Simple Interest Capital Investment Problem Solution

The question asks us to find the total capital invested by a man, given how he divided his capital and the simple interest earned from each part over one year. We are given the interest rates for each portion and the total interest earned.

Understanding the Problem

Let the total capital invested be \( P \). The capital is divided into three parts:

• Part 1: \( \frac{1}{4} \) of the capital at 7% per annum.
• Part 2: \( \frac{1}{4} \) of the capital at 8% per annum.
• Part 3: The remaining capital at 10% per annum.

The total interest earned in one year is Rs. 700.

Calculating the Third Part of the Capital

The first two parts together make up \( \frac{1}{4} + \frac{1}{4} = \frac{2}{4} = \frac{1}{2} \) of the total capital. The remaining capital is the total capital minus the sum of the first two parts:

Remaining Capital = \( P - \left( \frac{1}{4} P + \frac{1}{4} P \right) = P - \frac{1}{2} P = \frac{1}{2} P \)

So, the third part of the capital invested is \( \frac{1}{2} \) of the total capital.

Simple Interest Calculation for Each Part

The formula for Simple Interest (SI) is: \( SI = \frac{\text{Principal} \times \text{Rate} \times \text{Time}}{100} \)

Here, the time (T) is 1 year for all investments.

• Interest from Part 1: Principal \( = \frac{1}{4} P \), Rate \( = 7\% \), Time \( = 1 \) year.
  \( SI_1 = \frac{(\frac{1}{4} P) \times 7 \times 1}{100} = \frac{7P}{4 \times 100} = \frac{7P}{400} \)
• Interest from Part 2: Principal \( = \frac{1}{4} P \), Rate \( = 8\% \), Time \( = 1 \) year.
  \( SI_2 = \frac{(\frac{1}{4} P) \times 8 \times 1}{100} = \frac{8P}{4 \times 100} = \frac{8P}{400} \)
• Interest from Part 3: Principal \( = \frac{1}{2} P \), Rate \( = 10\% \), Time \( = 1 \) year.
  \( SI_3 = \frac{(\frac{1}{2} P) \times 10 \times 1}{100} = \frac{10P}{2 \times 100} = \frac{10P}{200} = \frac{20P}{400} \)

Calculating Total Simple Interest Earned

The total interest earned is the sum of the interest from all three parts:

Total SI = \( SI_1 + SI_2 + SI_3 \)
Total SI = \( \frac{7P}{400} + \frac{8P}{400} + \frac{20P}{400} \)
Total SI = \( \frac{7P + 8P + 20P}{400} = \frac{35P}{400} \)

Finding the Total Capital Invested

We are given that the total interest earned is Rs. 700. So, we can set up the equation:

\( \frac{35P}{400} = 700 \)

Now, we solve for \( P \):

\( 35P = 700 \times 400 \)
\( 35P = 280000 \)
\( P = \frac{280000}{35} \)

To simplify the division:

\( P = \frac{28000 \times 10}{35} \)

Divide 28000 by 35. \( 280 \div 35 = 8 \). So, \( 28000 \div 35 = 800 \).

\( P = 800 \times 10 \)
\( P = 8000 \)

The total capital invested is Rs. 8000.

Let's verify the interest for each part with P = 8000:

• Part 1 Capital: \( \frac{1}{4} \times 8000 = 2000 \). Interest: \( \frac{2000 \times 7 \times 1}{100} = 140 \)
• Part 2 Capital: \( \frac{1}{4} \times 8000 = 2000 \). Interest: \( \frac{2000 \times 8 \times 1}{100} = 160 \)
• Part 3 Capital: \( \frac{1}{2} \times 8000 = 4000 \). Interest: \( \frac{4000 \times 10 \times 1}{100} = 400 \)

Total Interest = \( 140 + 160 + 400 = 300 + 400 = 700 \), which matches the given information.

Conclusion

The total capital invested by the man is Rs. 8000.

Revision Table: Simple Interest Calculations

Additional Information: Simple Interest Concepts

What is Simple Interest?
Simple interest is a quick and easy method of calculating the interest charge on a loan or investment. It is calculated only on the principal amount, unlike compound interest which is calculated on the principal amount and the accumulated interest.

Key Components of Simple Interest:

• Principal (P): The initial amount of money borrowed or invested.
• Rate (R): The annual interest rate, usually expressed as a percentage.
• Time (T): The duration for which the money is borrowed or invested, usually in years.
• Interest (SI): The amount paid for using the principal amount over the given time period.

Calculating Remaining Capital:
When a capital is divided into fractions, the remaining fraction can be found by subtracting the sum of the known fractions from 1 (representing the whole capital). If fractions are \( f_1, f_2, ..., f_n \), the remaining fraction is \( 1 - (f_1 + f_2 + ... + f_n) \).

Weighted Average Interest Rate:
In problems like this, you can think of an effective weighted average interest rate for the entire capital. The total interest is \( \frac{35P}{400} \). Since Total SI \( = \frac{P \times R_{avg} \times 1}{100} \), we have \( \frac{35P}{400} = \frac{P \times R_{avg}}{100} \). This gives \( \frac{35}{4} = R_{avg} \), or \( R_{avg} = 8.75\% \). So, the total capital \( P \) invested at an effective rate of 8.75% for 1 year yields Rs. 700 interest. \( P = \frac{700 \times 100}{8.75} = \frac{70000}{8.75} = \frac{7000000}{875} = 8000 \).