Raj invested Rs. $10000$ in a fund. At the end of first year, he incurred a loss but his balance was more than Rs. $5000$. This balance, when invested for another year, grew and the percentage of growth in the second year was five times the percentage of loss in the first year. If the gain of Raj from the initial investment over the two year period is $35 \%$, then the percentage of loss in the first year is

Given Information:

Initial investment: Rs. 10000

First year: Loss of some percentage, but balance > Rs. 5000

Second year: Growth percentage = 5 × (first year loss percentage)

Overall gain after 2 years: 35%

Find: Percentage loss in the first year

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Let the loss percentage in the first year = x%

Since the balance after first year is more than Rs. 5000:

Balance after first year > Rs. 5000

This means: 10000 × $\tfrac{100-x}{100}$ > 5000

$\tfrac{100-x}{100}$ > $\tfrac{1}{2}$

Therefore: 100-x > 50

So x < 50%

This constraint will help us choose the correct answer later.

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After first year: Balance = 10000 × $\tfrac{100-x}{100}$ = 10000(1 - $\tfrac{x}{100}$)

Second year growth: 5x% (five times the first year loss)

After second year: Balance = [First year balance] × $\tfrac{100+5x}{100}$

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The overall gain is 35%, so final amount = 10000 × 1.35 = Rs. 13500

Using the compound interest formula:

Final Amount = Initial × (1 - $\tfrac{x}{100}$) × (1 + $\tfrac{5x}{100}$)

13500 = 10000 × (1 - $\tfrac{x}{100}$) × (1 + $\tfrac{5x}{100}$)

1.35 = (1 - $\tfrac{x}{100}$) × (1 + $\tfrac{5x}{100}$)

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Converting to simpler fractions:

1.35 = $\tfrac{100-x}{100}$ × $\tfrac{100+5x}{100}$

1.35 × 100 × 100 = (100-x)(100+5x)

13500 = (100-x)(100+5x)

Expanding the right side:

13500 = 100² + 500x - 100x - 5x²

13500 = 10000 + 400x - 5x²

3500 = 400x - 5x²

700 = 80x - x²

x² - 80x + 700 = 0

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We need two numbers that multiply to 700 and add to -80.

Those numbers are -70 and -10.

x² - 80x + 700 = 0

(x - 70)(x - 10) = 0

Therefore: x = 70 or x = 10

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From our constraint, we know x < 50%

Since 70 > 50, we reject x = 70%

Since 10 < 50, we accept x = 10%

Therefore, the percentage loss in the first year = 10%

Answer: 10%