The interest on sum of money at the end of ` 2(1/2) `years is `4/5` of the sum. The rate percent per year is

Options

A. `10%`
B. `20%`
C. `16%`
D. `32%`

Answer

To solve the problem step by step, we can follow these instructions:

### Step 1: Understand the problem
We need to find the rate percent per year given that the interest at the end of 2.5 years is \( \frac{4}{5} \) of the principal amount.

### Step 2: Define the variables
Let the principal amount be \( P \). The time given is \( 2 \frac{1}{2} \) years, which can be converted into an improper fraction:
\[ 2 \frac{1}{2} = \frac{5}{2} \text{ years} \]

### Step 3: Write the formula for Simple Interest
The formula for Simple Interest (SI) is:
\[ SI = \frac{P \times R \times T}{100} \]
Where:
- \( SI \) is the Simple Interest
- \( P \) is the principal amount
- \( R \) is the rate of interest per annum
- \( T \) is the time in years

### Step 4: Set up the equation
According to the problem, the Simple Interest is \( \frac{4}{5} \) of the principal:
\[ SI = \frac{4}{5} P \]
Substituting this into the Simple Interest formula gives:
\[ \frac{4}{5} P = \frac{P \times R \times \frac{5}{2}}{100} \]

### Step 5: Simplify the equation
We can cancel \( P \) from both sides (assuming \( P \neq 0 \)):
\[ \frac{4}{5} = \frac{R \times \frac{5}{2}}{100} \]

### Step 6: Cross-multiply to solve for \( R \)
Cross-multiplying gives:
\[ 4 \times 100 = R \times \frac{5}{2} \times 5 \]
\[ 400 = R \times \frac{25}{2} \]

### Step 7: Solve for \( R \)
To isolate \( R \), multiply both sides by \( \frac{2}{25} \):
\[ R = \frac{400 \times 2}{25} \]
\[ R = \frac{800}{25} \]
\[ R = 32 \]

### Step 8: Conclusion
Thus, the rate percent per year is:
\[ \text{Rate} = 32\% \text{ per annum} \]