A sum of ₹6900 was lent partly at 5% and the rest at 8% simple interest. Total interest received after 3 years from both was ₹1359. What was the ratio of money lent at 5% and 8%?

Options

A. 10 : 13
B. 11 : 12
C. 14 : 9
D. 31 : 38

Answer

To solve the problem step by step, let's denote the amount lent at 5% as \( x \) and the amount lent at 8% as \( 6900 - x \).

### Step 1: Set up the interest equations
The total interest earned from both amounts after 3 years is given as ₹1359. The interest from the amount lent at 5% for 3 years can be calculated as:
\[ \text{Interest from } x = \frac{5}{100} \times x \times 3 = \frac{15}{100} x = 0.15x \]
The interest from the amount lent at 8% for 3 years is:
\[ \text{Interest from } (6900 - x) = \frac{8}{100} \times (6900 - x) \times 3 = \frac{24}{100} (6900 - x) = 0.24(6900 - x) \]

### Step 2: Write the total interest equation
Now, we can write the equation for the total interest:
\[ 0.15x + 0.24(6900 - x) = 1359 \]

### Step 3: Simplify the equation
Expanding the equation:
\[ 0.15x + 0.24 \times 6900 - 0.24x = 1359 \]
Calculating \( 0.24 \times 6900 \):
\[ 0.24 \times 6900 = 1656 \]
So the equation becomes:
\[ 0.15x - 0.24x + 1656 = 1359 \]
Combining like terms:
\[ -0.09x + 1656 = 1359 \]

### Step 4: Solve for \( x \)
Now, isolate \( x \):
\[ -0.09x = 1359 - 1656 \]
Calculating the right side:
\[ 1359 - 1656 = -297 \]
So:
\[ -0.09x = -297 \]
Dividing both sides by -0.09:
\[ x = \frac{297}{0.09} = 3300 \]

### Step 5: Find the amount lent at 8%
Now, calculate the amount lent at 8%:
\[ 6900 - x = 6900 - 3300 = 3600 \]

### Step 6: Determine the ratio
Now we have:
- Amount lent at 5%: ₹3300
- Amount lent at 8%: ₹3600

The ratio of money lent at 5% to the money lent at 8% is:
\[ \text{Ratio} = \frac{3300}{3600} = \frac{11}{12} \]

### Final Answer
Thus, the ratio of money lent at 5% to that lent at 8% is \( 11:12 \).

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