P can finish a work in 18 days. When he had worked for 5 days, Q joined him. If both of them together completed the remaining work in \(\frac{13}{5} \) days, then in how many days can Q alone finish \(66\frac{2}{3} \) % of the same work?
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Understanding the Work and Time Problem
This problem involves concepts of work and time, specifically how individuals (P and Q) complete a task alone and when working together. We are given the time P takes to complete the entire work, the duration P works alone, the time P and Q work together on the remaining part, and we need to find the time Q takes to complete a specific percentage of the work alone.
Calculating P's Work Contribution
P can finish the entire work in 18 days. This means P's daily work rate is \( \frac{1}{18} \) of the total work per day.
P worked alone for 5 days. The amount of work completed by P in these 5 days is:
Work done by P in 5 days = Work rate of P \(\times\) Number of days worked
Work done by P in 5 days = \( \frac{1}{18} \times 5 = \frac{5}{18} \) of the total work.
Determining the Remaining Work
After P worked for 5 days, the amount of work remaining is the total work minus the work done by P:
Remaining Work = Total Work - Work done by P
Remaining Work = \( 1 - \frac{5}{18} = \frac{18 - 5}{18} = \frac{13}{18} \) of the total work.
Calculating the Combined Work Rate of P and Q
The remaining \( \frac{13}{18} \) of the work was completed by both P and Q working together in \( \frac{13}{5} \) days.
Their combined daily work rate is the remaining work divided by the time taken to complete it:
Combined Work Rate of P and Q = \( \frac{\text{Remaining Work}}{\text{Time taken by P and Q together}} \)
Combined Work Rate of P and Q = \( \frac{\frac{13}{18}}{\frac{13}{5}} = \frac{13}{18} \times \frac{5}{13} = \frac{5}{18} \) of the total work per day.
Finding Q's Individual Work Rate
We know P's daily work rate is \( \frac{1}{18} \) and the combined daily work rate of P and Q is \( \frac{5}{18} \).
Q's daily work rate can be found by subtracting P's work rate from the combined work rate:
Q's Work Rate = Combined Work Rate of P and Q - P's Work Rate
Q's Work Rate = \( \frac{5}{18} - \frac{1}{18} = \frac{5-1}{18} = \frac{4}{18} = \frac{2}{9} \) of the total work per day.
Calculating Time for Q to Complete 66\( \frac{2}{3} \) % of the Work
First, let's convert the percentage to a fraction:
\( 66\frac{2}{3} \% = \frac{(66 \times 3) + 2}{3} \% = \frac{198 + 2}{3} \% = \frac{200}{3} \% \)
To convert percentage to fraction, divide by 100: \( \frac{200}{3} \times \frac{1}{100} = \frac{200}{300} = \frac{2}{3} \).
So, we need to find the time Q takes to finish \( \frac{2}{3} \) of the same work.
Q's daily work rate is \( \frac{2}{9} \) of the work. This means Q takes \( \frac{1}{2/9} = \frac{9}{2} \) days to complete the entire work (100%).
The time Q takes to complete \( \frac{2}{3} \) of the work is:
Time taken by Q = \( \frac{\text{Fraction of work to be done}}{\text{Q's daily work rate}} \)
Time taken by Q = \( \frac{\frac{2}{3}}{\frac{2}{9}} = \frac{2}{3} \times \frac{9}{2} = \frac{18}{6} = 3 \) days.
Therefore, Q alone can finish \( 66\frac{2}{3} \) % of the same work in 3 days.
| Entity | Information |
|---|---|
| P's total time | 18 days |
| P's daily rate | \( \frac{1}{18} \) work/day |
| P works alone | 5 days |
| Work done by P | \( 5 \times \frac{1}{18} = \frac{5}{18} \) |
| Remaining work | \( 1 - \frac{5}{18} = \frac{13}{18} \) |
| Time P & Q together | \( \frac{13}{5} \) days |
| Combined rate (P+Q) | \( \frac{13/18}{13/5} = \frac{5}{18} \) work/day |
| Q's daily rate | \( \frac{5}{18} - \frac{1}{18} = \frac{4}{18} = \frac{2}{9} \) work/day |
| Percentage for Q | \( 66\frac{2}{3} \% = \frac{2}{3} \) |
| Time for Q (2/3 work) | \( \frac{2/3}{2/9} = \frac{2}{3} \times \frac{9}{2} = 3 \) days |
The final answer is 3 days.
Revision Table: Work and Time Concepts
| Concept | Formula/Explanation |
|---|---|
| Work Rate | If a person completes a work in 'n' days, their daily work rate is \( \frac{1}{n} \) of the work per day. |
| Work Done | Work Done = Work Rate \(\times\) Time |
| Time Taken | Time Taken = \( \frac{\text{Total Work}}{\text{Work Rate}} \). If doing a fraction of work, Time = \( \frac{\text{Fraction of Work}}{\text{Work Rate}} \). |
| Combined Work Rate | If A's rate is \(R_A\) and B's rate is \(R_B\), their combined rate is \(R_A + R_B\). |
| Remaining Work | Remaining Work = Total Work - Work Done |
Additional Information on Work and Time Problems
Work and Time problems are common in quantitative aptitude sections of various exams. They typically involve individuals or machines working at different rates to complete a task. The key is to convert the time taken to complete the whole work into a work rate (fraction of work done per unit time, usually a day).
- If a person completes work in 'D' days, they do \( \frac{1}{D} \) of the work each day.
- If multiple people work together, their individual work rates are added to find the combined work rate.
- If someone leaves or joins, calculate the work done by each person or group during their specific time period.
- Percentage of work can be converted into fractions for easier calculation. \(x\%\) of work is \( \frac{x}{100} \) of the work.
- Always ensure that the units (days, hours, etc.) are consistent throughout the calculation.
Understanding these fundamental principles helps in solving various complex Work and Time scenarios effectively.
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