The simple interest on a sum of ₹4,800 for `4(1)/(2)` years at a certain rate per annum ₹1,684.80. What will be the amount of the same sum at the same rate for `6(2)/(3)` years at simple interest?

Options

A. ₹7,096
B. ₹7,084
C. ₹7,296
D. ₹7,298

Answer

To solve the problem step by step, we will follow the process of calculating the simple interest and then finding the total amount for the new time period.

### Step 1: Identify the given values
- Principal (P) = ₹4,800
- Time (T) = 4.5 years (which is \(4 \frac{1}{2}\) years)
- Simple Interest (SI) = ₹1,684.80

### Step 2: Use the formula for Simple Interest
The formula for Simple Interest is given by:
\[ SI = \frac{P \times R \times T}{100} \]
Where:
- SI = Simple Interest
- P = Principal
- R = Rate of interest per annum
- T = Time in years

### Step 3: Rearranging the formula to find the Rate (R)
We can rearrange the formula to find the rate of interest (R):
\[ R = \frac{SI \times 100}{P \times T} \]

### Step 4: Substitute the known values to find R
Substituting the known values into the formula:
\[ R = \frac{1684.80 \times 100}{4800 \times 4.5} \]

### Step 5: Calculate the denominator
Calculating the denominator:
\[ 4800 \times 4.5 = 21600 \]

### Step 6: Calculate R
Now substituting back:
\[ R = \frac{168480}{21600} = 7.8\% \]

### Step 7: Find the new time period
The new time period is \(6 \frac{2}{3}\) years, which can be converted to an improper fraction:
\[ 6 \frac{2}{3} = \frac{20}{3} \text{ years} \]

### Step 8: Calculate the new Simple Interest (SI)
Using the same formula for Simple Interest for the new time period:
\[ SI_{new} = \frac{P \times R \times T_{new}}{100} \]
Substituting the values:
\[ SI_{new} = \frac{4800 \times 7.8 \times \frac{20}{3}}{100} \]

### Step 9: Calculate SI
Calculating:
\[ SI_{new} = \frac{4800 \times 7.8 \times 20}{300} \]
Calculating \(4800 \times 7.8 \times 20 = 752800\)
Now divide by 300:
\[ SI_{new} = \frac{752800}{300} = 2509.33 \]

### Step 10: Find the total amount
The total amount (A) after the new time period is given by:
\[ A = P + SI_{new} \]
Substituting the values:
\[ A = 4800 + 2509.33 = 7309.33 \]

### Final Answer
The total amount after \(6 \frac{2}{3}\) years at the same rate of simple interest is approximately ₹7309.33.

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