A certain sum lent out at simple interest amounts to Rs. 575 in 3 yr and to Rs. 625 in 5 yr. Then, the rate of interest is

Options

A. `3%`
B. `5%`
C. `4%`
D. `7%`

Answer

To solve the problem step by step, we need to find the rate of interest for a certain sum lent out at simple interest, given the amounts after 3 years and 5 years.

### Step 1: Understand the given information
We know:
- Amount after 3 years (A1) = Rs. 575
- Amount after 5 years (A2) = Rs. 625

### Step 2: Calculate the interest earned in the 2 years between 3 and 5 years
The difference in amounts over the 2 years is:
\[ A2 - A1 = 625 - 575 = 50 \]
This means that the interest earned in 2 years is Rs. 50.

### Step 3: Calculate the interest earned in 1 year
To find the interest earned in 1 year, we divide the total interest earned in 2 years by 2:
\[ \text{Interest per year} = \frac{50}{2} = 25 \]
So, the interest earned in 1 year is Rs. 25.

### Step 4: Calculate the total interest earned in 3 years
Now, we can calculate the total interest earned in 3 years:
\[ \text{Total Interest in 3 years} = 25 \times 3 = 75 \]

### Step 5: Calculate the principal amount
Using the formula for the amount:
\[ A = P + SI \]
We know:
- Amount after 3 years (A) = Rs. 575
- Total Interest (SI) in 3 years = Rs. 75

We can rearrange the formula to find the principal (P):
\[ P = A - SI = 575 - 75 = 500 \]
So, the principal amount is Rs. 500.

### Step 6: Use the simple interest formula to find the rate of interest
The formula for simple interest is:
\[ SI = \frac{P \times R \times T}{100} \]
Where:
- SI = Total Interest = Rs. 75
- P = Principal = Rs. 500
- T = Time = 3 years
- R = Rate of interest (which we need to find)

Substituting the known values into the formula:
\[ 75 = \frac{500 \times R \times 3}{100} \]

### Step 7: Solve for R
Rearranging the equation to solve for R:
\[ 75 = \frac{1500R}{100} \]
\[ 75 = 15R \]
\[ R = \frac{75}{15} = 5 \]

### Conclusion
The rate of interest is **5%**.

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