The interest charged on a certain sum is 720 for one year and 1.497.60 for two years. Find, whether the interest is simple or compound Also, calculate the rate per cent and the sum.

Answer

To solve the problem, we need to determine whether the interest is simple or compound, and then calculate the rate of interest and the principal sum. Here’s how we can do it step by step:

### Step 1: Identify the given information
- Interest for 1 year (I1) = 720
- Interest for 2 years (I2) = 1497.60

### Step 2: Calculate the interest for the second year
To determine if the interest is simple or compound, we need to find the interest for the second year.

\[ \text{Interest for the second year} (I2 - I1) = 1497.60 - 720 = 777.60 \]

### Step 3: Determine the type of interest
Since the interest for the second year (777.60) is different from the first year (720), this indicates that the interest is **compound interest**.

### Step 4: Use the formula for compound interest
For compound interest, the total amount after 2 years can be expressed as:

\[ A = P(1 + \frac{r}{100})^t \]

Where:
- \(A\) = Total amount after \(t\) years
- \(P\) = Principal amount
- \(r\) = Rate of interest per annum
- \(t\) = Time in years

For 2 years, we can express it as:

\[ A = P(1 + \frac{r}{100})^2 \]

### Step 5: Set up the equations
From the information given:
1. For 1 year:
\[ I1 = \frac{P \cdot r \cdot 1}{100} = 720 \quad \text{(Equation 1)} \]

2. For 2 years:
\[ I2 = P(1 + \frac{r}{100})^2 - P = 1497.60 \quad \text{(Equation 2)} \]

### Step 6: Solve Equation 1 for \(P \cdot r\)
From Equation 1:
\[ P \cdot r = 72000 \quad \text{(Multiply both sides by 100)} \]

### Step 7: Substitute into Equation 2
Substituting \(P \cdot r\) into Equation 2:
\[ P(1 + \frac{r}{100})^2 - P = 1497.60 \]
This can be simplified to:
\[ P \left( (1 + \frac{r}{100})^2 - 1 \right) = 1497.60 \]

### Step 8: Expand and simplify
Expanding \((1 + \frac{r}{100})^2\):
\[ (1 + \frac{r}{100})^2 = 1 + 2 \cdot \frac{r}{100} + \frac{r^2}{10000} \]
Thus,
\[ P \left( 2 \cdot \frac{r}{100} + \frac{r^2}{10000} \right) = 1497.60 \]

### Step 9: Substitute \(P\) from Equation 1
Using \(P = \frac{72000}{r}\):
\[ \frac{72000}{r} \left( 2 \cdot \frac{r}{100} + \frac{r^2}{10000} \right) = 1497.60 \]

### Step 10: Solve for \(r\)
This leads to:
\[ 72000 \left( \frac{2}{100} + \frac{r}{10000} \right) = 1497.60 \cdot r \]
Solving this will give us the value of \(r\).

### Step 11: Calculate \(P\) using \(r\)
Once \(r\) is found, substitute it back into \(P = \frac{72000}{r}\) to find the principal amount.

### Final Answers
After performing the calculations, we find:
- The rate of interest \(r = 8\%\)
- The principal amount \(P = 9000\)