The simple interest on a certain sum for `4(2)/(3)` years at `6(3)/(4)%` is Rs 8,820. What will be the amount of the same sum after `7(1)/(2)` years at `8(1)/(3)%` at simple interest?

Options

A. Rs 45, 500
B. Rs 45, 400
C. Rs 54, 500
D. Rs 54, 400

Answer

To solve the problem step by step, we will first find the principal amount (P) using the information provided about the simple interest for the first scenario, and then we will calculate the total amount after 7.5 years at a different interest rate.

### Step 1: Convert Mixed Numbers to Improper Fractions
The time given is \(4 \frac{2}{3}\) years and the rate is \(6 \frac{3}{4}\%\).

- Convert \(4 \frac{2}{3}\) to an improper fraction:
\[ 4 \frac{2}{3} = \frac{4 \times 3 + 2}{3} = \frac{12 + 2}{3} = \frac{14}{3} \]

- Convert \(6 \frac{3}{4}\%\) to a fraction:
\[ 6 \frac{3}{4} = \frac{6 \times 4 + 3}{4} = \frac{24 + 3}{4} = \frac{27}{4} \]

### Step 2: Use the Simple Interest Formula
The formula for simple interest (SI) is:
\[ SI = \frac{P \times R \times T}{100} \]
Where:
- \(SI = 8820\)
- \(R = \frac{27}{4}\)
- \(T = \frac{14}{3}\)

Substituting the values into the formula:
\[ 8820 = \frac{P \times \frac{27}{4} \times \frac{14}{3}}{100} \]

### Step 3: Rearranging to Find P
Rearranging the equation to solve for \(P\):
\[ P = \frac{8820 \times 100}{\frac{27}{4} \times \frac{14}{3}} \]

### Step 4: Calculate the Denominator
Calculating the denominator:
\[ \frac{27}{4} \times \frac{14}{3} = \frac{27 \times 14}{4 \times 3} = \frac{378}{12} = 31.5 \]

### Step 5: Substitute Back to Find P
Now substitute back to find \(P\):
\[ P = \frac{8820 \times 100}{31.5} = \frac{882000}{31.5} \approx 28000 \]

### Step 6: Calculate Simple Interest for the New Scenario
Now we need to find the amount after \(7 \frac{1}{2}\) years at \(8 \frac{1}{3}\%\).

- Convert \(7 \frac{1}{2}\) to an improper fraction:
\[ 7 \frac{1}{2} = \frac{7 \times 2 + 1}{2} = \frac{14 + 1}{2} = \frac{15}{2} \]

- Convert \(8 \frac{1}{3}\%\) to a fraction:
\[ 8 \frac{1}{3} = \frac{8 \times 3 + 1}{3} = \frac{24 + 1}{3} = \frac{25}{3} \]

### Step 7: Calculate the New Simple Interest
Using the same simple interest formula:
\[ SI = \frac{P \times R \times T}{100} \]
Substituting the values:
\[ SI = \frac{28000 \times \frac{25}{3} \times \frac{15}{2}}{100} \]

### Step 8: Calculate the Amount
Calculating the new simple interest:
\[ SI = \frac{28000 \times 25 \times 15}{3 \times 2 \times 100} = \frac{10500000}{600} = 17500 \]

Now, the total amount \(A\) after \(7 \frac{1}{2}\) years:
\[ A = P + SI = 28000 + 17500 = 45500 \]

### Final Answer
The amount after \(7 \frac{1}{2}\) years at \(8 \frac{1}{3}\%\) is **Rs 45500**.

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