Question

The income of A is \( \frac{3}{5} \) 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. 1 : 2
• B. 2 : 1
• C. 1 : 3
• D. 2 : 3

Hint:

Use trial values (like assuming income = 100) when equations get messy. It helps visualize ratio-based questions more clearly.

Answer 1

The Correct Option is A

Let B’s income = \( I_B \), B’s expenditure = \( E_B \) Then, A’s income = \( \frac{3}{5} I_B \) A’s expenditure = \( \frac{4}{5} E_B \) Also given: A's income = \( \frac{9}{10} E_B \) So, \[ \frac{3}{5} I_B = \frac{9}{10} E_B \Rightarrow I_B = \frac{3}{5} \div \frac{9}{10} E_B = \frac{3}{5} \cdot \frac{10}{9} E_B = \frac{6}{9} E_B = \frac{2}{3} E_B \] Now, B’s income = \( \frac{2}{3} E_B \Rightarrow \) B’s savings = Income - Expenditure \[ S_B = I_B - E_B = \frac{2}{3} E_B - E_B = -\frac{1}{3} E_B \] Wait, savings can't be negative. Let’s use concrete values instead: Assume B's expenditure \( = 10 \) units Then: - A's income = \( \frac{9}{10} \cdot 10 = 9 \) - A's expenditure = \( \frac{4}{5} \cdot 10 = 8 \) → A's savings = \( 9 - 8 = 1 \) From earlier: A's income = \( \frac{3}{5} I_B \Rightarrow 9 = \frac{3}{5} I_B \Rightarrow I_B = 15 \) So B's savings = \( 15 - 10 = 5 \) Therefore, A : B savings = 1 : 5 → Wait, this contradicts the selected answer! Oh! Let's double-check. If A's income is \( \frac{3}{5} I_B \) and also \( \frac{9}{10} E_B \), then: \[ \frac{3}{5} I_B = \frac{9}{10} E_B \Rightarrow I_B = \frac{3}{5} \cdot \frac{10}{9} E_B = \frac{2}{3} E_B \Rightarrow B's income = \( \frac{2}{3} E_B \) \Rightarrow S_B = \frac{2}{3} E_B - E_B = -\frac{1}{3} E_B \text{ — negative} \] Use trial method: Let \( B \)'s income = 100 Then A's income = \( 60 \), A's expenditure = \( \frac{4}{5} \cdot B_{exp} \), and A's income = \( \frac{9}{10} \cdot B_{exp} \Rightarrow B_{exp} = 66.67 \) Now, - A's expenditure = \( \frac{4}{5} \cdot 66.67 = 53.33 \Rightarrow \) A’s saving = \( 60 - 53.33 = 6.67 \) - B’s saving = \( 100 - 66.67 = 33.33 \Rightarrow \) A : B = \( 6.67 : 33.33 = 1 : 5 \) So final correct ratio is 1:5 — the selected answer in image is **wrong**.