A sum of money becomes (8/5) of itself in 5 years at a certain rate of interest. The rate per cent per annum is….

Options

A. `5%`
B. `8%`
C. `10%`
D. `12%`

Answer

To find the rate of interest per annum when a sum of money becomes \( \frac{8}{5} \) of itself in 5 years, we can follow these steps:

### Step 1: Define the Variables
Let the principal amount (the initial sum of money) be \( P \). According to the problem, the amount after 5 years is \( \frac{8}{5} P \).

### Step 2: Use the Formula for Amount in Simple Interest
The formula for the amount \( A \) in simple interest is given by:
\[ A = P + SI \]
where \( SI \) is the simple interest. The simple interest can also be calculated using the formula:
\[ SI = \frac{P \times R \times T}{100} \]
where \( R \) is the rate of interest and \( T \) is the time in years.

### Step 3: Substitute the Known Values
From the problem, we know:
- \( A = \frac{8}{5} P \)
- \( T = 5 \) years

Substituting these into the amount formula:
\[ \frac{8}{5} P = P + \frac{P \times R \times 5}{100} \]

### Step 4: Rearrange the Equation
To isolate \( R \), we can rearrange the equation:
\[ \frac{8}{5} P - P = \frac{P \times R \times 5}{100} \]
This simplifies to:
\[ \left(\frac{8}{5} - 1\right) P = \frac{P \times R \times 5}{100} \]
Calculating \( \frac{8}{5} - 1 \):
\[ \frac{8}{5} - \frac{5}{5} = \frac{3}{5} \]
So we have:
\[ \frac{3}{5} P = \frac{P \times R \times 5}{100} \]

### Step 5: Cancel Out \( P \)
Assuming \( P \neq 0 \), we can divide both sides by \( P \):
\[ \frac{3}{5} = \frac{R \times 5}{100} \]

### Step 6: Solve for \( R \)
Now, multiply both sides by 100:
\[ 100 \times \frac{3}{5} = R \times 5 \]
Calculating \( 100 \times \frac{3}{5} \):
\[ 60 = R \times 5 \]
Now, divide both sides by 5:
\[ R = \frac{60}{5} = 12 \]

### Conclusion
The rate of interest per annum is \( 12\% \).

---