On certain amount, the rate of interest for first year `14 (2)/(7)%`and second year is 12.5% and it becomes 630 in 2 years then find the amount?

Options

A. ₹ 480
B. ₹ 500
C. ₹483
D. ₹490

Answer

To solve the problem step by step, we need to find the principal amount (P) given the compound interest rates for two years and the total amount after two years.

### Step 1: Convert the rates of interest into decimal form
- The rate of interest for the first year is \( 14 \frac{2}{7} \% \).
- Convert \( 14 \frac{2}{7} \) to an improper fraction:
\[ 14 \frac{2}{7} = \frac{14 \times 7 + 2}{7} = \frac{98 + 2}{7} = \frac{100}{7} \% \]
- Convert this percentage to decimal:
\[ \frac{100}{7} \% = \frac{100}{7 \times 100} = \frac{1}{7} \approx 0.142857 \]

- The rate of interest for the second year is \( 12.5\% \).
- Convert this percentage to decimal:
\[ 12.5\% = \frac{12.5}{100} = 0.125 \]

### Step 2: Calculate the total amount after two years
We know that the total amount after two years is \( 630 \).

### Step 3: Use the formula for compound interest
The formula for the amount \( A \) after \( n \) years with principal \( P \) and rate \( r \) is:
\[ A = P(1 + r_1)(1 + r_2) \]
Where:
- \( r_1 \) is the rate for the first year,
- \( r_2 \) is the rate for the second year.

Substituting the values we have:
\[ 630 = P(1 + 0.142857)(1 + 0.125) \]

### Step 4: Simplify the equation
Calculate \( (1 + 0.142857) \) and \( (1 + 0.125) \):
\[ 1 + 0.142857 \approx 1.142857 \]
\[ 1 + 0.125 = 1.125 \]

Now substitute these values back into the equation:
\[ 630 = P \times 1.142857 \times 1.125 \]

### Step 5: Calculate \( 1.142857 \times 1.125 \)
\[ 1.142857 \times 1.125 \approx 1.284 \]

### Step 6: Solve for \( P \)
Now we have:
\[ 630 = P \times 1.284 \]
To find \( P \):
\[ P = \frac{630}{1.284} \approx 490.5 \]

### Conclusion
The principal amount \( P \) is approximately \( 490.5 \).