Ratio - Aptitude MCQ Questions and Solutions with Explanations

$$\eqalign{ & {\text{A}}:{\text{B}} = \frac{1}{2}:\frac{3}{8}, \cr & \frac{{\text{A}}}{{\text{B}}} = \frac{4}{3} \cr & {\text{B}}:{\text{C}} = \frac{1}{3}:\frac{5}{9}, \cr & \frac{{\text{B}}}{{\text{C}}} = \frac{1}{3} \times \frac{9}{5} = \frac{3}{5} \cr & {\text{C}}:{\text{D}} = \frac{5}{6}:\frac{3}{4}, \cr & \frac{{\text{C}}}{{\text{D}}} = \frac{{5 \times 4}}{{6 \times 3}} = \frac{{10}}{9} \cr & {\text{A}}:{\text{B}}:{\text{C}}:{\text{D}} \cr & {\text{4}}:{\text{3}} \cr & \,\,\,\,\,\,\,{\text{3}}:{\text{5}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,{\text{ 10}}:{\text{9}} \cr & \Rightarrow {\text{A}}:{\text{B}}:{\text{C}}:{\text{D}} \cr & \,\,\,\,\,\,\,\,\,8:6:10:9 \cr} $$

$$\eqalign{ & {\text{A}}:{\text{B}}:{\text{C}} = 2:3:4 \cr & {\text{Let }} \cr & {\text{A}} = 2x,{\text{B}} = 3x,{\text{C}} = 4x \cr & \therefore \frac{{\text{A}}}{{\text{B}}}:\frac{{\text{B}}}{{\text{C}}}:\frac{{\text{C}}}{{\text{A}}} = \frac{{2x}}{{3x}}:\frac{{3x}}{{4x}}:\frac{{4x}}{{2x}} \cr & \Rightarrow \frac{2}{3}:\frac{3}{4}:\frac{2}{1} \cr} $$
Multiply by the L.C.M of denominator to remove fraction
$$\eqalign{ & {\text{So,}} \cr & {\text{L}}{\text{.C}}{\text{.M of }}\left( {3,4,1} \right) = 12 \cr & \therefore \frac{{\text{A}}}{{\text{B}}}\,\,\,\,\,:\,\,\,\,\,\frac{{\text{B}}}{{\text{C}}}\,\,\,\,\,\,:\,\,\,\,\,\frac{{\text{C}}}{{\text{A}}} \cr & \frac{2}{3} \times 12:\frac{3}{4} \times 12:\frac{2}{1} \times 12 \cr & \,\,\,\,\,\,\,8\,\,\,\,\,\,\,:\,\,\,\,\,\,\,9\,\,\,\,\,\,\,:\,\,\,\,\,\,\,24 \cr} $$

$$\eqalign{ & = \frac{{\text{A}}}{{\text{B}}} = \frac{2}{3},\,\frac{{\text{B}}}{{\text{C}}} = \frac{2}{4},\,\frac{{\text{C}}}{{\text{D}}} = \frac{2}{5} \cr & \Rightarrow \frac{{\text{A}}}{{\text{D}}} = \left( {\frac{{\text{A}}}{{\text{B}}} \times \frac{{\text{B}}}{{\text{C}}} \times \frac{{\text{C}}}{{\text{D}}}} \right) \cr & \Rightarrow \frac{2}{3} \times \frac{2}{4} \times \frac{2}{5} = \frac{2}{{15}} \cr & \Rightarrow {\text{A}}:{\text{D}} = 2:15. \cr} $$

$$\eqalign{ & = {\text{3A}} = {\text{5B and 4B}} = {\text{6C}} \cr & \Rightarrow \frac{{\text{A}}}{{\text{B}}} = \frac{5}{3}{\text{ and }}\frac{{\text{B}}}{{\text{C}}} = \frac{6}{4} = \frac{3}{2} \cr & \Rightarrow \frac{{\text{A}}}{{\text{C}}} = \left( {\frac{{\text{A}}}{{\text{B}}} \times \frac{{\text{B}}}{{\text{C}}}} \right) = \frac{5}{3} \times \frac{3}{2} = \frac{5}{2} \cr & \Rightarrow {\text{A}}:{\text{C}} = {\text{5}}:{\text{2}} \cr} $$

$$\eqalign{ & = \frac{{\text{A}}}{{\text{B}}} = \frac{2}{1},\,\,\frac{{\text{B}}}{{\text{C}}} = \frac{4}{3},\,\,\frac{{\text{C}}}{{\text{D}}} = \frac{5}{6} \cr & \Rightarrow \frac{{\text{A}}}{{\text{D}}} = \left( {\frac{{\text{A}}}{{\text{B}}} \times \frac{{\text{B}}}{{\text{C}}} \times \frac{{\text{C}}}{{\text{D}}}} \right) \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\, = \left( {\frac{2}{1} \times \frac{4}{3} \times \frac{5}{6}} \right) \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{20}}{9} \cr & \Rightarrow {\text{A}}:{\text{D}} = {\text{20}}:{\text{9}} \cr} $$

a : b = 1 : 2
c : d = 1 : 2
e : f = 1 : 2
∴ a = x, b = 2x, c = x, d = 2x, e = x, f = 2x
$$\eqalign{ & \therefore \frac{{3 \times x + 5 \times 7 + 7 \times x}}{{3 \times 2x + 5 \times 2x + 7 \times 2x}} \times 1 \cr & \Rightarrow \frac{{15x}}{{30x}} \cr & \Rightarrow \frac{1}{2} \cr} $$

$$\eqalign{ & {\text{A}}:{\text{B}}:{\text{C}} \cr & {\text{ }}8:9 \cr & \,\,\,\,\,\,\,\,\,3:4 \cr & \boxed{8:9:12} \cr & 8x:9x:12x \cr & \therefore 8x \times 12x = 2400 \cr & \Rightarrow {x^2} = \frac{{2400}}{{12 \times 8}} \cr & \Rightarrow x = 5 \cr & \therefore {\text{B}} = 9 \times 5 = 45 \cr} $$

Income Ratio of A and B = 4 : 3
Expenditure ratio of A and B = 3 : 2
Saving Ratio of A and B = 1 : 1
∴1unit = 600
Income of A = 4 × 600 = Rs. 2400

$$\eqalign{ & = {\text{A}}:{\text{B}} = 7:9\,{\text{and}} \cr & {\text{B}}:{\text{C}} = 5:4 \cr & = \left( {5 \times \frac{9}{5}} \right):\left( {4 \times \frac{9}{5}} \right) \cr & = 9:\frac{{36}}{5}. \cr & \Rightarrow {\text{A}}:{\text{B}}:{\text{C}} = 7:9:\frac{{36}}{5} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 35:45:36. \cr} $$

$$\eqalign{ & = A:B = \frac{1}{2}:\frac{1}{3} = 3:2, \cr & = {\text{B}}:{\text{C}} = \frac{1}{2}:\frac{1}{3} = 3:2 \cr & = \left( {3 \times \frac{2}{3}} \right):\left( {2 \times \frac{2}{3}} \right) \cr & = 2:\frac{4}{3}. \cr & \Rightarrow {\text{A}}:{\text{B}}:{\text{C}} = 3:2:\frac{4}{3} \cr & = 9:6:4 \cr} $$