A sum of money becomes 13/5 times of itself in 32 years at r% of SI. What is the value of r?

Options

A. `6%`
B. `7%`
C. `5%`
D. `18%`

Answer

To solve the problem step by step, we will follow these calculations:

### Step 1: Understand the Problem
We are given that a sum of money becomes \( \frac{13}{5} \) times itself in 32 years at an interest rate of \( r\% \). We need to find the value of \( r \).

### Step 2: Define the Variables
Let the principal amount (initial sum of money) be \( P = x \).
The amount after 32 years is given as:
\[ A = \frac{13}{5} \times x \]

### Step 3: Calculate the Simple Interest
The formula for Simple Interest (SI) is:
\[ SI = A - P \]
Substituting the values we have:
\[ SI = \left(\frac{13}{5} \times x\right) - x \]
To simplify this, we can express \( x \) as \( \frac{5}{5} \times x \):
\[ SI = \frac{13x}{5} - \frac{5x}{5} = \frac{13x - 5x}{5} = \frac{8x}{5} \]

### Step 4: Use the Simple Interest Formula
The formula for Simple Interest is also given by:
\[ SI = \frac{P \times R \times T}{100} \]
Where:
- \( P = x \)
- \( R = r \)
- \( T = 32 \) years

Substituting these values into the formula:
\[ \frac{8x}{5} = \frac{x \times r \times 32}{100} \]

### Step 5: Cancel Out \( x \)
Since \( x \) is common on both sides (assuming \( x \neq 0 \)), we can cancel it out:
\[ \frac{8}{5} = \frac{r \times 32}{100} \]

### Step 6: Cross Multiply to Solve for \( r \)
Cross multiplying gives us:
\[ 8 \times 100 = 5 \times r \times 32 \]
\[ 800 = 160r \]

### Step 7: Isolate \( r \)
Now, divide both sides by 160 to find \( r \):
\[ r = \frac{800}{160} = 5 \]

### Final Answer
Thus, the value of \( r \) is:
\[ \boxed{5} \]