A sum of money becomes 8 times in 1.5 year if compounded half yearly. How much time it will take to become 81 times if compounded yearly.

Options

A. A)4 years
B. B)9 years
C. C)3 years
D. D)5years

Answer

To solve the problem step by step, we will break it down into manageable parts.

### Step 1: Understand the first case (Compounded Half-Yearly)
We know that a sum of money becomes 8 times in 1.5 years when compounded half-yearly.

### Step 2: Determine the number of compounding cycles
Since the interest is compounded half-yearly, we need to double the time given to find the number of compounding cycles:
- Time = 1.5 years
- Number of cycles = 2 * Time = 2 * 1.5 = 3 cycles

### Step 3: Set up the equation for the amount
The formula for compound interest is:
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
Where:
- \( A \) = final amount
- \( P \) = principal amount
- \( r \) = annual interest rate
- \( n \) = number of times interest is compounded per year
- \( t \) = time in years

In our case, since the amount becomes 8 times the principal:
\[ A = 8P \]

### Step 4: Find the ratio for one cycle
The ratio of the amount to the principal after 3 cycles is:
\[ \frac{A}{P} = 8 \]
Thus:
\[ \left(1 + \frac{r}{2}\right)^3 = 8 \]
Taking the cube root of both sides:
\[ 1 + \frac{r}{2} = 2 \]
So:
\[ \frac{r}{2} = 1 \]
This gives:
\[ r = 2 \text{ (or 200% per annum)} \]

### Step 5: Understand the second case (Compounded Yearly)
Now we need to find out how long it will take for the amount to become 81 times when compounded yearly.

### Step 6: Set up the equation for the second case
Using the same compound interest formula, we want:
\[ A = 81P \]
So:
\[ \frac{A}{P} = 81 \]
This can be expressed as:
\[ (1 + r)^t = 81 \]

### Step 7: Substitute the interest rate
From the first case, we found that the annual interest rate \( r = 2 \) (or 200%). Therefore:
\[ (1 + 2)^t = 81 \]
This simplifies to:
\[ 3^t = 81 \]

### Step 8: Solve for \( t \)
Since \( 81 = 3^4 \), we can set the exponents equal to each other:
\[ t = 4 \]

### Conclusion
Thus, it will take **4 years** for the sum of money to become 81 times when compounded yearly.

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