On certain amount, the rate of interest for first year is `14 (2)/(7)%` and second year is 10% and CI for second year is 180 then find the amount?

Options

A. ₹600
B. ₹ 700
C. ₹ 500
D. ₹800

Answer

To solve the problem, we need to find the principal amount (P) based on the given compound interest (CI) for the second year and the rates of interest for both years.

### Step-by-Step Solution:

1. **Convert the Rate of Interest for the First Year:**
The rate of interest for the first year is given as \( 14 \frac{2}{7}\% \).
To convert this to an improper fraction:
\[ 14 \frac{2}{7} = \frac{14 \times 7 + 2}{7} = \frac{98 + 2}{7} = \frac{100}{7}\% \]

2. **Convert the Rate of Interest for the Second Year:**
The rate of interest for the second year is given as \( 10\% \).

3. **Use the Compound Interest Formula:**
The formula for compound interest for two years is:
\[ A = P \left(1 + \frac{r_1}{100}\right) \left(1 + \frac{r_2}{100}\right) \]
where \( A \) is the amount after two years, \( P \) is the principal, \( r_1 \) is the rate for the first year, and \( r_2 \) is the rate for the second year.

4. **Substituting Values:**
Here, \( r_1 = \frac{100}{7} \) and \( r_2 = 10 \).
The amount after two years can be expressed as:
\[ A = P \left(1 + \frac{100/7}{100}\right) \left(1 + \frac{10}{100}\right) \]
Simplifying this:
\[ A = P \left(1 + \frac{1}{7}\right) \left(1 + 0.1\right) \]
\[ A = P \left(\frac{8}{7}\right) \left(\frac{11}{10}\right) \]

5. **Calculate the Amount for the Second Year:**
The compound interest for the second year is given as \( 180 \).
The interest for the second year can also be calculated as:
\[ \text{CI for 2nd year} = A_{2nd} - A_{1st} \]
Where:
- \( A_{1st} = P \left(1 + \frac{100/7}{100}\right) = P \left(\frac{8}{7}\right) \)
- \( A_{2nd} = A_{1st} \left(1 + \frac{10}{100}\right) = A_{1st} \left(\frac{11}{10}\right) \)

Therefore, we can express the CI for the second year as:
\[ 180 = A_{2nd} - A_{1st} \]
Substituting the values we have:
\[ 180 = \left(P \left(\frac{8}{7}\right) \left(\frac{11}{10}\right)\right) - \left(P \left(\frac{8}{7}\right)\right) \]
\[ 180 = P \left(\frac{8}{7}\right) \left(\frac{11}{10} - 1\right) \]
\[ 180 = P \left(\frac{8}{7}\right) \left(\frac{11 - 10}{10}\right) \]
\[ 180 = P \left(\frac{8}{7}\right) \left(\frac{1}{10}\right) \]
\[ 180 = \frac{8P}{70} \]
\[ 180 = \frac{4P}{35} \]

6. **Solve for P:**
\[ P = 180 \times \frac{35}{4} \]
\[ P = 180 \times 8.75 = 1575 \]

### Final Answer:
The principal amount (P) is \( 1575 \).