Ankit invests some money at 9% simple interest for 2 years, and the same sum for 4 years at 10% per annum. He eams Rs 1,740 in all. The sum invested in each case is:

Options

A. Rs 4,350
B. Rs 3,000
C. Rs 6,000
D. Rs 4,500

Answer

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

### Step 1: Define the Variables
Let the principal amount (the sum invested) be denoted as \( P \).

### Step 2: Calculate the Interest for the First Investment
Ankit invests this amount at a rate of 9% for 2 years. The formula for simple interest is:
\[ \text{Simple Interest} = \frac{P \times R \times T}{100} \]
For the first investment:
- Rate \( R = 9\% \)
- Time \( T = 2 \) years

Thus, the interest earned from the first investment is:
\[ \text{Interest}_1 = \frac{P \times 9 \times 2}{100} = \frac{18P}{100} = \frac{9P}{50} \]

### Step 3: Calculate the Interest for the Second Investment
Ankit invests the same amount at a rate of 10% for 4 years. Using the same formula:
- Rate \( R = 10\% \)
- Time \( T = 4 \) years

The interest earned from the second investment is:
\[ \text{Interest}_2 = \frac{P \times 10 \times 4}{100} = \frac{40P}{100} = \frac{2P}{5} \]

### Step 4: Set Up the Total Interest Equation
According to the problem, the total interest earned from both investments is Rs. 1,740. Therefore, we can set up the equation:
\[ \text{Interest}_1 + \text{Interest}_2 = 1740 \]
Substituting the expressions we derived:
\[ \frac{9P}{50} + \frac{2P}{5} = 1740 \]

### Step 5: Find a Common Denominator
To solve the equation, we need a common denominator. The least common multiple of 50 and 5 is 50. We rewrite \( \frac{2P}{5} \) as \( \frac{20P}{50} \):
\[ \frac{9P}{50} + \frac{20P}{50} = 1740 \]
Combining the fractions gives:
\[ \frac{29P}{50} = 1740 \]

### Step 6: Solve for \( P \)
To isolate \( P \), multiply both sides by 50:
\[ 29P = 1740 \times 50 \]
Calculating the right side:
\[ 29P = 87000 \]
Now, divide both sides by 29:
\[ P = \frac{87000}{29} \]
Calculating \( P \):
\[ P = 3000 \]

### Conclusion
The sum invested in each case is Rs. 3,000.

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