Rs. 1600 becomes Rs. 2000 in 2 years at a certain rate of compound interest. What will be the sum after 4 years ?

Answer

To solve the problem step by step, we need to find out how much the sum will be after 4 years, given that Rs. 1600 becomes Rs. 2000 in 2 years at a certain rate of compound interest.

### Step 1: Identify the given values
- Principal (P) = Rs. 1600
- Amount after 2 years (A) = Rs. 2000
- Time (t) = 2 years

### Step 2: Use the compound interest formula
The formula for compound interest is:
\[ A = P \left(1 + \frac{r}{100}\right)^t \]
Where:
- A = Amount after time t
- P = Principal
- r = Rate of interest per annum
- t = Time in years

### Step 3: Substitute the known values into the formula
Substituting the known values into the formula:
\[ 2000 = 1600 \left(1 + \frac{r}{100}\right)^2 \]

### Step 4: Simplify the equation
First, divide both sides by 1600:
\[ \frac{2000}{1600} = \left(1 + \frac{r}{100}\right)^2 \]
This simplifies to:
\[ \frac{5}{4} = \left(1 + \frac{r}{100}\right)^2 \]

### Step 5: Take the square root of both sides
Taking the square root of both sides gives:
\[ \sqrt{\frac{5}{4}} = 1 + \frac{r}{100} \]
This simplifies to:
\[ \frac{\sqrt{5}}{2} = 1 + \frac{r}{100} \]

### Step 6: Solve for r
Rearranging the equation to find r:
\[ \frac{r}{100} = \frac{\sqrt{5}}{2} - 1 \]
Multiplying by 100:
\[ r = 100 \left(\frac{\sqrt{5}}{2} - 1\right) \]

### Step 7: Calculate the amount after 4 years
Now we need to find the amount after 4 years using the same compound interest formula:
\[ A = 1600 \left(1 + \frac{r}{100}\right)^4 \]

### Step 8: Substitute the value of \(1 + \frac{r}{100}\)
From Step 5, we know that:
\[ 1 + \frac{r}{100} = \frac{\sqrt{5}}{2} \]
So we can substitute this into the formula:
\[ A = 1600 \left(\frac{\sqrt{5}}{2}\right)^4 \]

### Step 9: Calculate \(\left(\frac{\sqrt{5}}{2}\right)^4\)
Calculating \(\left(\frac{\sqrt{5}}{2}\right)^4\):
\[ \left(\frac{\sqrt{5}}{2}\right)^4 = \frac{5^2}{2^4} = \frac{25}{16} \]

### Step 10: Final calculation for the amount
Now substituting back into the amount formula:
\[ A = 1600 \times \frac{25}{16} \]
\[ A = 100 \times 25 = 2500 \]

### Final Answer
The sum after 4 years will be Rs. 2500.

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