A person invests money in two different schemes for 5 years and 8 years at 12% and 10% simple interest, respectively. At the completion of each scheme he gets the same interest. The ratio of his investments is

Options

A. 4 : 3
B. 1 : 3
C. 1 : 2
D. 2 : 4

Answer

To solve the problem step by step, we will use the formula for simple interest and set up equations based on the information provided.

### Step 1: Define the Variables
Let:
- \( P_1 \) = Principal amount invested in the first scheme
- \( P_2 \) = Principal amount invested in the second scheme

### Step 2: Write the Simple Interest Formula
The formula for simple interest is given by:
\[ \text{Simple Interest} = \frac{P \times R \times T}{100} \]
where \( P \) is the principal, \( R \) is the rate of interest, and \( T \) is the time in years.

### Step 3: Calculate Simple Interest for the First Scheme
For the first scheme:
- Time \( T_1 = 5 \) years
- Rate \( R_1 = 12\% \)

The simple interest \( SI_1 \) for the first scheme is:
\[ SI_1 = \frac{P_1 \times 12 \times 5}{100} \]

### Step 4: Calculate Simple Interest for the Second Scheme
For the second scheme:
- Time \( T_2 = 8 \) years
- Rate \( R_2 = 10\% \)

The simple interest \( SI_2 \) for the second scheme is:
\[ SI_2 = \frac{P_2 \times 10 \times 8}{100} \]

### Step 5: Set the Simple Interests Equal
According to the problem, the simple interests from both schemes are equal:
\[ SI_1 = SI_2 \]
This gives us the equation:
\[ \frac{P_1 \times 12 \times 5}{100} = \frac{P_2 \times 10 \times 8}{100} \]

### Step 6: Simplify the Equation
We can cancel \( 100 \) from both sides:
\[ P_1 \times 12 \times 5 = P_2 \times 10 \times 8 \]

### Step 7: Rearrange the Equation
Rearranging gives us:
\[ P_1 \times 60 = P_2 \times 80 \]

### Step 8: Find the Ratio of Investments
Dividing both sides by \( P_2 \) and \( 60 \):
\[ \frac{P_1}{P_2} = \frac{80}{60} = \frac{4}{3} \]

### Conclusion
The ratio of the investments \( P_1 : P_2 \) is \( 4 : 3 \).