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CAT 2021 Slot 2 — QA Question 12

Successive Percentage ChangeEasy

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 year period is 35%, then the percentage of loss in the first year is 

Answer & solution

  • 10

  • B

    15

  • C

    5

  • D

    70

Solution

Easy

Let the first-year loss be P%P\%. Then the second-year growth is 5P%5P\%. Apply the two successive multiplying factors to the principal and set the overall growth equal to 35%35\%. Solve the resulting quadratic, then use the "balance after year 1 exceeds 5000" condition to reject the extraneous root.

1

Write the two-year balance as a product of factors. A loss of P%P\% multiplies by (1−P100)\left(1-\tfrac{P}{100}\right); a growth of 5P%5P\% multiplies by (1+5P100)\left(1+\tfrac{5P}{100}\right).

Balance=10000(1−P100)(1+5P100)Overall gain 35% ⇒ Balance=10000×1.35\begin{aligned} &\text{Balance}=10000\left(1-\frac{P}{100}\right)\left(1+\frac{5P}{100}\right)\\ &\text{Overall gain }35\%\ \Rightarrow\ \text{Balance}=10000\times 1.35 \end{aligned}
2

Form and simplify the equation. Equate and expand.

10000(1−P100)(1+5P100)=13500⇒ 10000−5P2+400P=13500(expand)⇒ 5P2−400P+3500=0⇒ P2−80P+700=0(divide by 5)\begin{aligned} &10000\left(1-\frac{P}{100}\right)\left(1+\frac{5P}{100}\right)=13500\\ &\Rightarrow\ 10000-5P^{2}+400P=13500 \quad\text{(expand)}\\ &\Rightarrow\ 5P^{2}-400P+3500=0\\ &\Rightarrow\ P^{2}-80P+700=0 \quad\text{(divide by 5)} \end{aligned}
3

Solve and pick the valid root. Factor the quadratic.

P2−80P+700=0⇒ (P−10)(P−70)=0⇒ P=10  or  P=70\begin{aligned} &P^{2}-80P+700=0\\ &\Rightarrow\ (P-10)(P-70)=0\\ &\Rightarrow\ P=10 \ \text{ or } \ P=70 \end{aligned}

The balance after year 1 must exceed 50005000, i.e. 10000(1−P100)>500010000\left(1-\tfrac{P}{100}\right)>5000, so P<50P<50. Hence P=70P=70 is rejected.

P=10%P=10\%
CAT 2021 Slot 2 QA Q12: Raj invested ₹ 10000 in a fund. At the end of first year, he incurred a loss but his balance was more than ₹ 5 — Solution | TheCATExam