A certain sum of money amounts to Rs. 2900 at 4 % per annum in 4 years. In how many years will it amount to rupees 5000 at the same rate?

Options

A. a) 20
B. b) 22
C. c) 24
D. d) 25

Explanation

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

### Step 1: Understand the given information
We know that a certain sum of money amounts to Rs. 2900 at 4% per annum in 4 years. We need to find out how many years it will take for the same sum to amount to Rs. 5000 at the same interest rate.

### Step 2: Calculate the total interest earned in 4 years
Using the formula for Simple Interest (SI):
\[ \text{SI} = \frac{P \times R \times T}{100} \]

Where:
- \( P \) = Principal amount
- \( R \) = Rate of interest per annum
- \( T \) = Time in years

In this case, we need to first find the principal amount \( P \) using the total amount after 4 years.

### Step 3: Determine the total amount after 4 years
The total amount \( A \) after 4 years is given as Rs. 2900. The formula for the total amount is:
\[ A = P + \text{SI} \]

### Step 4: Express SI in terms of P
From the formula for SI:
\[ \text{SI} = A - P \]
Substituting the values, we have:
\[ \text{SI} = 2900 - P \]

### Step 5: Substitute SI into the SI formula
Now we can substitute SI into the SI formula:
\[ 2900 - P = \frac{P \times 4 \times 4}{100} \]
\[ 2900 - P = \frac{16P}{100} \]
\[ 2900 - P = 0.16P \]

### Step 6: Combine like terms
Rearranging gives:
\[ 2900 = P + 0.16P \]
\[ 2900 = 1.16P \]

### Step 7: Solve for P
Now, solving for \( P \):
\[ P = \frac{2900}{1.16} \]
Calculating this gives:
\[ P = 2500 \]

### Step 8: Find the time to amount to Rs. 5000
Now we need to find out how many years it will take for this principal amount \( P = 2500 \) to amount to Rs. 5000 at the same rate of 4%.

### Step 9: Calculate the new SI for Rs. 5000
Using the formula for the total amount again:
\[ A = P + \text{SI} \]
We can express SI as:
\[ \text{SI} = A - P = 5000 - 2500 = 2500 \]

### Step 10: Use the SI formula to find time
Now we can use the SI formula:
\[ 2500 = \frac{2500 \times 4 \times T}{100} \]
Simplifying gives:
\[ 2500 = 100T \]
\[ T = \frac{2500}{100} = 25 \]

### Final Answer
It will take **25 years** for the amount to reach Rs. 5000 at the same rate of interest.

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