A sum of Rs 1,950 was invested partly at 5% and partly at 8% per annum on simple interest. The total interest received after three years was Rs 300. The ratio of the money invested at 5% to that invested at 8% is .

Options

A. `113 : 5`
B. `112 : 5`
C. `111 : 7`
D. `113 : 7`

Answer

To solve the problem step by step, we will use the information given about the investments and the interest earned.

### Step 1: Define the variables
Let:
- \( x \) = amount invested at 5%
- \( y \) = amount invested at 8%

From the problem, we know that:
\[ x + y = 1950 \quad \text{(1)} \]

### Step 2: Use the interest formula
The formula for simple interest is:
\[ \text{Interest} = \text{Principal} \times \text{Rate} \times \text{Time} \]

For the amount invested at 5%:
\[ \text{Interest from } x = x \times \frac{5}{100} \times 3 = \frac{15x}{100} = 0.15x \]

For the amount invested at 8%:
\[ \text{Interest from } y = y \times \frac{8}{100} \times 3 = \frac{24y}{100} = 0.24y \]

### Step 3: Set up the total interest equation
According to the problem, the total interest received after 3 years is Rs 300:
\[ 0.15x + 0.24y = 300 \quad \text{(2)} \]

### Step 4: Solve the system of equations
Now we have two equations:
1. \( x + y = 1950 \)
2. \( 0.15x + 0.24y = 300 \)

From equation (1), we can express \( y \) in terms of \( x \):
\[ y = 1950 - x \]

Substituting \( y \) in equation (2):
\[ 0.15x + 0.24(1950 - x) = 300 \]

### Step 5: Simplify and solve for \( x \)
Expanding the equation:
\[ 0.15x + 0.24 \times 1950 - 0.24x = 300 \]
\[ 0.15x - 0.24x + 468 = 300 \]
Combine like terms:
\[ -0.09x + 468 = 300 \]
Subtract 468 from both sides:
\[ -0.09x = 300 - 468 \]
\[ -0.09x = -168 \]
Dividing both sides by -0.09:
\[ x = \frac{-168}{-0.09} = 1866.67 \]

### Step 6: Find \( y \)
Now substitute \( x \) back into equation (1):
\[ y = 1950 - 1866.67 = 83.33 \]

### Step 7: Find the ratio of investments
Now we need the ratio of the amounts invested at 5% to that invested at 8%:
\[ \text{Ratio} = \frac{x}{y} = \frac{1866.67}{83.33} \]

To simplify:
\[ \text{Ratio} = \frac{1866.67 \div 83.33}{83.33 \div 83.33} = \frac{22.4}{1} \]

### Step 8: Final ratio
Thus, the ratio of the money invested at 5% to that invested at 8% is approximately:
\[ \text{Ratio} = 22.4 : 1 \]