Question

The income of \( A \) is \( \frac{3}{4} \) of \( B \)'s income, and the expenditure of \( A \) is \( \frac{4}{5} \) of \( B \)'s expenditure. If \( A \)'s income is \( \frac{9}{10} \) of \( B \)'s expenditure, then the ratio of savings of \( A \) and \( B \) is:

• A. 2 : 1
• B. 3 : 5
• C. 1 : 3
• D. 2 : 3

Hint:

When dealing with ratios, express all terms in terms of one variable and then solve for the ratio.

Answer 1

The Correct Option is C

Let the income of \( B \) be \( x \), so the income of \( A \) will be \( \frac{3}{4}x \). Let the expenditure of \( B \) be \( y \), so the expenditure of \( A \) will be \( \frac{4}{5}y \). The savings of \( A \) will be: \[ \text{Savings of A} = \frac{3}{4}x - \frac{4}{5}y \] The savings of \( B \) will be: \[ \text{Savings of B} = x - y \] Now, using the relation that \( A \)'s income is \( \frac{9}{10} \) of \( B \)'s expenditure: \[ \frac{3}{4}x = \frac{9}{10}y \] Solving for \( x \) in terms of \( y \): \[ x = \frac{3}{4} \times \frac{10}{9}y = \frac{5}{6}y \] Now substitute this into the savings equations: \[ \text{Savings of A} = \frac{3}{4} \times \frac{5}{6}y - \frac{4}{5}y = \frac{15}{24}y - \frac{4}{5}y \] Simplifying the terms: \[ \text{Savings of A} = \frac{75}{120}y - \frac{96}{120}y = \frac{-21}{120}y = -\frac{7}{40}y \] Similarly, savings of \( B \) will be: \[ \text{Savings of B} = \frac{5}{6}y - y = \frac{-1}{6}y \] Thus, the ratio of savings of A and B is: \[ \frac{\text{Savings of A}}{\text{Savings of B}} = \frac{-7/40}{-1/6} = 1 : 3 \]