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
The simple interest earned on a certain sum of money for 3 years at 15% per annum is ₹2,700. Find the sum.
A sum of money invested at simple interest amounts to ₹21,500 in 5 years and ₹26,000 in 8 years. Find the principal amount (in ₹).
The interest earned on Rs. 1,600 at the rate of 5% simple interest per annum for 6 years would be:
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?
At 9.5% simple interest per annum, a sum of money became Rs. 942 in 6 years. The sum invested initially was:
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. 2250 at the rate 3% simple interest per annum for 2 years will be:
At 12% simple interest per annum a sum of money becomes Rs. 295 in \(1\frac{1}{2}\) years. What was the sum invested?
Rahi deposited Rs. 600 in a bank that promised 8% simple interest per annum. If Rahi kept the money with the bank for 5 years, she will earn an interest of:
At 8% simple interest per annum a sum of money becomes Rs. 300 in \(2\frac{1}{2}\) years. What was the sum invested?
Important Questions from Simple Interest
If ₹12,800 is invested in a bank for 5 years at the rate of 9% per annum simple interest. what amount is returned by the bank?
Somu has borrowed ₹10,000 from a money lender with simple interest at a rate of 7% half yearly. How much amount will he pay to the money lender after 3 years?
Find the Simple interest on Rs. 2,400 from 20 March 2019 to 31 may 2019 at \(6{1 \over 4}\) % rate?
If the simple interest for five years is equal is 35% of the principal, that rate of interest is:
A sum fetched a simple interest of Rs. 3,040 at the rate of 8% p.a in 5 years. what is the sum?
