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Raj invested ₹$10000$ in a fund. At the end of first year, he incurred a loss but his balance was more than ₹$5000.$ 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 period is $35 \%,$ then the percentage of loss in the first year is

  1. $15$
  2. $10$
  3. $70$
  4. $5$
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Let the percentage loss he had at the end of first be $x\%.$ Then at the end of the second year his gain is $5x\%.$

Now, $-x+5x+\frac{(-x) \times (5x)}{100} = 35 \quad [\because \color{Blue}{\text{Successive percentage}} ]$

$\Rightarrow 4x–\frac{x^{2}}{20} = 35$

$\Rightarrow 80x-x^{2} = 700$

$\Rightarrow x^{2}-80x+700 = 0$

$\Rightarrow x^{2}-70x-10x+700 = 0$

$\Rightarrow x(x-70)-10(x-70) = 0$

$\Rightarrow (x-70)(x-10) = 0$

$\Rightarrow x=10 \; \text{(or)}\; x=70\;\color{Red}{\text{(rejected)}}$

$\Rightarrow \boxed{x=10}$

$\therefore$ The percentage of loss in the first year is $10\%.$

Correct Answer $:\text{B}$

$\textbf{PS:}$ Let the successive increase in percentages be $a\%$ and $b\%$. Then, the total increase will be $\left(a+b+\frac{ab}{100}\right)\%.$

  • If there's an increase and a decrease, in that case, the decrease will be considered a negative value.
  • In the case of discounts, the value of discount percentages will be considered negative.
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