If a sum of money amounts to 12,900 and 14,250 at the end of the 4th year and 5th year respectively at a certain rate of simple interest, then rate of interest is

Options

A. 10%
B. 12%
C. 18%
D. 20%

Answer

To find the rate of interest given the amounts at the end of the 4th and 5th years, we can follow these steps:

### Step 1: Identify the amounts
The amount at the end of the 4th year is 12,900 and at the end of the 5th year is 14,250.

### Step 2: Calculate the simple interest for 1 year
The simple interest for one year can be calculated by subtracting the amount at the end of the 4th year from the amount at the end of the 5th year:
\[ \text{Simple Interest for 1 year} = \text{Amount at end of 5th year} - \text{Amount at end of 4th year} \]
\[ \text{Simple Interest for 1 year} = 14,250 - 12,900 = 1,350 \]

### Step 3: Calculate the total simple interest for 4 years
Since the interest is simple, the total simple interest for 4 years can be calculated as:
\[ \text{Total Simple Interest for 4 years} = \text{Simple Interest for 1 year} \times 4 \]
\[ \text{Total Simple Interest for 4 years} = 1,350 \times 4 = 5,400 \]

### Step 4: Calculate the principal amount
The principal amount can be found by subtracting the total simple interest from the amount at the end of the 4th year:
\[ \text{Principal} = \text{Amount at end of 4th year} - \text{Total Simple Interest for 4 years} \]
\[ \text{Principal} = 12,900 - 5,400 = 7,500 \]

### Step 5: Use the simple interest formula to find the rate
The formula for simple interest is:
\[ \text{Simple Interest} = \frac{\text{Principal} \times \text{Rate} \times \text{Time}}{100} \]
We can rearrange this to find the rate:
\[ \text{Rate} = \frac{\text{Simple Interest} \times 100}{\text{Principal} \times \text{Time}} \]
Substituting the values we have:
\[ \text{Rate} = \frac{5,400 \times 100}{7,500 \times 4} \]

### Step 6: Simplify the calculation
Calculating the denominator:
\[ 7,500 \times 4 = 30,000 \]
Now substituting back into the rate formula:
\[ \text{Rate} = \frac{5,400 \times 100}{30,000} \]
\[ \text{Rate} = \frac{540,000}{30,000} = 18 \]

### Final Answer
The rate of interest is **18% per annum**.

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