A lent Rs. 600 to B and some amount to c at the rate of `8(1)/(3)%` per annum simple interest. After 5 years he got the total interest of Rs. 400 from B and C together. The amount of money lent by A to C was

Options

A. Rs. 300
B. Rs. 360
C. Rs. 400
D. Rs. 420

Answer

To solve the problem step by step, we will follow these instructions:

### Step 1: Understand the Problem
A lent Rs. 600 to B and an unknown amount (let's call it Rs. X) to C at a simple interest rate of \(8 \frac{1}{3}\%\) per annum. After 5 years, the total interest earned from both B and C is Rs. 400.

### Step 2: Convert the Interest Rate
The interest rate given is \(8 \frac{1}{3}\%\), which can be converted to an improper fraction:
\[ 8 \frac{1}{3} = \frac{25}{3}\% \]

### Step 3: Write the Simple Interest Formula
The formula for simple interest is:
\[ \text{Simple Interest} = \frac{P \times R \times T}{100} \]
Where:
- \(P\) = Principal amount (the amount lent)
- \(R\) = Rate of interest per annum
- \(T\) = Time in years

### Step 4: Calculate Interest from B
For the amount lent to B (Rs. 600):
\[ \text{Interest from B} = \frac{600 \times \frac{25}{3} \times 5}{100} \]
Calculating this:
\[ = \frac{600 \times 25 \times 5}{3 \times 100} = \frac{75000}{300} = 250 \]

### Step 5: Calculate Interest from C
Now, for the amount lent to C (Rs. X):
\[ \text{Interest from C} = \frac{X \times \frac{25}{3} \times 5}{100} \]
Calculating this:
\[ = \frac{X \times 25 \times 5}{3 \times 100} = \frac{125X}{300} = \frac{5X}{12} \]

### Step 6: Set Up the Equation
According to the problem, the total interest from B and C together is Rs. 400:
\[ \text{Interest from B} + \text{Interest from C} = 400 \]
Substituting the values we calculated:
\[ 250 + \frac{5X}{12} = 400 \]

### Step 7: Solve for X
Subtract 250 from both sides:
\[ \frac{5X}{12} = 400 - 250 \]
\[ \frac{5X}{12} = 150 \]
Now, multiply both sides by 12 to eliminate the fraction:
\[ 5X = 150 \times 12 \]
\[ 5X = 1800 \]
Now, divide both sides by 5:
\[ X = \frac{1800}{5} = 360 \]

### Final Answer
The amount of money lent by A to C is Rs. 360.

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