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):
Rs. 1500
Solving Simple Interest Problems to Find the Principal Sum
This question involves calculating the original sum of money, also known as the Principal, based on simple interest calculations under different conditions.
Let the principal sum be P (in rupees).
The formula for Simple Interest (SI) is:
\( \text{SI} = \frac{\text{P} \times \text{R} \times \text{T}}{100} \)
Where:
- P is the Principal sum
- R is the Rate of interest per annum
- T is the Time period in years
Scenario 1: Simple Interest for 3 years at 8% p.a.
In this scenario:
- Time (T\(_1\)) = 3 years
- Rate (R\(_1\)) = 8% p.a.
- Principal = P
Simple Interest in Scenario 1 (SI\(_1\)) is:
\( \text{SI}_1 = \frac{\text{P} \times 8 \times 3}{100} = \frac{24\text{P}}{100} \)
Scenario 2: Simple Interest for 2 years at 9% p.a.
In this scenario:
- Time (T\(_2\)) = 2 years
- Rate (R\(_2\)) = 9% p.a.
- Principal = P
Simple Interest in Scenario 2 (SI\(_2\)) is:
\( \text{SI}_2 = \frac{\text{P} \times 9 \times 2}{100} = \frac{18\text{P}}{100} \)
Setting up the Equation based on the given condition
The problem states that the simple interest in Scenario 1 (SI\(_1\)) is Rs. 90 more than the simple interest in Scenario 2 (SI\(_2\)).
So, we can write the equation:
\( \text{SI}_1 = \text{SI}_2 + 90 \)
Solving for the Principal (P)
Substitute the expressions for SI\(_1\) and SI\(_2\) into the equation:
\( \frac{24\text{P}}{100} = \frac{18\text{P}}{100} + 90 \)
To eliminate the denominators, multiply the entire equation by 100:
\( 100 \times \left( \frac{24\text{P}}{100} \right) = 100 \times \left( \frac{18\text{P}}{100} \right) + 100 \times 90 \)
\( 24\text{P} = 18\text{P} + 9000 \)
Now, we need to isolate the term with P. Subtract 18P from both sides of the equation:
\( 24\text{P} - 18\text{P} = 9000 \)
\( 6\text{P} = 9000 \)
Finally, divide by 6 to find the value of P:
\( \text{P} = \frac{9000}{6} \)
\( \text{P} = 1500 \)
So, the principal sum is Rs. 1500.
Let's verify this:
- SI\(_1\) for P=1500, R=8%, T=3 years: \( \text{SI}_1 = \frac{1500 \times 8 \times 3}{100} = \frac{36000}{100} = 360 \)
- SI\(_2\) for P=1500, R=9%, T=2 years: \( \text{SI}_2 = \frac{1500 \times 9 \times 2}{100} = \frac{27000}{100} = 270 \)
- Difference = SI\(_1\) - SI\(_2\) = 360 - 270 = 90. This matches the condition given in the problem.
The sum is Rs. 1500.
Revision Table: Simple Interest Concepts
| Term | Definition | Formula Component |
|---|---|---|
| Principal (P) | The initial amount of money borrowed or invested. | P |
| Rate (R) | The percentage at which interest is charged or earned per year. | R (usually R/100 in formula) |
| Time (T) | The duration for which the money is borrowed or invested. | T (in years) |
| Simple Interest (SI) | Interest calculated only on the principal amount. | SI |
Additional Information: Simple vs. Compound Interest
Simple interest is a basic and quick method of calculating the interest charge on a loan or investment. Interest is calculated only on the principal amount.
Compound interest, on the other hand, is calculated on the principal amount and also on the accumulated interest from previous periods. This means compound interest grows faster than simple interest over time for the same rate and principal.
Key differences:
- Simple Interest: Interest is fixed each period (based on original principal).
- Compound Interest: Interest grows each period (based on principal + accumulated interest).
This problem specifically deals with simple interest, making the calculation straightforward based on the given formula.
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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 ?
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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?
