A man invests rupes 3000 for three years at compounds interest . After one years . The money amount to rupes 3,240 find the rate of interest and the amount (to the nearst rupee) due at the end of 3 years.

Answer

To solve the problem step by step, we will first find the rate of interest and then calculate the amount due at the end of three years.

### Step 1: Identify the given values
- Principal (P) = ₹3000
- Amount after 1 year (A) = ₹3240
- Time (t) = 1 year

### Step 2: Calculate the interest earned in the first year
The interest earned can be calculated using the formula:
\[ \text{Interest} = \text{Amount} - \text{Principal} \]
So,
\[ \text{Interest} = 3240 - 3000 = ₹240 \]

### Step 3: Use the simple interest formula to find the rate
Since the interest for the first year in compound interest is the same as in simple interest, we can use the formula for simple interest:
\[ \text{SI} = \frac{P \times r \times t}{100} \]
Where:
- SI = Simple Interest (which is ₹240)
- P = Principal (₹3000)
- r = Rate of interest (unknown)
- t = Time (1 year)

Substituting the known values into the formula:
\[ 240 = \frac{3000 \times r \times 1}{100} \]

### Step 4: Solve for r
Rearranging the equation:
\[ 240 = \frac{3000r}{100} \]
\[ 240 = 30r \]
Now, divide both sides by 30:
\[ r = \frac{240}{30} = 8 \]

So, the rate of interest is **8%**.

### Step 5: Calculate the amount at the end of 3 years
Now we will use the compound interest formula to find the amount after 3 years:
\[ A = P \left(1 + \frac{r}{100}\right)^n \]
Where:
- A = Amount after n years
- P = Principal (₹3000)
- r = Rate of interest (8%)
- n = Number of years (3)

Substituting the known values:
\[ A = 3000 \left(1 + \frac{8}{100}\right)^3 \]
\[ A = 3000 \left(1 + 0.08\right)^3 \]
\[ A = 3000 \left(1.08\right)^3 \]

### Step 6: Calculate \( (1.08)^3 \)
Calculating \( (1.08)^3 \):
\[ (1.08)^3 \approx 1.259712 \]

### Step 7: Calculate the final amount
Now substituting back into the amount formula:
\[ A = 3000 \times 1.259712 \]
\[ A \approx 3779.136 \]

Rounding to the nearest rupee, the amount is **₹3779**.

### Final Answer
- Rate of Interest = **8%**
- Amount due at the end of 3 years = **₹3779**