A sum was put at simple interest at certain rate for 2 years. Had it been put at 1% higher rate of interest, it would have fetched Rs. 24 more. What is the sum?
Rs. 1200
Calculating the Sum using Simple Interest Difference
This problem involves calculating the principal sum based on the difference in simple interest earned when the interest rate is changed. We are given the time period and the increase in interest amount due to a specific increase in the interest rate.
Understanding Simple Interest
Simple Interest (SI) is calculated only on the principal amount. The formula for simple interest is:
$\(\text{SI} = \frac{\text{Principal} \times \text{Rate} \times \text{Time}}{100}$\)
Where:
- Principal (P) is the initial sum of money
- Rate (R) is the annual interest rate
- Time (T) is the time period in years
Setting up the Problem
Let's define the variables from the question:
- Principal Sum = P (This is what we need to find)
- Original Interest Rate = R% per annum
- Time Period = T = 2 years
- New Interest Rate = (R + 1)% per annum
- Difference in Simple Interest = Rs. 24
We can write the simple interest formula for the two scenarios:
Scenario 1: Original Rate (R%)
Simple Interest (SI\(_1\)) = $\(\frac{P \times R \times 2}{100}$\)
Scenario 2: Higher Rate (R + 1%)
Simple Interest (SI\(_2\)) = $\(\frac{P \times (R+1) \times 2}{100}$\)
Calculating the Simple Interest Difference
According to the question, the simple interest fetched with the higher rate is Rs. 24 more than with the original rate. So, the difference is:
SI\(_2\) - SI\(_1\) = 24
Substitute the formulas for SI\(_1\) and SI\(_2\):
$\(\frac{P \times (R+1) \times 2}{100} - \frac{P \times R \times 2}{100} = 24$\)
Solving for the Principal Sum (P)
Now, let's simplify and solve the equation for P:
$\(\frac{2P(R+1)}{100} - \frac{2PR}{100} = 24$\)
Combine the terms on the left side since they have a common denominator:
$\(\frac{2P(R+1) - 2PR}{100} = 24$\)
Expand the term in the numerator:
$\(\frac{2PR + 2P - 2PR}{100} = 24$\)
The 2PR and -2PR terms cancel out:
$\(\frac{2P}{100} = 24$\)
Simplify the fraction on the left side:
$\(\frac{P}{50} = 24$\)
Multiply both sides by 50 to isolate P:
$\(P = 24 \times 50$\)
$\(P = 1200$\)
So, the principal sum is Rs. 1200.
Verification
Let's quickly check if this works. If the sum is Rs. 1200, the extra simple interest gained from a 1% higher rate over 2 years is:
Extra SI = $\(\frac{1200 \times 1 \times 2}{100} = \frac{2400}{100} = 24$\)
This matches the given difference of Rs. 24, confirming our calculated sum is correct.
The sum is Rs. 1200.
Revision Table: Simple Interest Concepts
| Concept | Description | Formula |
|---|---|---|
| Principal (P) | The initial amount of money invested or borrowed. | - |
| Rate (R) | The annual interest rate (usually in percent). | - |
| Time (T) | The duration for which the money is invested or borrowed (usually in years). | - |
| Simple Interest (SI) | Interest calculated only on the principal amount. | $\(\text{SI} = \frac{P \times R \times T}{100}$\) |
| Amount (A) | The total sum after adding simple interest to the principal. | A = P + SI |
Additional Information: Simple Interest vs. Compound Interest
It's important to distinguish between simple interest and compound interest when solving problems.
- Simple Interest: Interest is calculated only on the initial principal amount. The interest earned each period is the same if the rate and principal are constant.
- Compound Interest: Interest is calculated on the principal amount plus any accumulated interest from previous periods. This means interest earns interest, leading to faster growth of the sum over time compared to simple interest at the same rate. The formula for compound interest is more complex, involving exponentiation.
This specific question clearly states it is a "simple interest" problem, so we correctly used the simple interest formula.
Similar Questions
If a sum of money at a certain rate of simple interest per year doubles in 5 years and at a different rate of simple interest per year becomes three times in 12 years, then the difference in the two rates of simple interest per year is
A sum of money was invested at simple interest at a certain rate for 5 years. Had it been invested at a 5% higher rate, it would have fetched Rs.500 more. What was the principal amount?
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A lent Rs. 25000 to B and at the same time lent some amount to C at the same 7% simple interest. After 4 years a received Rs. 11200 as interest from B and C. How much did A lend to C?
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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?
A sum of Rs.1200 becomes Rs.1560 at a rate of simple interest in 3 years. In how many years will the sum of Rs.800 amount to Rs.1120 at the same rate of simple interest?
