A sum of money becomes 9 times in 1 year if compounded half yearly. How much time it will take to become 125 times if compounded yearly.

Answer

To solve the problem step by step, we will break it down into two parts: first, we will find the interest rate when the sum becomes 9 times in 1 year compounded half-yearly, and then we will determine how long it will take for the same sum to become 125 times when compounded yearly.

### Step 1: Understanding the first scenario
We know that a sum of money becomes 9 times in 1 year when compounded half-yearly.

Let:
- \( P \) = initial amount
- After 1 year, the amount becomes \( 9P \)
- Compounded half-yearly means it is compounded 2 times a year, so \( N = 2 \)
- Time \( T = 1 \) year

### Step 2: Setting up the compound interest formula
The formula for compound interest is given by:
\[ A = P \left(1 + \frac{R}{100N}\right)^{NT} \]
Where:
- \( A \) = final amount
- \( R \) = rate of interest per annum
- \( N \) = number of times interest is compounded per year
- \( T \) = time in years

In our case:
\[ 9P = P \left(1 + \frac{R}{200}\right)^{2 \times 1} \]

### Step 3: Simplifying the equation
Dividing both sides by \( P \):
\[ 9 = \left(1 + \frac{R}{200}\right)^{2} \]

### Step 4: Taking the square root
Taking the square root of both sides:
\[ 3 = 1 + \frac{R}{200} \]

### Step 5: Solving for \( R \)
Subtracting 1 from both sides:
\[ 2 = \frac{R}{200} \]
Multiplying both sides by 200:
\[ R = 400\% \]

### Step 6: Moving to the second scenario
Now, we need to find out how long it will take for the same sum to become 125 times when compounded yearly.

Let:
- New amount \( A = 125P \)
- Compounded yearly means \( N = 1 \)

### Step 7: Setting up the new compound interest equation
Using the compound interest formula again:
\[ 125P = P \left(1 + \frac{400}{100}\right)^{1 \times T} \]

### Step 8: Simplifying the new equation
Dividing both sides by \( P \):
\[ 125 = \left(1 + 4\right)^{T} \]
\[ 125 = 5^{T} \]

### Step 9: Solving for \( T \)
We can express 125 as \( 5^3 \):
\[ 5^3 = 5^{T} \]
Since the bases are the same, we can equate the exponents:
\[ T = 3 \]

### Final Answer
It will take **3 years** for the sum to become 125 times when compounded yearly.

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