A sum of money at simple interest amounts to Rs. 6,000 in 4 years and to Rs. 6,750 in 7 years at the same rate per cent p.a. of interest. The sum (in Rs.) is:
5,000
Simple Interest Calculation: Finding the Original Sum
This question asks us to find the original sum of money (the principal) that was invested at a simple interest rate. We are given the amounts the sum grew to after two different periods of time.
Understanding Simple Interest
Simple interest is calculated only on the initial principal amount. The interest earned each year remains the same.
The total amount at the end of a period is the sum of the principal and the total simple interest earned:
\( \text{Amount} = \text{Principal} + \text{Simple Interest} \)
The simple interest earned for a period is directly proportional to the time period.
Analyzing the Given Information
We are given two pieces of information about the amount the sum grows to:
- Amount after 4 years = Rs. 6,000
- Amount after 7 years = Rs. 6,750
Both amounts include the original principal plus the simple interest earned over the respective time periods.
Calculating the Interest Earned Over the Difference in Time
The difference between the amounts at 7 years and 4 years is the simple interest earned during the period from the end of year 4 to the end of year 7. This period is \( 7 - 4 = 3 \) years.
The increase in amount in 3 years is:
\( \text{Increase} = \text{Amount after 7 years} - \text{Amount after 4 years} \)
\( \text{Increase} = 6750 - 6000 = 750 \)
So, the simple interest earned in 3 years is Rs. 750.
Calculating the Annual Simple Interest
Since simple interest is constant every year, we can find the simple interest earned in one year:
\( \text{Annual Simple Interest} = \frac{\text{Simple Interest in 3 years}}{\text{Number of years}} \)
\( \text{Annual Simple Interest} = \frac{750}{3} = 250 \)
The simple interest earned per year is Rs. 250.
Calculating the Total Simple Interest in 4 Years
To find the original sum, we can use the amount after 4 years (Rs. 6,000). This amount consists of the principal plus the simple interest earned over 4 years.
\( \text{Total Simple Interest in 4 years} = \text{Annual Simple Interest} \times \text{Number of years} \)
\( \text{Total Simple Interest in 4 years} = 250 \times 4 = 1000 \)
The simple interest earned over 4 years is Rs. 1,000.
Finding the Original Sum (Principal)
We know that:
\( \text{Amount after 4 years} = \text{Principal} + \text{Total Simple Interest in 4 years} \)
We can rearrange this to find the Principal:
\( \text{Principal} = \text{Amount after 4 years} - \text{Total Simple Interest in 4 years} \)
\( \text{Principal} = 6000 - 1000 = 5000 \)
The original sum (principal) is Rs. 5,000.
Verification (Optional)
Let's quickly verify using the amount after 7 years:
Total Simple Interest in 7 years = Annual Simple Interest \(\times\) 7 = \(250 \times 7 = 1750\)
Principal + Total Simple Interest in 7 years = \(5000 + 1750 = 6750\)
This matches the given amount after 7 years, confirming our principal calculation is correct.
| Description | Value (Rs.) | Time (Years) |
|---|---|---|
| Amount after 4 years | 6,000 | 4 |
| Amount after 7 years | 6,750 | 7 |
| Interest earned in (7-4) years | \(6750 - 6000 = 750\) | 3 |
| Annual Simple Interest | \(750 / 3 = 250\) | 1 |
| Total Interest in 4 years | \(250 \times 4 = 1000\) | 4 |
| Principal (Sum) | \(6000 - 1000 = 5000\) | 0 |
The sum of money is Rs. 5,000.
Revision Table: Simple Interest Concepts
| Term | Definition | Formula (Simple Interest) |
|---|---|---|
| Principal (P) | The initial amount of money invested or borrowed. This is the sum we calculated. | N/A (Base amount) |
| Interest (I) | The extra money earned on the principal or paid for borrowing. In simple interest, it's constant per year based on P. | \( I = \frac{P \times R \times T}{100} \) (where R is rate, T is time) |
| Rate (R) | The percentage at which interest is calculated, usually per year (p.a.). | N/A |
| Time (T) | The duration for which the money is invested or borrowed. | N/A |
| Amount (A) | The total money at the end of the period, including the principal and the total interest. | \( A = P + I \) or \( A = P \left( 1 + \frac{R \times T}{100} \right) \) |
Additional Information: Simple vs. Compound Interest
It's important to distinguish simple interest from compound interest.
- Simple Interest: Interest is calculated only on the original principal amount for the entire duration. The interest amount is the same for each period.
- Compound Interest: Interest is calculated on the principal amount AND also on the accumulated interest from previous periods. This means the interest amount grows over time, leading to faster growth of the investment or debt.
This problem specified simple interest, which is why the interest earned between year 4 and year 7 was constant per year.
Important Questions from Simple Interest
Anil lent a sum of Rs. 5,000 on simple interest for 10 years in such a way that the rate of interest is 6% per annum for the first 2 years, 8% per anmum for the next 2 years and 10% per annum beyond 4 years. How much interest (in Rs.) will he earn at the end of 10 years?
What will be the simple interest on a sum of Rs. 12000 at the rate of 15 percent per annum for three years ?
If in 13 years fixed sum doubles at simple interest, what will be the interest rate per year? (correct to two decimal places)
On simple interest a sum of Rs. 640 becomes Rs. 832 in 2 years. What will Rs. 860 become in 4 years at the same rate of simple interest?
A certain sum amounts to Rs. 81840 in 3 years and to Rs. 92400 in 5 years at x% p.a. under simple interest. If the rate of interest is becomes (x + 2)%, then in how many years will the same sum double itself?
