A man invests a total sum of Rs. 10,000 in a company. A part of the sum was invested at 10% simple interest per annum and the remaining part at 15% simple interest per annum. If the total interest accrued to him in two years equals Rs. 2,400, the sum invested at 15% simple interest per annum is:
Solving Simple Interest Investment Problems
This problem involves a man investing a total sum in two different parts at different simple interest rates. We need to find the amount invested at the 15% simple interest rate given the total investment, the time period, and the total interest earned.
Understanding the Concepts
Simple interest is calculated only on the principal amount, or on the portion of the principal that remains unpaid. The formula for simple interest is:
\( I = \frac{P \times R \times T}{100} \)
Where:
- \( I \) is the Simple Interest
- \( P \) is the Principal amount (the initial investment)
- \( R \) is the Rate of Interest per annum
- \( T \) is the Time period in years
Setting up the Problem
Let the total investment be \( P_{total} = \text{Rs. } 10,000 \).
The investment is split into two parts:
- Part 1: Invested at a rate \( R_1 = 10\% \) per annum.
- Part 2: Invested at a rate \( R_2 = 15\% \) per annum.
The time period for both investments is \( T = 2 \) years.
The total interest accrued in two years is \( I_{total} = \text{Rs. } 2,400 \).
Let the amount invested at 10% per annum be \( x \).
Then, the remaining amount invested at 15% per annum is \( 10,000 - x \).
Calculating Interest from Each Part
The simple interest earned from the first part (invested at 10%) is \( I_1 \):
\( I_1 = \frac{x \times 10 \times 2}{100} \)
\( I_1 = \frac{20x}{100} \)
\( I_1 = \frac{x}{5} \)
The simple interest earned from the second part (invested at 15%) is \( I_2 \):
\( I_2 = \frac{(10000 - x) \times 15 \times 2}{100} \)
\( I_2 = \frac{30(10000 - x)}{100} \)
\( I_2 = \frac{3(10000 - x)}{10} \)
Formulating the Total Interest Equation
The total interest is the sum of the interest from both parts:
\( I_1 + I_2 = I_{total} \)
\( \frac{x}{5} + \frac{3(10000 - x)}{10} = 2400 \)
Solving the Equation for x
To solve for \( x \), we can find a common denominator for the fractions, which is 10.
\( \frac{2x}{10} + \frac{3(10000 - x)}{10} = 2400 \)
Combine the fractions on the left side:
\( \frac{2x + 3(10000 - x)}{10} = 2400 \)
\( \frac{2x + 30000 - 3x}{10} = 2400 \)
\( \frac{30000 - x}{10} = 2400 \)
Multiply both sides by 10:
\( 30000 - x = 2400 \times 10 \)
\( 30000 - x = 24000 \)
Now, solve for \( x \):
\( x = 30000 - 24000 \)
\( x = 6000 \)
So, the amount invested at 10% per annum is Rs. 6,000.
Finding the Amount Invested at 15%
The amount invested at 15% per annum is \( 10000 - x \).
Amount at 15% = \( 10000 - 6000 \)
Amount at 15% = \( 4000 \)
Therefore, the sum invested at 15% simple interest per annum is Rs. 4,000.
Checking the Answer
Let's verify if the total interest is Rs. 2,400 with these amounts.
- Interest from Rs. 6,000 at 10% for 2 years: \( I_1 = \frac{6000 \times 10 \times 2}{100} = \frac{120000}{100} = 1200 \)
- Interest from Rs. 4,000 at 15% for 2 years: \( I_2 = \frac{4000 \times 15 \times 2}{100} = \frac{120000}{100} = 1200 \)
Total interest = \( I_1 + I_2 = 1200 + 1200 = 2400 \).
This matches the given total interest, confirming our calculation is correct.
| Investment Details | Amount Invested | Rate | Time | Simple Interest |
|---|---|---|---|---|
| Part 1 (10%) | Rs. 6,000 | 10% | 2 years | Rs. 1,200 |
| Part 2 (15%) | Rs. 4,000 | 15% | 2 years | Rs. 1,200 |
| Total | Rs. 10,000 | Rs. 2,400 |
Revision Table: Simple Interest Concepts
| Term | Description | Formula Component |
|---|---|---|
| Principal (P) | The initial amount of money invested or borrowed. | \(P\) in \( \frac{P \times R \times T}{100} \) |
| Rate (R) | The percentage at which interest is calculated per year. | \(R\) in \( \frac{P \times R \times T}{100} \) |
| Time (T) | The duration for which the money is invested or borrowed, usually in years. | \(T\) in \( \frac{P \times R \times T}{100} \) |
| Simple Interest (I) | The interest calculated only on the principal amount. | \(I\) in \( I = \frac{P \times R \times T}{100} \) |
| Total Amount (A) | The sum of the principal and the simple interest. | \( A = P + I \) |
Additional Information on Simple Interest
Simple interest problems often involve calculating interest for different principals, rates, or times, or finding one of the variables when others are known. In this case, we had a total principal split into parts with different rates, and we used the total interest to find the individual amounts.
Contrast this with compound interest, where interest is calculated on the initial principal plus all accumulated interest from previous periods. Simple interest is a more straightforward calculation, primarily used for short-term loans or basic financial calculations.
When solving investment problems with split amounts and total interest, setting up equations based on the simple interest formula for each part and summing the interest is a common approach. The problem then reduces to solving a linear equation or a system of linear equations.
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Important Questions from Simple Interest
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