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The income of Amala is $20\%$ more than that of Bimala and $20\%$ less than that of Kamala. If Kamala's income goes down by $4\%$ and Bimala’s goes up by $10\%$, then the percentage by which Kamala's income would exceed Bimala’s is nearest to

1. $31$
2. $28$
3. $32$
4. $29$

Let the income of Bimala $= 100x$

The income of Amala is $20\%$ more than that of Bimala.

So, the income of Amala $= 100x \times \frac{120}{100} = 120x$

The ratio of Bimala and Amala $= 100x : 120x = 5:6 \quad \longrightarrow (1)$

Let the income of Kamala $= 100y$

The income of Amala $20\%$ less than that of Kamala.

So, the income of Amala $= 100y \times \frac{80}{100} = 80x$

The ratio fo Kamala and Amala $= 100y : 80y = 5:4 \quad \longrightarrow (2)$

Now, combining equation $(1)$ and $(2)$, we get

• Bimala : Amala $= (5 : 6) \times 4 = 20: 24$
• Kamala : Amala $= (5 : 4) \times 6 = 30: 24$

The ratio of Bimala : Amala : Kamala $= 20 : 24 : 30 = 10 : 12 : 15$

If Kamala's income goes down by $4\%$ and Bimala’s goes up by $10\%$. Then,

• Kamala’s income $= 15 \times \frac{96}{100} = 14 . 4$
• Bimala’s income $= 10 \times \frac {110}{100} = 11$

$\therefore$ The percentage by which kamala’s income would exceed Bimala’s income $= \frac {(14 . 4 – 11)}{11} \times 100 = \frac {3 .4}{11} \times 100 = 30 . 90 \approx 31.$

Correct Answer$: \text{A}$

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