Working together B and C take `50%` more number of days than A,B and C together take and A and B working together, take `8/3` more number of days than A, B and C take together. If A, B and C all have worked together till the completion of the work and B has received ₹ 120 out of the total earning of ₹ 450, then in how many days did A, B and C together complete the whole work?

Options

A. 10
B. 6
C. 4
D. 2

Answer

To solve the problem, we will break it down into manageable steps:

### Step 1: Define Variables
Let the number of days taken by A, B, and C together to complete the work be \( D \).

### Step 2: Establish Relationships
From the problem, we know:
1. B and C together take 50% more days than A, B, and C together.
- This means:
\[ \text{Days taken by B and C} = D + 0.5D = 1.5D \]
2. A and B together take \( \frac{8}{3} \) more days than A, B, and C together.
- This means:
\[ \text{Days taken by A and B} = D + \frac{8}{3} \]

### Step 3: Calculate Work Efficiency
Let’s denote the work done by A, B, and C in one day as \( a, b, c \) respectively. The total work done in one day by A, B, and C together is:
\[ a + b + c = \frac{1}{D} \]
The work done by B and C together in one day is:
\[ b + c = \frac{1}{1.5D} = \frac{2}{3D} \]
The work done by A and B together in one day is:
\[ a + b = \frac{1}{D + \frac{8}{3}} = \frac{1}{\frac{3D + 8}{3}} = \frac{3}{3D + 8} \]

### Step 4: Set Up Equations
From the relationships established, we can set up equations:
1. From \( a + b + c = \frac{1}{D} \) and \( b + c = \frac{2}{3D} \):
\[ a = \frac{1}{D} - \frac{2}{3D} = \frac{3 - 2}{3D} = \frac{1}{3D} \]
2. From \( a + b = \frac{3}{3D + 8} \):
\[ \frac{1}{3D} + b = \frac{3}{3D + 8} \]
Rearranging gives:
\[ b = \frac{3}{3D + 8} - \frac{1}{3D} \]

### Step 5: Solve for D
Now we can substitute \( b \) back into the equations and solve for \( D \). However, we also know that B received ₹120 out of ₹450, which gives us the ratio of their efficiencies:
\[ \frac{b}{a + b + c} = \frac{120}{450} = \frac{4}{15} \]
This implies:
\[ b = \frac{4}{15}(a + b + c) = \frac{4}{15} \cdot 1 = \frac{4}{15} \]

### Step 6: Calculate Total Work
Using the ratios of efficiencies:
- If \( a = 5k \), \( b = 4k \), and \( c = 6k \) (from the earnings ratio), then:
\[ 5k + 4k + 6k = 15k \]
Thus, the total work can be expressed as:
\[ D = \frac{15}{(5 + 4 + 6)} = \frac{15}{15} = 1 \]

### Step 7: Find Days Taken
Since we have established that \( D \) is the total number of days taken by A, B, and C together, we can conclude that:
\[ D = 15 \text{ days} \]

### Final Answer
Thus, A, B, and C together completed the whole work in **15 days**.