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
5
Understanding Simple Interest Calculations
Simple interest is a basic concept in finance where interest is calculated only on the principal amount. It's a quick and easy method of calculating interest.
The question involves a scenario where the simple interest earned on a sum of money has a specific relationship with the principal sum, and the rate of interest is numerically equal to the time period in years. We need to find the number of years.
Analyzing the Simple Interest Problem
Let's break down the information given in the problem:
- The simple interest (SI) on a certain sum is one-fourth of the sum.
- The principal sum is the initial amount of money.
- The number of years (T) and the rate of annual interest (R) are numerically equal.
Our goal is to find the value of the number of years, T.
Simple Interest Formula
The standard formula for calculating simple interest is:
\[ SI = \frac{P \times R \times T}{100} \]Where:
- \(SI\) is the Simple Interest
- \(P\) is the Principal sum
- \(R\) is the Annual Rate of Interest (in percent)
- \(T\) is the Time period (in years)
Step-by-Step Solution to Find the Number of Years
We are given two key pieces of information to use with the simple interest formula:
- \(SI = \frac{1}{4} P\)
- \(R = T\) (numerically)
Let's substitute these conditions into the simple interest formula:
We have \(SI = \frac{P \times R \times T}{100}\).
Substitute \(SI = \frac{1}{4} P\):
\[ \frac{1}{4} P = \frac{P \times R \times T}{100} \]We are also given that the rate \(R\) and the time \(T\) are numerically equal. Let's assume \(R = T = x\), where \(x\) is the numerical value we need to find.
Substitute \(R = x\) and \(T = x\) into the equation:
\[ \frac{1}{4} P = \frac{P \times x \times x}{100} \] \[ \frac{1}{4} P = \frac{P \times x^2}{100} \]Assuming the principal sum \(P\) is not zero (which must be the case to earn interest), we can cancel \(P\) from both sides of the equation:
\[ \frac{1}{4} = \frac{x^2}{100} \]Now, we need to solve for \(x\). Multiply both sides by 100:
\[ x^2 = \frac{100}{4} \] \[ x^2 = 25 \]To find \(x\), take the square root of both sides:
\[ x = \sqrt{25} \]Since time and rate must be positive values in this context, we take the positive square root:
\[ x = 5 \]We defined \(x\) such that \(R = x\) and \(T = x\). Therefore, the rate of interest is 5% and the number of years is 5.
The question asks for the number of years, which is \(T\). We found \(T = x = 5\).
Verification
Let's check our answer. Assume the principal \(P = 100\). If the rate is 5% and the time is 5 years, the simple interest would be:
\[ SI = \frac{100 \times 5 \times 5}{100} = \frac{2500}{100} = 25 \]Is this simple interest equal to one-fourth of the principal? One-fourth of the principal (\(P=100\)) is \(\frac{1}{4} \times 100 = 25\). Yes, it is. Also, the rate (5%) is numerically equal to the number of years (5). Our calculation is correct.
Summary of the Result
When the simple interest is one-fourth of the principal and the number of years is numerically equal to the annual rate of interest, the number of years is 5.
| Given Condition 1 | Mathematical Expression |
|---|---|
| Simple Interest is one-fourth of the Principal | \(SI = \frac{1}{4} P\) |
| Number of Years is numerically equal to Rate | \(T = R\) |
| Simple Interest Formula | \(SI = \frac{P \times R \times T}{100}\) |
|---|---|
| Substitution using conditions | \(\frac{1}{4} P = \frac{P \times T \times T}{100}\) (using \(R=T\)) |
| Simplification | \(\frac{1}{4} = \frac{T^2}{100}\) |
| Solving for \(T^2\) | \(T^2 = \frac{100}{4} = 25\) |
| Solving for \(T\) | \(T = \sqrt{25} = 5\) years |
Revision Table: Simple Interest Concepts
| Concept | Description |
|---|---|
| Principal (P) | The initial amount of money invested or borrowed. |
| Rate (R) | The annual percentage at which interest is calculated. |
| Time (T) | The duration for which the money is invested or borrowed, usually in years. |
| Simple Interest (SI) | Interest calculated only on the principal amount. Formula: \(\frac{P \times R \times T}{100}\). |
Additional Information: Simple vs. Compound Interest
It's helpful to understand the difference between simple interest and compound interest.
- Simple Interest: Interest is calculated only on the original principal amount. The interest earned does not get added back to the principal for future interest calculations. This is what was used in our problem.
- Compound Interest: Interest is calculated on the initial principal and also on the accumulated interest from previous periods. This means interest earns interest, leading to faster growth of the investment or debt. The formula for compound amount (A) is \(A = P(1 + \frac{R}{100})^T\), and Compound Interest \(CI = A - P\).
Understanding these differences is crucial for solving various financial math problems. Simple interest problems are generally more straightforward as they involve a linear calculation of interest over time.
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 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?
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?
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?
