If Rs 5600 amounts to Rs 6678 in `3(1)/(2)` years, what will Rs 9600 amount to in `5(1)/(4)` years at the same rate of interset ?

Options

A. Rs 12732
B. Rs 12372
C. Rs 12722
D. Rs 12237

Answer

To solve the problem step by step, we will first determine the rate of interest using the information provided and then calculate the amount for Rs 9600 in the specified time period.

### Step 1: Identify the given values
- Principal (P1) = Rs 5600
- Amount (A1) = Rs 6678
- Time (T1) = 3.5 years (which is 3 years and 6 months or \( \frac{7}{2} \) years)

### Step 2: Calculate the Simple Interest (SI) for the first scenario
Using the formula for the total amount:
\[ A = P + SI \]
We can rearrange this to find the Simple Interest:
\[ SI = A - P \]
Substituting the known values:
\[ SI = 6678 - 5600 = 1078 \]

### Step 3: Use the Simple Interest formula to find the rate of interest (R)
The formula for Simple Interest is:
\[ SI = \frac{P \times R \times T}{100} \]
Rearranging to find R gives:
\[ R = \frac{SI \times 100}{P \times T} \]
Substituting the known values:
\[ R = \frac{1078 \times 100}{5600 \times \frac{7}{2}} \]
Calculating the denominator:
\[ 5600 \times \frac{7}{2} = 5600 \times 3.5 = 19600 \]
Now substituting this back into the equation for R:
\[ R = \frac{107800}{19600} \]
Calculating R:
\[ R = 5.5\% \]

### Step 4: Calculate the Simple Interest for the second scenario
Now we need to find the amount for Rs 9600 in \( 5.25 \) years (which is \( 5 \frac{1}{4} \) years or \( \frac{21}{4} \) years).
Using the same formula for Simple Interest:
\[ SI = \frac{P \times R \times T}{100} \]
Substituting the values:
\[ SI = \frac{9600 \times 5.5 \times \frac{21}{4}}{100} \]
Calculating:
1. First calculate \( 9600 \times 5.5 = 52800 \)
2. Then calculate \( 52800 \times \frac{21}{4} = 52800 \times 5.25 = 27720 \)
3. Finally, divide by 100:
\[ SI = \frac{27720}{100} = 277.2 \]

### Step 5: Calculate the total amount for Rs 9600
Using the total amount formula again:
\[ A = P + SI \]
Substituting the values:
\[ A = 9600 + 277.2 = 9877.2 \]

### Final Answer
The total amount for Rs 9600 after \( 5 \frac{1}{4} \) years at the same rate of interest is Rs 9877.2.

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