Mohan lends Rs. 4,800 to John for `4""1/2` years and Rs. 2,500 to Shyam for 6 years and receives a total sum of Rs. 2,196 as interest. Find the rate per cent per annum, it being the same in both the cases.

Answer

To find the rate of interest per annum that Mohan receives from John and Shyam, we will follow these steps:

### Step 1: Identify the given values
- Principal lent to John (P1) = Rs. 4,800
- Time lent to John (T1) = 4.5 years (which can be expressed as \( \frac{9}{2} \) years)
- Principal lent to Shyam (P2) = Rs. 2,500
- Time lent to Shyam (T2) = 6 years
- Total interest received (I) = Rs. 2,196

### Step 2: Write the formula for simple interest
The formula for simple interest is:
\[ I = \frac{P \times R \times T}{100} \]
where:
- \( I \) = Interest
- \( P \) = Principal
- \( R \) = Rate of interest per annum
- \( T \) = Time in years

### Step 3: Calculate the interest from both loans
1. Interest from John:
\[ I_1 = \frac{P_1 \times R \times T_1}{100} = \frac{4800 \times R \times \frac{9}{2}}{100} \]
Simplifying this:
\[ I_1 = \frac{4800 \times R \times 9}{200} = 216R \]

2. Interest from Shyam:
\[ I_2 = \frac{P_2 \times R \times T_2}{100} = \frac{2500 \times R \times 6}{100} \]
Simplifying this:
\[ I_2 = \frac{2500 \times R \times 6}{100} = 150R \]

### Step 4: Write the equation for total interest
The total interest received from both loans is:
\[ I_1 + I_2 = 2196 \]
Substituting the values we calculated:
\[ 216R + 150R = 2196 \]

### Step 5: Combine like terms
Combine the terms on the left:
\[ 366R = 2196 \]

### Step 6: Solve for R
To find the rate of interest \( R \):
\[ R = \frac{2196}{366} \]
Calculating this gives:
\[ R = 6 \]

### Conclusion
The rate of interest per annum is **6%**.

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