A sum of money amounts to Rs 2000 in 3 years and Rs 2500 in 5 years at simple interest. Find the rate of interest per annum.

Options

A. `33(1)/(3)%`
B. `12(1)/(3)%`
C. `25%`
D. `20%`

Answer

To find the rate of interest per annum based on the given information, we can follow these steps:

### Step 1: Identify the amounts and the time periods
We know that:
- The amount after 3 years (A1) = Rs 2000
- The amount after 5 years (A2) = Rs 2500

### Step 2: Calculate the interest earned in the 2-year period
The interest earned between the 3rd year and the 5th year can be calculated as:
\[ \text{Interest for 2 years} = A2 - A1 = 2500 - 2000 = Rs 500 \]

### Step 3: Determine the annual interest
Since this interest of Rs 500 is earned over 2 years, we can find the annual interest (I) by dividing this amount by 2:
\[ I = \frac{500}{2} = Rs 250 \]

### Step 4: Find the principal amount (P)
Now, we can use the formula for simple interest to find the principal amount. The formula for the amount (A) in simple interest is:
\[ A = P + I \]
For the first scenario (after 3 years):
\[ 2000 = P + (3 \times 250) \]
\[ 2000 = P + 750 \]
\[ P = 2000 - 750 = Rs 1250 \]

### Step 5: Calculate the rate of interest (R)
Now that we have the principal, we can use the simple interest formula again to find the rate of interest:
\[ I = \frac{P \times R \times T}{100} \]
Using the interest earned in 1 year (Rs 250), we can rearrange this formula to find R:
\[ 250 = \frac{1250 \times R \times 1}{100} \]
Multiplying both sides by 100:
\[ 25000 = 1250 \times R \]
Now, divide both sides by 1250:
\[ R = \frac{25000}{1250} = 20 \]

### Conclusion
The rate of interest per annum is **20%**.

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