A man puts $P in a bank which offers n% interest compounded annually. After 2 years , the amount of money in the bank is $M. Which of the following ist the value of r if M =$1728 and P=1200?

Answer

To solve the problem, we need to use the formula for compound interest, which is:

\[ M = P \left(1 + \frac{n}{100}\right)^t \]

Where:
- \( M \) is the amount of money after time \( t \),
- \( P \) is the principal amount (initial investment),
- \( n \) is the interest rate,
- \( t \) is the time in years.

Given:
- \( M = 1728 \)
- \( P = 1200 \)
- \( t = 2 \) years

We need to find the value of \( n \).

### Step 1: Substitute the known values into the formula

Substituting \( M \), \( P \), and \( t \) into the formula gives us:

\[ 1728 = 1200 \left(1 + \frac{n}{100}\right)^2 \]

### Step 2: Divide both sides by 1200

To isolate the compound interest term, divide both sides by 1200:

\[ \frac{1728}{1200} = \left(1 + \frac{n}{100}\right)^2 \]

### Step 3: Simplify the left side

Calculating \( \frac{1728}{1200} \):

\[ \frac{1728}{1200} = 1.44 \]

So, we have:

\[ 1.44 = \left(1 + \frac{n}{100}\right)^2 \]

### Step 4: Take the square root of both sides

Taking the square root of both sides gives:

\[ \sqrt{1.44} = 1 + \frac{n}{100} \]

Calculating the square root:

\[ 1.2 = 1 + \frac{n}{100} \]

### Step 5: Isolate \( n \)

Now, subtract 1 from both sides:

\[ 1.2 - 1 = \frac{n}{100} \]

This simplifies to:

\[ 0.2 = \frac{n}{100} \]

### Step 6: Solve for \( n \)

To find \( n \), multiply both sides by 100:

\[ n = 0.2 \times 100 = 20 \]

Thus, the value of \( n \) is:

\[ n = 20\% \]

### Final Answer:
The interest rate \( n \) is **20%**.

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