Understanding the Time and Work Problem

This question involves a classic Time and Work scenario. We are given the time it takes for two individuals, A and B, to complete a task individually. They start working together, one person leaves, and the other person finishes the remaining work. We need to find out after how many days the first person left the job.

To solve this type of problem, we first determine the per-day work rate of each person. The work rate is the fraction of the total work completed in one day.

Calculating Individual Work Rates

• A can complete the work in 45 days.
• Therefore, A's work rate per day is $\frac{1}{45}$ of the total work.

• B can complete the work in 40 days.
• Therefore, B's work rate per day is $\frac{1}{40}$ of the total work.

Setting Up the Problem Equation

Let's assume A and B worked together for 'x' days before A left. During these 'x' days, both A and B contributed to the work.

• Work done by A in 'x' days = $x \times (\text{A's work rate}) = x \times \frac{1}{45} = \frac{x}{45}$.
• Work done by B in 'x' days = $x \times (\text{B's work rate}) = x \times \frac{1}{40} = \frac{x}{40}$.
• Total work done by A and B together in 'x' days = $\frac{x}{45} + \frac{x}{40}$.

After A left, B completed the remaining work alone in 23 days.

• Work done by B alone in 23 days = $23 \times (\text{B's work rate}) = 23 \times \frac{1}{40} = \frac{23}{40}$.

The total work is considered as 1 unit (or 100%). The work done together plus the work done by B alone equals the total work.

So, the equation is:
$\text{(Work done together)} + \text{(Work done by B alone)} = \text{Total work}$
$\left(\frac{x}{45} + \frac{x}{40}\right) + \frac{23}{40} = 1$

Solving the Equation to Find When A Left

Now, let's solve the equation for 'x'.

Combine the terms with 'x':

$x \left(\frac{1}{45} + \frac{1}{40}\right) + \frac{23}{40} = 1$

Find a common denominator for $\frac{1}{45}$ and $\frac{1}{40}$. The least common multiple (LCM) of 45 and 40 is 360.

$\frac{1}{45} = \frac{1 \times 8}{45 \times 8} = \frac{8}{360}$

$\frac{1}{40} = \frac{1 \times 9}{40 \times 9} = \frac{9}{360}$

Substitute these values back into the equation:

$x \left(\frac{8}{360} + \frac{9}{360}\right) + \frac{23}{40} = 1$

$x \left(\frac{8+9}{360}\right) + \frac{23}{40} = 1$

$x \left(\frac{17}{360}\right) + \frac{23}{40} = 1$

Subtract $\frac{23}{40}$ from both sides of the equation:

$x \left(\frac{17}{360}\right) = 1 - \frac{23}{40}$

$1 - \frac{23}{40} = \frac{40}{40} - \frac{23}{40} = \frac{40-23}{40} = \frac{17}{40}$

So, the equation becomes:

$x \left(\frac{17}{360}\right) = \frac{17}{40}$

To find 'x', divide both sides by $\frac{17}{360}$:

$x = \frac{\frac{17}{40}}{\frac{17}{360}}$

$x = \frac{17}{40} \times \frac{360}{17}$

Cancel out the 17 in the numerator and denominator:

$x = \frac{1}{40} \times 360$

$x = \frac{360}{40}$

$x = 9$

So, A and B worked together for 9 days. This means A left after 9 days.

Final Answer

A left the work after 9 days.

Revision Table: Key Concepts

Additional Information: Time and Work Problems

Time and Work problems often involve calculating how quickly people or machines can complete a task. Key principles include:

• Inverse Relationship: Time taken is inversely proportional to the work rate (efficiency). More efficient workers take less time.
• Consistent Rate: It's usually assumed that individuals work at a constant rate unless stated otherwise.
• Fractional Work: Work done is often represented as a fraction of the total work. Completing the entire work means the fraction is 1.
• Multiple Workers: When multiple people work together, their individual work rates are added to find the combined work rate.
• Work Left: If a portion of work is completed, the remaining work is $1 - (\text{work completed})$.

Understanding these core concepts helps in setting up the correct equations for different scenarios, such as people starting together and one leaving, people joining later, or comparing the efficiencies of different workers.