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

Options

A. 0.06
B. 0.075
C. 0.1
D. `12.5%`

Answer

To solve the problem step by step, we will use the formula for Simple Interest (SI) and the information provided in the question.

### Step 1: Understand the given information
- P's investment (Principal, P1) = Rs 5400
- Q's investment (Principal, P2) = Rs 7200
- Time for P (T1) = 4 years
- Time for Q (T2) = 3 years
- Total Simple Interest (SI_total) = Rs 3456
- Rate of interest for P and Q is in the ratio of 3:5.

### Step 2: Set up the equations for Simple Interest
The formula for Simple Interest is:
\[ \text{SI} = \frac{P \times R \times T}{100} \]

For P:
\[ \text{SI}_P = \frac{5400 \times R_1 \times 4}{100} \]
For Q:
\[ \text{SI}_Q = \frac{7200 \times R_2 \times 3}{100} \]

### Step 3: Express the rates in terms of a variable
Let the rate for P be \( R_1 = 3x \) and for Q be \( R_2 = 5x \) (since the ratio is 3:5).

### Step 4: Substitute the rates into the Simple Interest equations
Substituting \( R_1 \) and \( R_2 \) into the SI equations:
\[ \text{SI}_P = \frac{5400 \times 3x \times 4}{100} = \frac{64800x}{100} = 648x \]
\[ \text{SI}_Q = \frac{7200 \times 5x \times 3}{100} = \frac{108000x}{100} = 1080x \]

### Step 5: Set up the total Simple Interest equation
Now, we know that:
\[ \text{SI}_P + \text{SI}_Q = 3456 \]
Substituting the values we found:
\[ 648x + 1080x = 3456 \]
\[ 1728x = 3456 \]

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

### Step 7: Find the rates for P and Q
Now substitute \( x \) back to find the rates:
- For P:
\[ R_1 = 3x = 3 \times 2 = 6\% \]
- For Q:
\[ R_2 = 5x = 5 \times 2 = 10\% \]

### Final Answer
The rate at which P invested the money is **6%**.

---