Ratio - Aptitude MCQ Questions and Solutions with Explanations

$$\eqalign{ & {\text{A}} + {\text{B}} = 94 \cr & \therefore \frac{{\text{A}}}{5}:\frac{{\text{B}}}{8} = 3:4 \cr & \Rightarrow \frac{{{\text{A}} \times 8}}{{5 \times {\text{B}}}} = \frac{3}{4} \cr & \Rightarrow \frac{{\text{A}}}{{\text{B}}} = \frac{3}{4} \times \frac{5}{8} \cr & \Rightarrow \frac{{\text{A}}}{{\text{B}}} = \frac{{15}}{{32}} \cr & {\text{ A}}:{\text{B}} \cr & {\text{ }}15:32 \cr & {\text{Let }}15x:32x \cr & \therefore 15x + 32x = 47x \cr & \Rightarrow 47x = 94 \cr & \Rightarrow x = 2 \cr & \therefore {\text{A}} = 2 \times 15 = 30 \cr & {\text{B}} = 32 \times 2 = 64 \cr} $$

$$\eqalign{ & {\text{Let the fourth proportional to}} \cr & \left( {{a^2} - {b^2}} \right),\left( {{a^2} - ab} \right),\left( {{a^3} + {b^3}} \right)\,{\text{be}}\,x \cr & {\text{Then,}} \cr & = \left( {{a^2} - {b^2}} \right):\left( {{a^2} - ab} \right)::\left( {{a^3} + {b^3}} \right):x \cr & \Rightarrow \left( {{a^2} - {b^2}} \right)x = \left( {{a^3} + {b^3}} \right)\left( {{a^2} - ab} \right) \cr & \Rightarrow x = \frac{{\left( {{a^3} + {b^3}} \right)\left( {{a^2} - ab} \right)}}{{\left( {{a^2} - {b^2}} \right)}} \cr & \Rightarrow x = \frac{{\left( {a + b} \right)\left( {{a^2} - ab + {b^2}} \right)a\left( {a - b} \right)}}{{\left( {a - b} \right)\left( {a + b} \right)}} \cr & \Rightarrow x = a\left( {{a^2} - ab + b} \right) \cr} $$

Let the third proportional to 38 and 15 be x
$$\eqalign{ & {\text{Then,}} \cr & \Rightarrow 38:15::15:x \cr & \Rightarrow 38x = 15 \times 15 \cr & \Rightarrow x = \frac{{15 \times 15}}{{38}} \cr} $$

Required mean proportional
$$\eqalign{ & = \sqrt {\left( {3 + \sqrt 2 } \right)\left( {12 - \sqrt {32} } \right)} \cr & = \sqrt {\left( {3 + \sqrt 2 } \right)\left( {12 - 4\sqrt 2 } \right)} \cr & = \sqrt {36 - 8} \cr & = \sqrt {28} \cr & = 2\sqrt {7} \cr} $$

$$\frac{{3x + 5}}{{5x - 2}} = \frac{2}{3}$$
⇒ Cross multiply the equation
$$\eqalign{ & 9x + 15 = 10x - 4 \cr & x = 19 \cr} $$

7, 16 , 43, 79
If a, b, c, d are in proportion
⇒ a : b :: c : d
Then ad = bc
Let x is added to make it a proportion
⇒ (7 + x) : (16 + x) :: (43 + x) : (79 + x)
⇒ (7 + x)(79 + x) = (16 + x)(43 + x)
∴ x² + 86x + 553 = x² + 59x + 688
⇒ 27x = 135
⇒ x = 5
∴ Hence, it 5 is added to make it a proportion

A : B
4x : 7x
Now 4 is added to each number
$$\eqalign{ & \frac{{4x + 4}}{{7x + 4}} = \frac{3}{5} \cr & \Rightarrow 20x + 20 = 21x + 12 \cr & \Rightarrow x = 8 \cr} $$
∴ Smaller number is 4 × 8 = 32
Larger number is 7 × 8 = 56

Let the third proportional to 12 and 30 be x.
Then,
$$\eqalign{ & = 12:30::30:x \cr & \Rightarrow 12x = 30 \times 30 \cr & \Rightarrow x = \frac{{\left( {30 \times 30} \right)}}{{12}} = 75 \cr} $$
∴ Third proportional to 12 and 30 = 75
Mean proportional between 9 and 25
$$\eqalign{ & = \sqrt {9 \times 25} = 15 \cr & \therefore {\text{Required ratio}} = 75:15 \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 5:1 \cr} $$

$$\eqalign{ & {\text{A}}:{\text{B}}:{\text{C}} = 8:14:22, \cr & {\text{B}}:{\text{C}}:{\text{D}} = 21:33:44 \cr & = \left( {21 \times \frac{2}{3}} \right):\left( {33 \times \frac{2}{3}} \right):\left( {44 \times \frac{2}{3}} \right) \cr & = 14:22:\frac{{88}}{3}. \cr & \therefore A:B:C:D = 8:14:22:\frac{{88}}{3} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 24:42:66:88 \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 12:21:33:44 \cr} $$

$$\eqalign{ & {\text{Abhinav's investment}} \cr & = {\text{Rs}}{\text{. 6000}} \cr & {\text{Sunil's investment}} \cr & = \left( {100 - 30} \right)\% {\text{ of Rs}}{\text{. 6000}} \cr & = {\text{70}}\% {\text{ of Rs}}{\text{. 6000}} \cr & = {\text{Rs}}{\text{.}}\left( {\frac{{70}}{{100}} \times 6000} \right) \cr & = {\text{Rs}}.4200 \cr & {\text{Rita's investment}} \cr & = \left( {{\text{100 + 25}}} \right)\% {\text{ of Rs}}{\text{. 4200}} \cr & = {\text{125}}\% {\text{ of Rs}}{\text{. }}42{\text{00}} \cr & = {\text{Rs}}{\text{.}}\left( {\frac{{125}}{{100}} \times 4200} \right) \cr & = {\text{Rs}}{\text{. }}5250 \cr & {\text{Total amount invested}} \cr & = {\text{Rs}}{\text{.}}\left( {{\text{6000}} + 42{\text{00}} + 5250} \right) \cr & = {\text{Rs}}{\text{. }}15450. \cr & \therefore {\text{Required Ratio}} \cr & = 5250:15450 \cr & = 35:103 \cr} $$