Understanding the Work and Time Problem

This problem involves the concept of work and time, specifically dealing with the rates at which individuals or groups complete a task. The key idea is that if someone can complete a work in \( d \) days, their work rate is \( \frac{1}{d} \) of the work per day. When people work together, their individual work rates add up to form the combined work rate.

Setting up the Problem Equations

We are given the following information:

• P and Q together can complete the work in 6 days.
• P alone can complete the work in 18 days.

Let's denote the number of days Q takes to complete the work alone as \( Q_{days} \).

Based on the work rate concept:

• Work rate of (P + Q) together = \( \frac{1}{6} \) of the work per day.
• Work rate of P alone = \( \frac{1}{18} \) of the work per day.
• Work rate of Q alone = \( \frac{1}{Q_{days}} \) of the work per day.

Calculating the Work Rate of Q

When P and Q work together, their combined work rate is the sum of their individual work rates. So, we can write the equation:

Work rate of P + Work rate of Q = Work rate of (P + Q)

Substituting the values we know:

\( \frac{1}{18} + \frac{1}{Q_{days}} = \frac{1}{6} \)

To find the work rate of Q, we need to isolate \( \frac{1}{Q_{days}} \):

\( \frac{1}{Q_{days}} = \frac{1}{6} - \frac{1}{18} \)

To subtract the fractions, we find a common denominator, which is 18. We convert \( \frac{1}{6} \) to an equivalent fraction with a denominator of 18:

\( \frac{1}{6} = \frac{1 \times 3}{6 \times 3} = \frac{3}{18} \)

Now, substitute this back into the equation:

\( \frac{1}{Q_{days}} = \frac{3}{18} - \frac{1}{18} \)

Subtract the fractions:

\( \frac{1}{Q_{days}} = \frac{3 - 1}{18} = \frac{2}{18} \)

Simplify the fraction \( \frac{2}{18} \):

\( \frac{2}{18} = \frac{1}{9} \)

So, we have:

\( \frac{1}{Q_{days}} = \frac{1}{9} \)

This means that Q's work rate is \( \frac{1}{9} \) of the work per day. If Q completes \( \frac{1}{9} \) of the work in one day, Q will take 9 days to complete the entire work alone.

Therefore, the number of days required for Q to finish the work alone is 9 days.

Conclusion on Q's Time to Finish Work

Based on our calculation, Q takes 9 days to complete the work alone. This matches one of the given options.

Revision Table: Key Concepts

Additional Information on Work Problems

Work and time problems often involve calculating rates and combining them. Understanding fractions is crucial, as work rates are typically expressed as fractions of the total work completed per unit of time. Problems can become more complex with varying work rates, breaks, or multiple groups working simultaneously.

• Always ensure the time units are consistent (e.g., all in days or all in hours).
• The total work is often considered '1 unit' or 'the whole work'.
• If someone works for \( t \) days at a rate of \( r \) per day, the amount of work done is \( r \times t \).