A person invested a total amount of Rs $15$ lakh. A part of it was invested in a fixed deposit earning $6\%$ annual interest, and the remaining amount was invested in two other deposits in the ratio $2: 1$, earning annual interest at the rates of $4\%$ and $3\%$, respectively. If the total annual interest income is Rs $76000$ then the amount (in Rs lakh) invested in the fixed deposit was

We have a person who invested Rs 15 lakh in three different deposits:

Part 1: Fixed deposit earning 6% annual interest

Part 2: Two other deposits in ratio 2:1, earning 4% and 3% respectively

Total annual interest: Rs 76000

We need to find how much was invested in the fixed deposit.

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Let's say the amount invested in the fixed deposit = x lakh

This means the remaining amount = (15 - x) lakh was split between the other two deposits.

When an amount is divided in the ratio 2:1, it means:

First part gets 2/(2+1) = 2/3 of the total

Second part gets 1/(2+1) = 1/3 of the total

So:

Amount at 4% interest = (2/3) × (15 - x) lakh

Amount at 3% interest = (1/3) × (15 - x) lakh

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The total interest is given as Rs 76000, which equals 0.76 lakh.

Since our principal amounts are in lakh, keeping interest in lakh makes calculations cleaner.

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Simple Interest = (Principal × Rate × Time) / 100

For 1 year, the formula becomes: Interest = (Principal × Rate) / 100

Total Annual Interest = Sum of interests from all three deposits

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Interest from fixed deposit + Interest from 4% deposit + Interest from 3% deposit = 0.76

$\dfrac{x \times 6}{100} + \dfrac{\frac{2}{3}(15-x) \times 4}{100} + \dfrac{\frac{1}{3}(15-x) \times 3}{100} = 0.76$

$\dfrac{6x}{100} + \dfrac{8(15-x)}{300} + \dfrac{3(15-x)}{300} = 0.76$

$\dfrac{6x}{100} + \dfrac{8(15-x) + 3(15-x)}{300} = 0.76$

$\dfrac{6x}{100} + \dfrac{11(15-x)}{300} = 0.76$

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Multiply everything by 300:

$\dfrac{6x \times 300}{100} + 11(15-x) = 0.76 \times 300$

$18x + 11(15-x) = 228$

$18x + 165 - 11x = 228$

$7x = 228 - 165$

$7x = 63$

$x = 9$

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The amount invested in the fixed deposit was 9 lakh.

When dealing with investment problems involving ratios, we always break down the remaining amount according to the given ratio before applying interest calculations. Converting all units to the same denomination makes the arithmetic much cleaner.