After two successive increments, Gopal's salary became $187.5 \%$ of his initial salary. If the percentage of salary increase in the second increment was twice of that in the first increment, then the percentage of salary increase in the first increment was

27.5

✅ Correct Option: 2

Let the first increment $=x\%$

Then the second increment $=2x\%$

The key insight here is that Gopal's salary became $187.5\%$ of his initial salary.

If his original salary was $100\%$ , then his salary increased by $87.5\%$ in total.

When we have successive percentage increases, we cannot simply add them together.

If your salary increases by $10\%$ first, then by $10\%$ next, the second increase is calculated on the already increased salary, not the original one. Hence, there is a compounding effect, and the total increase would be more than $20\%$ .

For two successive percentage changes $a\%$ and $b\%$ , the net percentage change is:

Net $\%$ change $= a + b + \tfrac{a \times b}{100}$

The third term $\tfrac{a \times b}{100}$ accounts for the compound effect.

We substitute our values:

$a = x$

$b = 2x$

Net change $= 87.5\%$

$87.5 = x + 2x + \tfrac{x \times 2x}{100}$

$87.5 = 3x + \tfrac{x \times 2x}{100}$

$87.5 = 3x + \tfrac{x \times x}{100}$

If we expand the equation, we will have to solve for the root. We can see there is $3x$ and can reasonably use $x = 25$ to test.

$75 + \tfrac{25 \times 25}{50} = 75 + 12.5 = 87.5$

Therefore, the percentage of salary increase in the first increment was 25%.

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After two successive increments, Gopal's salary became $187.5 \%$ of his initial salary. If the percentage of salary increase in the second increment was twice of that in the first increment, then the percentage of salary increase in the first increment was