What is the ratio of the simple interest earned on a certain amount at the rate of 21% per annum for 8 years to that earned on the same sum at the same rate for 21 years?
8 ∶ 21
Solving Simple Interest Ratio Problems
The question asks for the ratio of the simple interest earned on a certain amount for two different time periods, given that the principal amount and the rate of interest are the same in both cases. Let's break down the steps using the simple interest formula.
Understanding Simple Interest
Simple interest (SI) is calculated on the initial 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 amount of money.
- Rate (R) is the annual interest rate (as a percentage).
- Time (T) is the time period for which the money is borrowed or invested (in years).
Applying the Formula to the Given Scenario
We are given the following information:
- Principal amount is the same for both cases. Let's call it \(P\).
- Rate of interest is the same for both cases, 21% per annum. So, \(R = 21\).
- Time period for the first case is 8 years. Let's call it \(T_1 = 8\).
- Time period for the second case is 21 years. Let's call it \(T_2 = 21\).
Now, let's calculate the simple interest earned in each case.
Simple Interest for 8 Years (\(SI_1\))
Using the formula \( \text{SI} = \frac{P \times R \times T}{100} \), we get:
\( SI_1 = \frac{P \times 21 \times 8}{100} \)
Simple Interest for 21 Years (\(SI_2\))
Using the same formula, we get:
\( SI_2 = \frac{P \times 21 \times 21}{100} \)
Calculating the Ratio of Simple Interests
The question asks for the ratio of the simple interest earned for 8 years to that earned for 21 years, i.e., \( SI_1 : SI_2 \).
The ratio is:
\( \frac{SI_1}{SI_2} = \frac{\frac{P \times 21 \times 8}{100}}{\frac{P \times 21 \times 21}{100}} \)
We can simplify this expression by cancelling out the common terms from the numerator and the denominator. The common terms are \(P\), \(21\), and \(100\).
\( \frac{SI_1}{SI_2} = \frac{\cancel{P} \times \cancel{21} \times 8}{\cancel{P} \times \cancel{21} \times 21} \)
\( \frac{SI_1}{SI_2} = \frac{8}{21} \)
Therefore, the ratio of the simple interest earned for 8 years to that earned for 21 years is \(8 : 21\).
Conclusion
The ratio of the simple interest earned is directly proportional to the time period when the principal and rate are constant. The ratio of the time periods is \(T_1 : T_2 = 8 : 21\), and thus the ratio of the simple interests is also \(8 : 21\).
Comparing this result with the given options, we find that the correct ratio is 8 ∶ 21.
Revision Table: Simple Interest Ratio
| Aspect | Description |
|---|---|
| Formula Used | \( \text{SI} = \frac{P \times R \times T}{100} \) |
| Case 1 (Time = 8 years) | \( SI_1 = \frac{P \times 21 \times 8}{100} \) |
| Case 2 (Time = 21 years) | \( SI_2 = \frac{P \times 21 \times 21}{100} \) |
| Required Ratio | \( SI_1 : SI_2 \) |
| Calculation | \( \frac{SI_1}{SI_2} = \frac{P \times 21 \times 8}{P \times 21 \times 21} = \frac{8}{21} \) |
| Final Ratio | 8 ∶ 21 |
Additional Information: Simple Interest Concepts
Simple interest is a fundamental concept in finance and is often used for short-term loans or investments. Here are some key points:
- Principal Remains Constant: Unlike compound interest, simple interest is always calculated on the original principal amount throughout the loan or investment period.
- Linear Growth: The interest earned each year is the same amount, leading to linear growth of the total amount (Principal + Interest).
- Factors Affecting Simple Interest: The amount of simple interest earned depends on the principal amount, the annual interest rate, and the duration (time) of the investment or loan. If the principal and rate are fixed, the simple interest is directly proportional to the time. This is evident in our ratio calculation where the ratio of interests was simply the ratio of the times.
- Total Amount: The total amount (A) at the end of the time period is given by \(A = P + SI\), or \(A = P \left(1 + \frac{R \times T}{100}\right)\).
Understanding simple interest is crucial before learning about compound interest, which involves earning interest on previously accumulated interest as well.
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