A,B and C can do a work in 20,25 and 30, respectively. They together started the work, but B adn A left the work 7 days and 4 days, respectively before completion of the work. How many days did C work for?

Answer

To solve the problem step by step, we will first determine the work done by A, B, and C, and then calculate how many days C worked for.

### Step 1: Determine the work rates of A, B, and C
- A can complete the work in 20 days, so A's work rate is \( \frac{1}{20} \) of the work per day.
- B can complete the work in 25 days, so B's work rate is \( \frac{1}{25} \) of the work per day.
- C can complete the work in 30 days, so C's work rate is \( \frac{1}{30} \) of the work per day.

### Step 2: Set up the equation for total work done
Let the total number of days taken to complete the work be \( x \).

- A works for \( x - 4 \) days (since A leaves 4 days before completion).
- B works for \( x - 7 \) days (since B leaves 7 days before completion).
- C works for all \( x \) days.

The equation for the total work done can be expressed as:
\[ \left( \frac{1}{20} \right)(x - 4) + \left( \frac{1}{25} \right)(x - 7) + \left( \frac{1}{30} \right)(x) = 1 \]

### Step 3: Find a common denominator
The least common multiple (LCM) of 20, 25, and 30 is 300. We will multiply the entire equation by 300 to eliminate the fractions:
\[ 300 \left( \frac{1}{20} \right)(x - 4) + 300 \left( \frac{1}{25} \right)(x - 7) + 300 \left( \frac{1}{30} \right)(x) = 300 \]
This simplifies to:
\[ 15(x - 4) + 12(x - 7) + 10x = 300 \]

### Step 4: Expand and simplify the equation
Expanding the equation gives:
\[ 15x - 60 + 12x - 84 + 10x = 300 \]
Combining like terms:
\[ (15x + 12x + 10x) - 144 = 300 \]
\[ 37x - 144 = 300 \]

### Step 5: Solve for \( x \)
Adding 144 to both sides:
\[ 37x = 444 \]
Dividing by 37:
\[ x = \frac{444}{37} = 12 \]

### Step 6: Calculate how many days C worked
Since \( x = 12 \), C worked for all 12 days.

### Final Answer
C worked for **12 days**.

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