At 5% simple interest per annum a certain sum yields a total amount of ₹2,790 at the end of 3\(\frac{1}{4}\) years. The sum invested was:
Understanding the Simple Interest Problem
This problem asks us to find the original sum of money (the Principal) that was invested. We are given the simple interest rate, the time period, and the total amount received at the end of the investment period. The total amount is the sum of the original principal and the simple interest earned.
Given Information:
- Simple Interest Rate (\(R\)): 5% per annum
- Time Period (\(T\)): 3\(\frac{1}{4}\) years
- Total Amount (\(A\)): ₹2,790
We need to find the Principal (\(P\)).
Key Formulas for Simple Interest
The formulas we will use are:
- Simple Interest (\(SI\)) = \(\frac{P \times R \times T}{100}\)
- Amount (\(A\)) = Principal (\(P\)) + Simple Interest (\(SI\))
Step-by-Step Solution to Find the Sum Invested
Step 1: Convert the Time Period into a Decimal or Fraction
The time given is 3\(\frac{1}{4}\) years. We can write this as:
\(T = 3\frac{1}{4} = 3 + \frac{1}{4} = 3 + 0.25 = 3.25\) years.
Alternatively, as an improper fraction: \(T = 3\frac{1}{4} = \frac{(3 \times 4) + 1}{4} = \frac{12 + 1}{4} = \frac{13}{4}\) years.
Using the decimal form (\(T = 3.25\) years) is often easier for calculation.
Step 2: Express Simple Interest in terms of Principal
Using the simple interest formula, we can write \(SI\) as:
\(SI = \frac{P \times R \times T}{100}\)
Substitute the given values for \(R\) and \(T\):
\(SI = \frac{P \times 5 \times 3.25}{100}\)
\(SI = \frac{P \times 16.25}{100}\)
\(SI = 0.1625P\)
Step 3: Use the Amount Formula to Solve for Principal
The total amount is the sum of the principal and the simple interest:
\(A = P + SI\)
Substitute the given value for \(A\) and the expression for \(SI\) from Step 2:
\(2790 = P + 0.1625P\)
Step 4: Combine the Principal terms and Solve for P
Combine the \(P\) terms on the right side of the equation:
\(2790 = P(1 + 0.1625)\)
\(2790 = 1.1625P\)
Now, isolate \(P\) by dividing the amount by 1.1625:
\(P = \frac{2790}{1.1625}\)
\(P = 2400\)
So, the sum invested (the Principal) was ₹2,400.
Verification
Let's check our answer. If \(P = ₹2400\), \(R = 5\%\), and \(T = 3.25\) years:
\(SI = \frac{2400 \times 5 \times 3.25}{100}\)
\(SI = \frac{2400 \times 16.25}{100}\)
\(SI = \frac{39000}{100}\)
\(SI = ₹390\)
Amount \(A = P + SI = 2400 + 390 = ₹2790\). This matches the given amount, so our calculation is correct.
| Item | Value |
|---|---|
| Principal (\(P\)) | ₹2,400 |
| Rate (\(R\)) | 5% per annum |
| Time (\(T\)) | 3.25 years |
| Simple Interest (\(SI\)) | ₹390 |
| Amount (\(A\)) | ₹2,790 |
Revision Table: Simple Interest Components
| Term | Definition | Formula (Simple Interest) |
|---|---|---|
| Principal (P) | The initial sum of money invested or borrowed. | \(P = \frac{100 \times SI}{R \times T}\) or \(P = \frac{100 \times A}{100 + (R \times T)}\) |
| Rate (R) | The percentage at which interest is calculated per period (usually per year). | \(R = \frac{100 \times SI}{P \times T}\) |
| Time (T) | The duration for which the money is invested or borrowed. Must be in years for annual rate. | \(T = \frac{100 \times SI}{P \times R}\) |
| Simple Interest (SI) | The interest calculated only on the principal amount. | \(SI = \frac{P \times R \times T}{100}\) |
| Amount (A) | The total sum at the end of the period, including the principal and interest. | \(A = P + SI\) or \(A = P \left(1 + \frac{R \times T}{100}\right)\) |
Additional Information: Simple vs. Compound Interest
It's important to distinguish between simple interest and compound interest.
- Simple Interest: As seen in this problem, interest is calculated only on the initial principal amount. It remains constant for each period.
- Compound Interest: Interest is calculated on the initial principal and also on the accumulated interest from previous periods. This means the interest earned grows over time, leading to a higher total amount compared to simple interest over longer periods.
The formulas and calculations differ significantly between simple and compound interest problems.
Similar Questions
Saathi deposited Rs. 825 in a bank that promised 8% simple interest per annum. If Saathi kept the money with the bank for 5 years, she will earn an interest of:
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A sum of money invested for 2 years and 9 months at the rate of 8% simple interest per annum became Rs. 732 at the end of the period. What was the sum that was initially invested?
The simple interest on a certain sum for 3 years at 8% p.a is Rs. 90 more than simple interest at 9% p.a for 2 years on the same sum. The sum is (in rupees):
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A sum of money was invested at the rate of 7.5% simple interest per annuum for 4 years. If the investments were for 5 years, the interest earned would have been Rs. 375 more. What was the initial sum invested?
Rs. 750 invested for 3 months gave an interest of Rs. 18. What was the simple rate of interest per annum?
The interest earned on Rs. 1,600 at the rate of 5% simple interest per annum for 6 years would be:
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Important Questions from Simple Interest
How much time will it take for an amount of Rs. 450 to yield Rs. 81 as interest at 4.5% per annum of simple interest ?
Nirav and Mehul borrowed Rs.4000 and Rs.5000 respectively for 2.5 years at the rate of x% per annum. Mehul paid Rs 125 more interest than Nirav. Find x.
If the interest on a sum of Rs.1200 is more than the interest on Rs.1000 by Rs.120 in three years, then what is the rate of interest per annum?.
The difference between the simple interest received from two banks on Rs. 500 for two years is Rs. 2.50. What is the difference between their rates?
P borrowed an amount from Q at a simple interest of 10% p.a. At the end of the year, P paid back Rs 99000 which was 90% of what she owed. How much money did P borrow?
