × 4 + 4 = 28
$$\eqalign{ & \therefore {\text{Required ratio}} \cr & {\text{ = }}\frac{{24}}{{28}} \cr & = \frac{6}{7} \cr & = 6:7 \cr} $$

Ratio of A's age to B's age is 4 : 3 = 4x : 3x
After 6 years A's age = 4x + 6
∴ According to question,
$$\eqalign{ & 4x + 6 = 26 \cr & \boxed{x = 5} \cr & \therefore {\text{B's age}} = 3x \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 3 \times 5 \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 15 \cr} $$

Let numbers are = 3x and 5x
∴ According to question,
$$\eqalign{ & \Rightarrow \frac{{3x + 6}}{{5x + 6}} = \frac{2}{3} \cr & \Rightarrow 9x + 18 = 10x + 12 \cr & \Rightarrow x = 6 \cr & {\text{So, numbers are }} \cr & 3x = 3 \times 6 = 18 \cr & 5x = 5 \times 6 = 30 \cr} $$

Let the number of boys and girls be 3x and 2x respectively
Then,
Number of boys and girls who are adults
$$\eqalign{ & = 20\% {\text{ of }}3x + 25\% {\text{ of }}2x \cr & = \left( {\frac{{20}}{{100}} \times 3x} \right) + \left( {\frac{{25}}{{100}} \times 2x} \right) \cr & = \frac{3}{5}x + \frac{x}{2} = \frac{{11x}}{{10}} \cr} $$
∴ Number of boys and girls who are not adults
$$\eqalign{ & = \left[ {\left( {3x + 2x} \right) - \frac{{11x}}{{10}}} \right] \cr & = 5x - \frac{{11x}}{{10}} \cr & = \frac{{39x}}{{10}} \cr & {\text{Required percentage}} \cr & = \left( {\frac{{39x}}{{10}} \times \frac{1}{{5x}} \times 100} \right)\% \cr & = 78\% \cr} $$

Let the cost of the table and chair be Rs. 5x and Rs. 7x respectively
$$\eqalign{ & {\text{New cost of chair}} \cr & = 120\% {\text{ of }}7x \cr & = {\text{Rs}}{\text{.}}\left( {\frac{6}{5} \times 7x} \right) \cr & = {\text{Rs}}{\text{.}}\frac{{42x}}{5} \cr & {\text{New cost of table}} \cr & = 110\% {\text{ of }}5x \cr & = {\text{Rs}}{\text{.}}\left( {\frac{{11}}{{10}} \times 5x} \right) \cr & = {\text{Rs}}{\text{.}}\frac{{55x}}{{10}}. \cr & \therefore {\text{New ratio}} \cr & = \frac{{55x}}{{10}}:\frac{{42x}}{5} \cr & = 55:84 \cr} $$

Let the earnings of A and B Rs. 8x and Rs. 9x respectively.
Then,
$$\eqalign{ & = \frac{{150\% {\text{ of }}8x}}{{75\% {\text{ of }}9x}} = \frac{{16}}{9} \cr & \Rightarrow \frac{{\frac{3}{2} \times 8x}}{{\frac{3}{4} \times 9x}} = \frac{{16}}{9} \cr & \Rightarrow \frac{{16}}{9} = \frac{{16}}{9} \cr} $$
Hence, A's earnings cannot be determined

Spirit and Water in the ratio of 3 : 2
There are 3 liters more spirit in the mixture.
∴ 3x - 2x = 3
⇒ x = 3
∴ Quantity of spirit in the mixture = 3 × 3 = 9

Total Profit = Rs. 960/-
A : B = $$\frac{1}{3}$$ : $$\frac{1}{5}$$   = 5 : 3
Profit share of A = $$\frac{5}{8} \times 960 = 600$$
Profit share of B = $$\frac{3}{8} \times 960 = 360$$
Difference of their profit = 600 - 360 = Rs. 240