A takes 7 days more than B and 16 days more than C to complete a work. C works as much as (A + B) works together. In how many days will A, B and C alone complete this work ?

Answer

To solve the problem step by step, let's denote the number of days taken by each person to complete the work as follows:

- Let C take \( x \) days to complete the work.
- Then, A takes \( x + 16 \) days to complete the work.
- B takes \( x + 7 \) days to complete the work.

From the problem, we also know that C works as much as A and B together. This means that the work done by C in one day is equal to the combined work done by A and B in one day.

### Step 1: Establish the relationship between their work rates
The work done in one day by each person can be expressed as:
- Work rate of A = \( \frac{1}{x + 16} \)
- Work rate of B = \( \frac{1}{x + 7} \)
- Work rate of C = \( \frac{1}{x} \)

According to the problem:
\[ \frac{1}{x} = \frac{1}{x + 16} + \frac{1}{x + 7} \]

### Step 2: Solve the equation
To solve the equation, we can find a common denominator and simplify:
\[ \frac{1}{x} = \frac{(x + 7) + (x + 16)}{(x + 16)(x + 7)} \]
This simplifies to:
\[ \frac{1}{x} = \frac{2x + 23}{(x + 16)(x + 7)} \]

Cross-multiplying gives us:
\[ (x + 16)(x + 7) = x(2x + 23) \]

### Step 3: Expand both sides
Expanding both sides:
\[ x^2 + 23x + 112 = 2x^2 + 23x \]

### Step 4: Rearranging the equation
Rearranging the equation gives:
\[ x^2 + 112 = 0 \]
This leads to:
\[ x^2 = 112 \]

### Step 5: Finding the value of x
Taking the square root:
\[ x = \sqrt{112} = 4\sqrt{7} \]

### Step 6: Calculate A and B's days
Now, substituting \( x \) back to find A and B's days:
- C's days = \( x = 4\sqrt{7} \)
- A's days = \( x + 16 = 4\sqrt{7} + 16 \)
- B's days = \( x + 7 = 4\sqrt{7} + 7 \)

### Final Answer
Thus, the number of days taken by A, B, and C to complete the work alone are:
- A: \( 4\sqrt{7} + 16 \) days
- B: \( 4\sqrt{7} + 7 \) days
- C: \( 4\sqrt{7} \) days