A takes 24 days more than B and 32 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 A, B, and C to complete the work as follows:

- Let the number of days taken by C to complete the work be \( x \).
- According to the problem, A takes 32 days more than C, so A takes \( x + 32 \) days.
- A also takes 24 days more than B, so B takes \( (x + 32) - 24 = x + 8 \) days.

Now we have:
- Days taken by C = \( x \)
- Days taken by A = \( x + 32 \)
- Days taken by B = \( x + 8 \)

Next, we know from the problem that C works as much as A and B together. This means that the work done by C in \( x \) days is equal to the work done by A and B together in the same time.

The work done can be expressed in terms of their rates:
- Rate of A = \( \frac{1}{x + 32} \) (work per day)
- Rate of B = \( \frac{1}{x + 8} \) (work per day)
- Rate of C = \( \frac{1}{x} \) (work per day)

According to the problem, the work done by C in \( x \) days is equal to the work done by A and B in the same time:
\[ \frac{x}{x} = \frac{x}{x + 32} + \frac{x}{x + 8} \]
This simplifies to:
\[ 1 = \frac{x}{x + 32} + \frac{x}{x + 8} \]

To combine the fractions on the right-hand side, we find a common denominator:
\[ 1 = \frac{x(x + 8) + x(x + 32)}{(x + 32)(x + 8)} \]
This simplifies to:
\[ 1 = \frac{x^2 + 8x + x^2 + 32x}{(x + 32)(x + 8)} \]
\[ 1 = \frac{2x^2 + 40x}{(x + 32)(x + 8)} \]

Cross-multiplying gives:
\[ (x + 32)(x + 8) = 2x^2 + 40x \]
Expanding the left-hand side:
\[ x^2 + 40x + 256 = 2x^2 + 40x \]
Subtracting \( 2x^2 + 40x \) from both sides:
\[ x^2 + 40x + 256 - 2x^2 - 40x = 0 \]
This simplifies to:
\[ -x^2 + 256 = 0 \]
Thus, we have:
\[ x^2 = 256 \implies x = 16 \]

Now we can find the days taken by A, B, and C:
- Days taken by C = \( x = 16 \) days
- Days taken by B = \( x + 8 = 16 + 8 = 24 \) days
- Days taken by A = \( x + 32 = 16 + 32 = 48 \) days

**Final Answers:**
- A takes 48 days to complete the work.
- B takes 24 days to complete the work.
- C takes 16 days to complete the work.