Arya and Sonu invested Rs 5400 and Rs 7200 for 4 years and 3 years respectively. It the total simple interest received by both of them is 3456 and the rate charged by them is in the ratio of 3 : 5. At what rate did Arya invest the money

Options

A. `11%`
B. `8%`
C. `6%`
D. `9%`

Answer

To solve the problem step by step, we will follow the information given in the question and apply the formula for simple interest.

### Step 1: Identify the given values
- Arya's investment (Principal, P1) = Rs 5400
- Sonu's investment (Principal, P2) = Rs 7200
- Time for Arya (T1) = 4 years
- Time for Sonu (T2) = 3 years
- Total Simple Interest (SI) = Rs 3456
- Rate of interest ratio = 3:5

### Step 2: Define the rates of interest
Let Arya's rate of interest be \(3x\) and Sonu's rate of interest be \(5x\).

### Step 3: Write the formula for Simple Interest
The formula for Simple Interest is:
\[ SI = \frac{P \times R \times T}{100} \]
Using this formula, we can calculate the Simple Interest for both Arya and Sonu.

### Step 4: Calculate Simple Interest for Arya and Sonu
- Simple Interest for Arya (SI1):
\[ SI1 = \frac{5400 \times (3x) \times 4}{100} = \frac{64800x}{100} = 648x \]

- Simple Interest for Sonu (SI2):
\[ SI2 = \frac{7200 \times (5x) \times 3}{100} = \frac{108000x}{100} = 1080x \]

### Step 5: Set up the equation for total Simple Interest
According to the problem, the total Simple Interest from both Arya and Sonu is Rs 3456:
\[ SI1 + SI2 = 3456 \]
Substituting the values we calculated:
\[ 648x + 1080x = 3456 \]
\[ 1728x = 3456 \]

### Step 6: Solve for \(x\)
To find \(x\), divide both sides by 1728:
\[ x = \frac{3456}{1728} = 2 \]

### Step 7: Calculate Arya's rate of interest
Now that we have \(x\), we can find Arya's rate of interest:
\[ \text{Arya's rate} = 3x = 3 \times 2 = 6\% \]

### Final Answer
Arya invested the money at a rate of **6%**.

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