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A man invested one-third of his capital at $7\mathrm{%}$$7\%$, one-fourth at $8\mathrm{%}$$8\%$ and the remainder at $10\mathrm{%}$$10\%$ simple interest, respectively. If his annual income is Rs. 561, the capital is worth of:A) Rs. 5400B) Rs. 6000C) Rs. 6600D) Rs. 7200

Last updated date: 14th Jun 2024
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Hint: In these types of questions we need to assume some variable value of the capital and form an equation according to the given condition in the question.

Complete step by step solution: Let, the worth of the capital be Rs x.

Given, the man invests one-third of his capital at $7\%$, one-fourth at $8\%$ and the remaining at $10\%$.

The remaining capital would be , $1 - \left( {\dfrac{1}{3} + \dfrac{1}{4}} \right) = 1 - \dfrac{7}{{12}} = \dfrac{5}{{12}}$, i.e., $\dfrac{5}{{12}}$th at $10\%$.

If, the annual income is Rs 561.

Let the total worth of the capital be x.

Therefore, according to the question,

$\;7\% \,{\text{of}}\,\dfrac{1}{3}x + 8\% \,{\text{of}}\,\dfrac{1}{4}x + 10\% \,{\text{of}}\,\dfrac{5}{{12}}x = 561$.

So,

$\;\dfrac{7}{{100}}\left( {\dfrac{1}{3}x} \right) + \dfrac{8}{{100}}\left( {\dfrac{1}{4}x} \right) + \dfrac{{10}}{{100}}\left( {\dfrac{5}{{12}}x} \right) = 561$

Now, take x common from the equation and simplify the equation,

$x\left( {\dfrac{7}{{300}} + \dfrac{8}{{400}} + \dfrac{{50}}{{1200}}} \right) = 561$

$x\left( {\dfrac{{28 + 24 + 50}}{{1200}}} \right) = 561$

$x\left( {\dfrac{{102}}{{1200}}} \right) = 561$

$x = 561 \times \dfrac{{1200}}{{102}}$

$x = 6600$

Therefore, the worth of the capital is Rs 6600.

1. Man has invested $7\%$ of his one-third capital, i.e., $0.07$ times $\dfrac{x}{3}$.

2. Man has invested $8\%$ of his one-third capital, i.e., $0.08$ times $\dfrac{x}{4}$.

3. Man has also invested remaining capitals, i.e., $\dfrac{5}{{12}}x$ at $10\%$ which is $0.1$ times

$\dfrac{{5x}}{{12}}$.

4. According to the question, if we sum up all the above values it will come out to be 561, which will lead

to the value of x, and that will be the value of the capital that is possessed by the man.

Note: In this case, if we assume some value for the worth of capital, we can easily interpret its value, here in this question we have assumed it to be x, as it is given in the question that one-third of his capital at $7\%$, one-fourth at $8\%$ and the remainder(i.e., $\dfrac{5}{{12}}th$) at $10\%$ gives annual income as Rs 561. So, if we add $7\%$ of one-third x, $8\%$ of one-fourth x and $10\%$ of $\dfrac{5}{{12}}th$x and equate it to 561, then the value of x, can be calculated easily, which will come out to be Rs 6600.