A person invested $$\frac{1}{4}$$​ of his capital at $$3\frac{1}{2}$$​%, $$\frac{1}{3}$$​ at $$7\frac{1}{2}$$​% and the remaining at 8% simple interest. If the amount of the yearly simple interest is ₹2,576 then what is the amount of the capital?

Correct option is C

​Given:

Let the total capital be ₹x.

​$$\frac{1}{4}$$ of capital invested at $$3\frac{1}{2}\%$$​ per annum.

​$$\frac{1}{3}$$​ of capital invested at $$7\frac{1}{2}\%$$​ per annum.

The remaining capital invested at 8% per annum.

Total yearly simple interest = ₹ 2,576.

Time period = 1 year (since it is yearly interest).

Concept Used:

The formula for simple interest:

​$$\text{SI} = \frac{P \times R \times T}{100}$$​

where P = Principal, R = Rate of interest, and T = Time.

Solution:

Expressing Principal Amounts

First investment:$$P_1 = \frac{x}{4}, Rate = 3.5\%$$​​

​$$SI_1 = \frac{\frac{x}{4} \times 3.5 \times 1}{100} = \frac{3.5x}{400}$$​

Second investment:$$P_2 = \frac{x}{3}, Rate = 7.5\%$$​​

​$$SI_2 = \frac{\frac{x}{3} \times 7.5 \times 1}{100} = \frac{7.5x}{300} = \frac{2.5x}{100}$$​

Remaining investment:

​$$P_3 = x - \left(\frac{x}{4} + \frac{x}{3}\right)$$​

​$$P_3 = x - \frac{7x}{12} = \frac{5x}{12}$$​

Third investment Rate $$= 8\%$$​​

​$$SI_3 = \frac{\frac{5x}{12} \times 8 \times 1}{100} = \frac{40x}{1200} = \frac{x}{30}$$​

​$$SI_1 + SI_2 + SI_3 = 2,576$$​

$$\frac{3.5x}{400} + \frac{2.5x}{100} + \frac{x}{30} = 2,576$$

​$$\frac{5.25x + 15x + 20x}{600} = 2,576$$​​​

x = $$\frac{15,45,600}{40.25} = 38,400$$​​

The total capital is ₹38,400.