A takes 6 hours more than (A + B + C) together to complete a work. B takes 1 hours more than (A + B + C) together to complete a work. C takes twice as (A + B + C) together to complete a work. In how many days will A and B together complete this work ?

Answer

To solve the problem, let's define the variables and set up the equations based on the information given.

### Step 1: Define Variables
Let \( T \) be the time taken by \( A + B + C \) together to complete the work.

From the problem:
- \( A \) takes \( T + 6 \) hours to complete the work.
- \( B \) takes \( T + 1 \) hours to complete the work.
- \( C \) takes \( 2T \) hours to complete the work.

### Step 2: Calculate Work Rates
The work done by each person in one hour can be expressed as:
- Work rate of \( A \) = \( \frac{1}{T + 6} \)
- Work rate of \( B \) = \( \frac{1}{T + 1} \)
- Work rate of \( C \) = \( \frac{1}{2T} \)

The combined work rate of \( A + B + C \) is:
\[ \text{Work rate of } A + B + C = \frac{1}{T + 6} + \frac{1}{T + 1} + \frac{1}{2T} \]

### Step 3: Set Up the Equation
Since \( A + B + C \) together can also be expressed as:
\[ \text{Work rate of } A + B + C = \frac{1}{T} \]

We can equate the two expressions:
\[ \frac{1}{T + 6} + \frac{1}{T + 1} + \frac{1}{2T} = \frac{1}{T} \]

### Step 4: Solve the Equation
To solve this equation, we will find a common denominator and simplify:
1. The common denominator for the left side is \( (T + 6)(T + 1)(2T) \).
2. Multiply through by this common denominator to eliminate the fractions.

This leads to:
\[ 2T(T + 1) + 2T(T + 6) + (T + 6)(T + 1) = (T + 6)(T + 1)(2T) \]

Now, we will expand and simplify this equation to find \( T \).

### Step 5: Calculate Time Taken by A and B Together
Once we find \( T \), we can calculate the work rates of \( A \) and \( B \):
\[ \text{Work rate of } A + B = \frac{1}{T + 6} + \frac{1}{T + 1} \]

The time taken by \( A + B \) together to complete the work is:
\[ \text{Time} = \frac{1}{\text{Work rate of } A + B} \]

### Step 6: Convert Time into Days
If the time calculated is in hours, convert it into days by dividing by 24.