The total amount of Rs. 14,500 with simple interest in 6 years is Rs. 21,460, find out the Rate of interest per annum ?

Question

Accepted Answer

To determine the rate of interest per annum for the given problem, we need to use the formula for simple interest:

$$ \text{Simple Interest (SI)} = \frac{\text{Principal (P)} \times \text{Rate (R)} \times \text{Time (T)}}{100} $$

Given:
- Principal (P) = Rs. 14,500
- Total amount after 6 years = Rs. 21,460
- Time (T) = 6 years

First, we calculate the Simple Interest (SI):

$$ \text{SI} = \text{Total Amount} - \text{Principal} = 21,460 - 14,500 = 6,960 $$

Now, we substitute the values into the simple interest formula to find the rate (R):

$$ 6,960 = \frac{14,500 \times R \times 6}{100} $$

Solving for R:

$$ 6,960 = \frac{87,000 \times R}{100} $$

$$ 6,960 \times 100 = 87,000 \times R $$

$$ 696,000 = 87,000 \times R $$

$$ R = \frac{696,000}{87,000} $$

$$ R = 8 $$

Therefore, the rate of interest per annum is 8%, which corresponds to Option 4.

**Explanation for why other options are incorrect:**

1. **Option 1 (4%)**: If the rate were 4%, the simple interest would be calculated as follows:
   $$ \text{SI} = \frac{14,500 \times 4 \times 6}{100} = 3,480 $$
   The total amount would then be $ 14,500 + 3,480 = 17,980 $, which does not match the given total amount of Rs. 21,460.

2. **Option 2 (10%)**: If the rate were 10%, the simple interest would be:
   $$ \text{SI} = \frac{14,500 \times 10 \times 6}{100} = 8,700 $$
   The total amount would then be $ 14,500 + 8,700 = 23,200 $, which exceeds the given total amount of Rs. 21,460.

3. **Option 3 (6%)**: If the rate were 6%, the simple interest would be:
   $$ \text{SI} = \frac{14,500 \times 6 \times 6}{100} = 5,220 $$
   The total amount would then be $ 14,500 + 5,220 = 19,720 $, which is less than the given total amount of Rs. 21,460.

Thus, the only rate that results in the correct total amount is 8%, making Option 4 the correct choice.