P is 30% more efficient than Q. How much time will they, working together, take to complete a job that P alone could have done in 23 days?
13 days
Efficiency Comparison: P and Q
The question states that P is 30% more efficient than Q. This means that for every unit of work Q can do, P can do 1 unit plus 30% of that unit. We can represent this mathematically:
- Let the efficiency of Q be represented by \( E_Q \).
- The efficiency of P, \( E_P \), is 30% more than \( E_Q \).
- So, \( E_P = E_Q + 0.30 \times E_Q \).
- Simplifying this, we get \( E_P = 1.30 \times E_Q \).
To make calculations easier, we can express the relationship as a ratio of their efficiencies:
\( \frac{E_P}{E_Q} = \frac{1.30 \times E_Q}{E_Q} = 1.3 = \frac{13}{10} \)
This ratio tells us that if Q's efficiency is 10 units, then P's efficiency is 13 units. We can use these values (or any multiple) for our calculations.
Calculating Total Work Required
We know that P alone can complete the job in 23 days. The total amount of work required for the job can be calculated using the formula:
Work = Efficiency × Time
Using P's efficiency and the time P takes:
- Assume \( E_P = 13 \) units/day.
- Time taken by P, \( T_P = 23 \) days.
- Total Work \( W = E_P \times T_P = 13 \times 23 \).
- \( W = 299 \) units.
So, the total work required to complete the job is 299 units.
Combined Efficiency of P and Q
When P and Q work together, their efficiencies add up. We found the ratio of their efficiencies is \( \frac{E_P}{E_Q} = \frac{13}{10} \).
- Let \( E_P = 13 \) units/day.
- Let \( E_Q = 10 \) units/day.
- The combined efficiency \( E_{P+Q} = E_P + E_Q \).
- \( E_{P+Q} = 13 + 10 = 23 \) units/day.
Working together, P and Q complete 23 units of work per day.
Time Taken by P and Q Together
To find the time it takes for them to complete the job working together, we use the formula:
Time = Total Work / Combined Efficiency
- Total Work \( W = 299 \) units.
- Combined Efficiency \( E_{P+Q} = 23 \) units/day.
- Time taken together \( T_{P+Q} = \frac{W}{E_{P+Q}} = \frac{299}{23} \).
- \( T_{P+Q} = 13 \) days.
Therefore, P and Q, working together, will take 13 days to complete the job.
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