A and B can do a piece of work in 40 days and 50 days , respectively. Both begin together but after a certain time , A leaves off . In this case B finished the remaining work in 20 days . After how many days did A leave ?

Answer

To solve the problem step by step, we will first determine the work done by A and B, then calculate how long A worked before leaving.

### Step 1: Determine the work rates of A and B
- A can complete the work in 40 days, so A's work rate is \( \frac{1}{40} \) of the work per day.
- B can complete the work in 50 days, so B's work rate is \( \frac{1}{50} \) of the work per day.

**Hint:** Work rate is calculated as the total work divided by the time taken to complete that work.

### Step 2: Set up the equation for the total work done
Let A work for \( x \) days before leaving. In those \( x \) days, A and B work together. The amount of work done by A in \( x \) days is:
\[ \text{Work done by A} = A's \text{ rate} \times x = \frac{x}{40} \]

The amount of work done by B in \( x \) days is:
\[ \text{Work done by B} = B's \text{ rate} \times x = \frac{x}{50} \]

Thus, the total work done by A and B together in \( x \) days is:
\[ \text{Total work done in } x \text{ days} = \frac{x}{40} + \frac{x}{50} \]

**Hint:** To combine fractions, find a common denominator.

### Step 3: Find a common denominator and simplify
The least common multiple of 40 and 50 is 200. Therefore, we can rewrite the work done as:
\[ \frac{x}{40} = \frac{5x}{200} \]
\[ \frac{x}{50} = \frac{4x}{200} \]

So, the total work done in \( x \) days is:
\[ \frac{5x}{200} + \frac{4x}{200} = \frac{9x}{200} \]

**Hint:** Combine the fractions by adding the numerators while keeping the common denominator.

### Step 4: Calculate the remaining work done by B
After A leaves, B finishes the remaining work in 20 days. The amount of work done by B in 20 days is:
\[ \text{Work done by B in 20 days} = B's \text{ rate} \times 20 = \frac{20}{50} = \frac{2}{5} \]

**Hint:** Remember that the total work is considered as 1 unit.

### Step 5: Set up the equation for total work
The total work is 1 unit, so we can set up the equation:
\[ \text{Total work} = \text{Work done by A and B in } x \text{ days} + \text{Work done by B in 20 days} \]
\[ 1 = \frac{9x}{200} + \frac{2}{5} \]

**Hint:** Convert \( \frac{2}{5} \) to have a common denominator with \( \frac{9x}{200} \).

### Step 6: Solve for \( x \)
Convert \( \frac{2}{5} \) to a fraction with a denominator of 200:
\[ \frac{2}{5} = \frac{80}{200} \]

Now, substitute back into the equation:
\[ 1 = \frac{9x}{200} + \frac{80}{200} \]
\[ 1 = \frac{9x + 80}{200} \]

Multiply both sides by 200:
\[ 200 = 9x + 80 \]

Now, isolate \( x \):
\[ 9x = 200 - 80 \]
\[ 9x = 120 \]
\[ x = \frac{120}{9} = \frac{40}{3} \]

### Step 7: Convert \( x \) into days
To convert \( \frac{40}{3} \) into days:
\[ \frac{40}{3} = 13 \frac{1}{3} \text{ days} \]

### Final Answer
A leaves after \( 13 \frac{1}{3} \) days.

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