Rahul invests Rs 12,000 as fixed deposit at a bank at the rate of 10% per annum simple interest. But due to some pressing needs he has to withdraw the entire money after 3 years, for which the bank allowed him a lower rate of interest. If he gets Rs 3,320 less than what he would have got at the end of 5 years, the rate of interest allowed by the bank is:

Options

A. `5(9)/(5)%`
B. `5(4)/(9)%`
C. `9(9)/(5)%`
D. `7(4)/(9)%`

Answer

To solve the problem step by step, we need to calculate the simple interest that Rahul would have earned after 5 years and then find out the lower rate of interest he received after withdrawing his money after 3 years.

### Step 1: Calculate the total amount Rahul would have received after 5 years at 10% per annum.

**Formula for Simple Interest (SI)**:
\[ \text{SI} = \frac{P \times R \times T}{100} \]
Where:
- \( P \) = Principal amount (Rs 12,000)
- \( R \) = Rate of interest (10%)
- \( T \) = Time (5 years)

**Calculation**:
\[ \text{SI} = \frac{12000 \times 10 \times 5}{100} = \frac{600000}{100} = 6000 \]

**Total amount after 5 years**:
\[ \text{Total Amount} = P + \text{SI} = 12000 + 6000 = 18000 \]

### Step 2: Determine the amount Rahul received after 3 years.

According to the problem, he received Rs 3,320 less than what he would have received after 5 years.

**Calculation**:
\[ \text{Amount after 3 years} = 18000 - 3320 = 14680 \]

### Step 3: Calculate the simple interest earned after 3 years.

Using the same formula for simple interest, we can find the interest earned in 3 years.

**Total amount after 3 years**:
\[ \text{Total Amount} = P + \text{SI} \]
Let the simple interest earned after 3 years be \( \text{SI}_3 \).

So,
\[ 12000 + \text{SI}_3 = 14680 \]

**Calculation**:
\[ \text{SI}_3 = 14680 - 12000 = 2680 \]

### Step 4: Calculate the lower rate of interest allowed by the bank.

Using the simple interest formula again, we can find the rate of interest for the 3-year period.

**Rearranging the formula for Rate (R)**:
\[ R = \frac{\text{SI} \times 100}{P \times T} \]

**Substituting the values**:
\[ R = \frac{2680 \times 100}{12000 \times 3} \]

**Calculation**:
\[ R = \frac{268000}{36000} \]
\[ R = \frac{2680}{360} = \frac{67}{9} \]

### Step 5: Convert the fraction to a percentage.

To convert \( \frac{67}{9} \) to a percentage:
\[ R \approx 7.44\% \]

### Final Answer:
The rate of interest allowed by the bank is approximately \( 7 \frac{4}{9}\% \).

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