Ankit , Bhavya and chiruy together starts a business . Bhavya invested 25 % more amount than Ankit as well as `33 1/3 ` % less amount than Chiru . If ratio of profit sharing of Ankit , Bhavya and Chiru is `8:15:25` , then find for what time Ankit invested .if Chiru invested for 20 months .

Options

A. A)6 months
B. B)9 months
C. C)12 months
D. D)15 months

Answer

To solve the problem step by step, we will follow the information provided in the question and the video transcript.

### Step 1: Define Variables for Investments
Let the investment of Ankit be denoted as \( A \). According to the problem, Bhavya invested 25% more than Ankit, and Chiru invested 33⅓% more than Bhavya.

- Therefore, Bhavya's investment \( B \) can be expressed as:
\[ B = A + 0.25A = 1.25A \]

- Let Chiru's investment be \( C \). Since Bhavya invested 33⅓% less than Chiru, we can express Chiru's investment as:
\[ B = C - \frac{1}{3}C = \frac{2}{3}C \implies C = \frac{3}{2}B \]

### Step 2: Substitute Bhavya's Investment
Substituting \( B \) from the first equation into the equation for \( C \):
\[ C = \frac{3}{2}(1.25A) = \frac{3.75}{2}A = 1.875A \]

### Step 3: Establish Investment Ratios
Now we have the investments in terms of \( A \):
- Ankit's investment \( A \)
- Bhavya's investment \( 1.25A \)
- Chiru's investment \( 1.875A \)

To find the ratio of their investments, we can express them in a simpler form:
\[ A : 1.25A : 1.875A = 1 : 1.25 : 1.875 \]

To eliminate decimals, we can multiply each term by 8:
\[ 1 \times 8 : 1.25 \times 8 : 1.875 \times 8 = 8 : 10 : 15 \]

### Step 4: Compare with Profit Sharing Ratio
The profit-sharing ratio is given as \( 8 : 15 : 25 \). We will focus on Ankit and Chiru:
- Ankit's profit share = 8
- Chiru's profit share = 25

### Step 5: Set Up the Time Investment Equation
Let \( K \) be the time Ankit invested. Chiru invested for 20 months. The profit is proportional to the investment and the time for which it is invested:
\[ \frac{8K}{15 \times 20} = \frac{8}{25} \]

### Step 6: Cross Multiply and Solve for \( K \)
Cross-multiplying gives:
\[ 8K \times 25 = 8 \times (15 \times 20) \]
\[ 200K = 2400 \]
\[ K = \frac{2400}{200} = 12 \]

### Conclusion
Ankit invested for **12 months**.