Disha invested an amount of Rs. 20,000 for 9 years at a simple interest per annum and obtain the total amount of Rs. 33,500. What is the rate of interest?
7.5%
Understanding the Simple Interest Problem
The question asks us to find the annual rate of simple interest applied to an investment. We are given the initial investment amount, the time period, and the final amount received after the investment period.
Identifying the Given Values
We are provided with the following information:
- Principal Amount (P): The initial amount invested by Disha is Rs. 20,000.
- Time Period (T): The duration for which the amount was invested is 9 years.
- Total Amount (A): The final amount received after 9 years is Rs. 33,500.
We need to calculate the Rate of Interest (R) per annum.
Calculating the Simple Interest Earned
The total amount received is the sum of the principal amount and the simple interest earned over the time period. Therefore, we can find the simple interest (SI) by subtracting the principal from the total amount.
The formula for Simple Interest is:
\(\text{Simple Interest (SI)} = \text{Total Amount (A)} - \text{Principal Amount (P)}\)
Substituting the given values:
\(\text{SI} = \text{Rs. } 33,500 - \text{Rs. } 20,000\)
\(\text{SI} = \text{Rs. } 13,500\)
So, the simple interest earned over 9 years is Rs. 13,500.
Finding the Simple Interest Rate
The formula relating Simple Interest, Principal, Rate, and Time is:
\(\text{SI} = \frac{\text{P} \times \text{R} \times \text{T}}{100}\)
We need to find the Rate (R). We can rearrange the formula to solve for R:
\(\text{R} = \frac{\text{SI} \times 100}{\text{P} \times \text{T}}\)
Now, we substitute the values we know:
- SI = Rs. 13,500
- P = Rs. 20,000
- T = 9 years
Substituting these into the formula for R:
\(\text{R} = \frac{13,500 \times 100}{20,000 \times 9}\)
\(\text{R} = \frac{1,350,000}{180,000}\)
We can simplify this expression by cancelling out common zeros:
\(\text{R} = \frac{1350}{180}\)
\(\text{R} = \frac{135}{18}\)
Both 135 and 18 are divisible by 9:
\(\text{R} = \frac{135 \div 9}{18 \div 9} = \frac{15}{2}\)
Converting the fraction to a decimal:
\(\text{R} = 7.5\)
Since R is the rate as a percentage, the rate of interest is 7.5% per annum.
Summary of Calculation Steps
| Step | Description | Calculation |
|---|---|---|
| 1 | Calculate Simple Interest (SI) | \(A - P = 33500 - 20000 = 13500\) |
| 2 | Use SI formula to find Rate (R) | \(SI = \frac{P \times R \times T}{100} \Rightarrow R = \frac{SI \times 100}{P \times T}\) |
| 3 | Substitute values and calculate R | \(R = \frac{13500 \times 100}{20000 \times 9} = \frac{1350000}{180000}\) |
| 4 | Simplify to find Rate | \(R = \frac{135}{18} = 7.5\) |
Conclusion on the Rate of Interest
Based on the calculations, the annual rate of simple interest is 7.5%.
Revision Table: Simple Interest Concepts
| Term | Definition | Formula |
|---|---|---|
| Principal (P) | The initial amount of money invested or borrowed. | - |
| Time (T) | The duration for which the principal is invested or borrowed, usually in years. | - |
| Rate (R) | The percentage at which interest is charged or earned per year. | - |
| Simple Interest (SI) | Interest calculated only on the principal amount. | \(SI = \frac{P \times R \times T}{100}\) |
| Amount (A) | The total sum of the principal and the interest earned or paid. | \(A = P + SI\) |
Additional Information on Simple Interest Calculations
Simple interest is a straightforward way to calculate interest on a principal amount. It is most commonly used for short-term loans or investments. Unlike compound interest, simple interest does not involve adding the earned interest back to the principal for subsequent calculations. The interest is always calculated on the original principal amount.
Understanding the relationship between Principal, Rate, Time, Simple Interest, and Amount is crucial for solving simple interest problems. By rearranging the basic formula, you can find any missing variable if the others are known.
For example, if you knew the Simple Interest, Rate, and Time, you could find the Principal using the formula:
\(P = \frac{SI \times 100}{R \times T}\)
Similarly, you could find the Time:
\(T = \frac{SI \times 100}{P \times R}\)
These rearrangements are derived directly from the main simple interest formula \(SI = \frac{P \times R \times T}{100}\).
Important Questions from Simple Interest
Anil lent a sum of Rs. 5,000 on simple interest for 10 years in such a way that the rate of interest is 6% per annum for the first 2 years, 8% per anmum for the next 2 years and 10% per annum beyond 4 years. How much interest (in Rs.) will he earn at the end of 10 years?
What will be the simple interest on a sum of Rs. 12000 at the rate of 15 percent per annum for three years ?
If in 13 years fixed sum doubles at simple interest, what will be the interest rate per year? (correct to two decimal places)
On simple interest a sum of Rs. 640 becomes Rs. 832 in 2 years. What will Rs. 860 become in 4 years at the same rate of simple interest?
A certain sum amounts to Rs. 81840 in 3 years and to Rs. 92400 in 5 years at x% p.a. under simple interest. If the rate of interest is becomes (x + 2)%, then in how many years will the same sum double itself?
