X took a loan of Rs.1400 with simple interest for as many years as the rate of interest. If he paid Rs.1210 as interest at the end of that period, what was the rate of interest?
9.29 percent
Understanding the Simple Interest Problem
This problem involves calculating the rate of interest for a loan taken on simple interest. We are given the principal amount, the total simple interest paid, and a unique condition: the number of years is equal to the rate of interest (in percent).
Given Information:
- Principal Amount (P) = Rs. 1400
- Simple Interest (SI) = Rs. 1210
- Time (T) = Rate of Interest (R), where R is in percent. So, T = R years.
Simple Interest Formula
The formula for calculating simple interest is:
\( \text{SI} = \frac{P \times R \times T}{100} \)
Where:
- SI is the Simple Interest
- P is the Principal Amount
- R is the Rate of Interest per annum (in percent)
- T is the Time period in years
Solving for the Rate of Interest
We need to find the value of R. We can substitute the given values and the condition T = R into the simple interest formula:
\( 1210 = \frac{1400 \times R \times R}{100} \)
Now, let's simplify and solve for R:
\( 1210 = \frac{1400 \times R^2}{100} \)
Cancel out the 100 from the denominator and two zeros from 1400:
\( 1210 = 14 \times R^2 \)
Now, isolate \( R^2 \):
\( R^2 = \frac{1210}{14} \)
Simplify the fraction:
\( R^2 = \frac{605}{7} \)
To find R, take the square root of both sides:
\( R = \sqrt{\frac{605}{7}} \)
Let's calculate the approximate value:
\( \frac{605}{7} \approx 86.42857 \)
\( R \approx \sqrt{86.42857} \)
\( R \approx 9.2967 \)
The rate of interest is approximately 9.2967 percent.
Matching with Options
Comparing the calculated rate with the given options:
| Option | Rate |
|---|---|
| 1 | 4.5 percent |
| 2 | 7.5 percent |
| 3 | 7.85 percent |
| 4 | 9.29 percent |
Our calculated value, approximately 9.2967%, is very close to 9.29%. Therefore, option 4 is the correct rate of interest.
Conclusion
By using the simple interest formula and the given condition that the time period is equal to the rate of interest, we calculated the rate to be approximately 9.29 percent.
Revision Table: Simple Interest Concepts
| Concept | Definition/Formula |
|---|---|
| Simple Interest (SI) | Interest calculated only on the principal amount. |
| Principal (P) | The initial amount of money borrowed or invested. |
| Rate of Interest (R) | The percentage at which interest is charged or earned per annum. |
| Time (T) | The duration for which the money is borrowed or invested, usually in years. |
| Formula | \( \text{SI} = \frac{P \times R \times T}{100} \) |
| Amount (A) | Total money paid back (Principal + SI). \( A = P + SI \) |
Additional Information: Variations in Simple Interest Problems
Simple interest problems can come in various forms. Sometimes you might need to find the principal, the time, or the total amount paid back. Here are some common variations:
- Finding Principal (P) when SI, R, T are known: \( P = \frac{SI \times 100}{R \times T} \)
- Finding Rate (R) when SI, P, T are known: \( R = \frac{SI \times 100}{P \times T} \)
- Finding Time (T) when SI, P, R are known: \( T = \frac{SI \times 100}{P \times R} \)
- Problems involving calculating the total Amount (A): \( A = P + \frac{P \times R \times T}{100} \)
- Problems like the one solved here, where there's a relationship between Rate and Time (e.g., T = R, T = 2R, T = R/2, etc.). These require substituting the relationship into the main formula and solving for the unknown variable.
Always ensure that the rate is in percent per annum and the time is in years before applying the formula.
Important Questions from Simple and Compound Both
The S.I. on a certain sum of money for 4 years at 4 percent per annum exceeds the C.I. on the same sum for 3 years at 5 percent per annum by Rs. 57. Find the approximate sum.
Compound interest on a certain sum of money for 2 years at a rate of 'r' per cent per annum (compounding annually) is Rs. 8385. Simple interest on the same sum at the same rate for 2 years is Rs 7800. What is the value of r?
Find the difference between compound interest and simple interest on a sum of Rs. 5000 at 10% per annum for 3 years.
The difference between the compound interest and simple interest on a certain sum of money in 3 years at 5% p.a. is Rs. 6,100. What is the initial amount?
A sum of money becomes two times of itself in 8 years at simple interest, and it becomes four times of itself in 2 years at compound interest, when interest is compounded annually. Find the ratio of the rate of simple interest to the rate of compound interest offered per year.
