Ratio - Aptitude MCQ Questions and Solutions with Explanations

$$\eqalign{ & \Rightarrow \frac{x}{y} = \frac{{\left( {\frac{a}{b}} \right)}}{{\left( { - \frac{b}{a}} \right)}} = - \frac{{{a^2}}}{{{b^2}}} \cr & \Rightarrow y = \left( { - \frac{{{b^2}}}{{{a^2}}}} \right)x \cr & \therefore x - y = \frac{{\text{a}}}{{\text{b}}}{\text{ + }}\frac{{\text{b}}}{{\text{a}}} \cr & \Rightarrow x + \frac{{{b^2}}}{{{a^2}}}x = \frac{{{a^2} + {b^2}}}{{ab}} \cr & \Rightarrow x\left( {\frac{{{a^2} + {b^2}}}{{{a^2}}}} \right) = \frac{{{a^2} + {b^2}}}{{ab}} \cr & \Rightarrow x = \frac{{{a^2}}}{{ab}} = \frac{a}{b} \cr} $$

a + b = ck,
b + c = ak,
c + a = bk
Adding, we get : 2(a + b + c) = ak + bk + ck
⇒ k(a + b + c) or k = 2

$$\eqalign{ & x:y = 3:4 \cr & \therefore \frac{x}{y} = \frac{3}{4} \cr & \Rightarrow \frac{{4x - y}}{{2x + 3y}} \cr & \Rightarrow \frac{{y\left( {4\frac{x}{y} - 1} \right)}}{{y\left( {2\frac{x}{y} + 3} \right)}} \cr & \Rightarrow \frac{{4 \times \frac{3}{4} - 1}}{{2 \times \frac{3}{4} + 3}} \cr & \Rightarrow \frac{{\left( {3 - 1} \right) \times 2}}{{3 + 6}} \cr & \Rightarrow \frac{4}{9} \cr & \Rightarrow 4:9 \cr} $$

	A	:	B	:	C
	3	:	4	:	5
Let	3x	:	4x	:	5x

∴ Sum of (smallest + largest)
A + C = 8x
According of the question,
8x = 4x + 52
4x = 52
x = 13
∴ Smallest number is = A
⇒ 13 × 3 = 39

Quantity of A in mixture 7x and B in Mixture in 5x
Quantity of A in Mixture left after 9 liter Drawn = $$\left( {7x - \frac{7}{{12}} \times 9} \right) = \left( {7x - \frac{{21}}{4}} \right)$$     liters
Quantity of B in Mixture left after 9 liter Drawn = $$\left( {7x - \frac{5}{{12}} \times 9} \right) = \left( {7x - \frac{{15}}{4}} \right)$$     liters
$$\eqalign{ & \therefore \frac{{\left( {7x - \frac{{21}}{4}} \right)}}{{\left( {5x - \frac{{15}}{4}} \right) + 9}} = \frac{1}{1} \cr & \Rightarrow \frac{{28x - 21}}{{20x + 21}} = 1 \cr & \Rightarrow 28x - 21 = 20x + 21 \cr & \Rightarrow x = \frac{{42}}{8} = \frac{{21}}{4} \cr} $$

∴Quantity of A in mixture = $$7x$$
$$ = 7 \times \frac{{21}}{4} = \frac{{147}}{4} = 36\frac{3}{4}$$     liter.

$$\eqalign{ & {\text{a}}:{\text{b}} = {\text{2}}:{\text{3}}, \cr & {\text{b}}:{\text{c}} = {\text{4}}:{\text{5}} = {\text{4}} \times \frac{3}{4}:5 \times \frac{3}{4} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 3:\frac{{15}}{4} \cr & So,a:b:c = 2:3:\frac{{15}}{4} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 8:12:15. \cr & {\text{Let}} \cr & {\text{a}} = 8{\text{k}}, \cr & {\text{b}} = 12{\text{k}}, \cr & {\text{c}} = 15{\text{k}}. \cr & {\text{Then,}} \cr & \frac{{\left( {{\text{a}} + {\text{b}}} \right)}}{{\left( {{\text{b}} + {\text{c}}} \right)}} = \frac{{\left( {8{\text{k}} + 12{\text{k}}} \right)}}{{\left( {12{\text{k}} + 15{\text{k}}} \right)}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{20{\text{k}}}}{{27{\text{k}}}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{20}}{{27}} \cr} $$

Let the fourth proportional to 5, 8, 15 be x.
Then,
$$\eqalign{ & \Rightarrow 5:8:15:x \cr & \Rightarrow 5x = 8 \times 15 \cr & \Rightarrow x = \frac{{8 \times 15}}{5} = 24. \cr} $$

Let the fourth proportional to 0.12, 0.21 and 8 be x
$$\eqalign{ & {\text{Then,}} \cr & = {\text{0}}{\text{.12}}:{\text{0}}{\text{.21}}::{\text{8}}:x \cr & \Rightarrow {\text{0}}{\text{.12}}x = 0.21 \times 8 \cr & \Rightarrow x = \frac{{0.21 \times 8}}{{0.12}} \cr & \,\,\,\,\,\,\,\,\,\,\,\, = \frac{{21 \times 8}}{{12}} \cr & \,\,\,\,\,\,\,\,\,\,\,\, = 14. \cr} $$

$$\eqalign{ & {\text{Acid}}:{\text{Water}} \cr & \,\,\,\,\,4\,\,\,\,\,:\,\,\,\,\,\,\,1 \cr & {\text{Let }}4x + x = 5x \cr & \because 4x + x = 25 \cr & \Rightarrow x = 5 \cr & \therefore {\text{Acid }} = {\text{ }}5 \times 4 = 20 \cr & {\text{Water }} = 5 \times 1 = 5 \cr & \therefore {\text{3 Litres water is added }} \cr & {\text{So, new quantity of water}} \cr & = 5 + 3 = 8 \cr & \therefore {\text{Acid}}:{\text{Water}} \cr & \,\,\,\,\,\,\,\,\,{\text{20}}\,\,\,:\,\,\,\,\,\,\,8 \cr & \,\,\,\,\,\,\,\,\,\,\,5\,\,\,\,:\,\,\,\,\,\,\,2 \cr} $$

	A	:	B
Income	9	:	7
Expense	4	:	3

Income - Savings + Expenditure
$$\eqalign{ & \therefore \frac{{9x - 200}}{{7x - 200}} = \frac{4}{3} \cr & \Rightarrow 27x - 600 = 28x - 800 \cr & \Rightarrow x = 200 \cr & {\text{Sum of weekly income}} \cr & = 9x + 7x = 16x \cr & = 16 \times 200 = {\text{Rs}}.\,3200 \cr} $$