The simple interest earned on a sum is `(1)/(25)` of the sum, where the number of years of investment is equal to the rate percentage. For how many years was the sum invested?

Options

A. 2
B. 5
C. 3
D. 4

Answer

To solve the problem step by step, we will use the formula for simple interest and the information provided in the question.

### Step 1: Understand the Problem
We know that:
- The simple interest (SI) earned on a sum is \( \frac{1}{25} \) of the sum (let's denote the sum as \( P \)).
- The number of years of investment (let's denote it as \( T \)) is equal to the rate of interest (let's denote it as \( R \)).

### Step 2: Write the Formula for Simple Interest
The formula for simple interest is given by:
\[ SI = \frac{P \times R \times T}{100} \]

### Step 3: Substitute the Given Values
From the problem, we have:
\[ SI = \frac{1}{25} P \]
And since \( T = R \), we can denote both as \( R \). Thus, we can rewrite the simple interest formula as:
\[ \frac{1}{25} P = \frac{P \times R \times R}{100} \]
This simplifies to:
\[ \frac{1}{25} P = \frac{P \times R^2}{100} \]

### Step 4: Cancel \( P \) from Both Sides
Assuming \( P \neq 0 \), we can divide both sides by \( P \):
\[ \frac{1}{25} = \frac{R^2}{100} \]

### Step 5: Cross Multiply to Solve for \( R^2 \)
Cross multiplying gives us:
\[ 1 \times 100 = 25 \times R^2 \]
This simplifies to:
\[ 100 = 25R^2 \]

### Step 6: Solve for \( R^2 \)
Now, divide both sides by 25:
\[ R^2 = \frac{100}{25} \]
\[ R^2 = 4 \]

### Step 7: Take the Square Root
Taking the square root of both sides gives us:
\[ R = 2 \]

### Step 8: Conclusion
Since \( R \) is equal to the number of years \( T \), we find that:
\[ T = 2 \]
Thus, the sum was invested for **2 years**.

### Final Answer
The number of years the sum was invested is **2 years**.

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