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?
Rs.2,000
Solving the Simple Interest Principal Amount Problem
This problem asks us to find the initial amount of money that was invested, which is called the principal amount. We are given information about how a change in the simple interest rate affects the total interest earned over a specific period.
Let's define the key terms involved in simple interest calculations:
- Principal (P): The initial sum of money invested or borrowed. This is what we need to find.
- Rate (R): The percentage at which interest is charged per year (per annum).
- Time (T): The duration for which the money is invested or borrowed, in years.
- Simple Interest (SI): The interest calculated only on the principal amount.
The formula for calculating simple interest is:
\( \text{SI} = \frac{\text{P} \times \text{R} \times \text{T}}{100} \)
In this problem, the time period (\( T \)) is given as 5 years.
Setting up the Scenarios
We have two scenarios based on the interest rate:
Scenario 1: Original Rate
Let the original simple interest rate be \( R \) % per annum.
The simple interest earned in 5 years (\( \text{SI}_1 \)) would be:
\( \text{SI}_1 = \frac{P \times R \times 5}{100} = \frac{5PR}{100} \)
Scenario 2: Higher Rate
The new rate is 5% higher than the original rate, so the new rate is \( (R+5) \) % per annum.
The simple interest earned in 5 years (\( \text{SI}_2 \)) with this higher rate would be:
\( \text{SI}_2 = \frac{P \times (R+5) \times 5}{100} = \frac{5P(R+5)}{100} \)
Using the Difference in Simple Interest
The problem states that if the rate had been 5% higher, the interest would have fetched Rs. 500 more. This means the difference between the interest earned in Scenario 2 and Scenario 1 is Rs. 500.
\( \text{SI}_2 - \text{SI}_1 = 500 \)
Now, let's substitute the formulas for \( \text{SI}_1 \) and \( \text{SI}_2 \) into this equation:
\( \frac{5P(R+5)}{100} - \frac{5PR}{100} = 500 \)
Since both terms have a common denominator (100) and a common factor (\( 5P \)) in the numerator, we can simplify:
\( \frac{5P \times (R+5 - R)}{100} = 500 \)
\( \frac{5P \times 5}{100} = 500 \)
\( \frac{25P}{100} = 500 \)
Simplify the fraction on the left side:
\( \frac{P}{4} = 500 \)
To find \( P \), multiply both sides of the equation by 4:
\( P = 500 \times 4 \)
\( P = 2000 \)
So, the principal amount was Rs. 2000.
Simple Interest Formula Revision
| Term | Symbol | Description |
|---|---|---|
| Principal | \( P \) | The initial sum of money. |
| Rate | \( R \) | Annual interest rate (in percentage). |
| Time | \( T \) | Duration in years. |
| Simple Interest | SI | Interest calculated only on \( P \). Formula: \( \frac{P \times R \times T}{100} \) |
Additional Information on Principal Calculation
This problem highlights a useful concept in simple interest: the change in interest is directly proportional to the change in rate (or time), assuming other factors remain constant. The extra 5% rate for 5 years only applies to the principal amount \( P \), as simple interest is calculated solely on the principal.
The extra interest received (Rs. 500) is simply the simple interest on the principal \( P \) for 5 years at the extra rate of 5%.
Using the SI formula for the extra interest:
\( \text{Extra SI} = \frac{P \times (\text{Extra Rate}) \times T}{100} \)
Here, \( \text{Extra SI} = 500 \), \( \text{Extra Rate} = 5 \), and \( T = 5 \).
\( 500 = \frac{P \times 5 \times 5}{100} \)
\( 500 = \frac{25P}{100} \)
\( 500 = \frac{P}{4} \)
\( P = 500 \times 4 \)
\( P = 2000 \)
This quicker approach confirms that the principal amount is indeed Rs. 2000. This method is useful when dealing with changes in rate or time in simple interest problems.
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 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?
Two equal amounts were borrowed at 5% and 4% simple interest. The total interest after 4 years amounted to Rs. 405. What was the total amount borrowed?
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?
The annual income of a person decreases by Rs. 64 if the rate of interest decreases from 4% to 3.75%. What is his original annual income?
A person borrows Rs. 5000 at 5% rate of interest per annum and immediately lent it at 5.5%. After two years he collected the amount and settled his loan. What is the amount gained by him this transaction?
A person divided a sum of Rs. 17, 200 into three parts and invested at 5%, 6% and 9% per annum simple interest. At the end of two years, he got the same interest on each part of money. What is the money invested at 9%?
A person borrowed ₹9,000 at 7%, ₹12,000 at 8% and ₹15,000 at 9% simple interest per annum. He had to pay ₹50,700 at the end of n years. What is the value of n?
The simple interest on a certain sum is one-fourth of the sum. If the number of years and the rate of annual interest are numerically equal, then the number of years is
Important Questions from Simple Interest
At what rate percent per annum will the simple interest on a sum of money be 2/5 of the principal in 10 years?
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?
