P is 30 percent more efficient than Q. How much time will P and Q working together take to complete a job that P alone could complete in 23 days?

Introduction / Context:
This question explores relative efficiency and combined work rates. P is given to be more efficient than Q by a certain percentage, and we know how long P alone would take to do a job. We are asked how long both of them together will need. Such questions are common in competitive exams to check whether the candidate can correctly interpret percentage increases in efficiency and translate them into work rates and combined times.

Given Data / Assumptions:

• P is 30 percent more efficient than Q.
• P alone can complete the job in 23 days.
• Both P and Q work at constant rates when working together.
• We must find the time required when P and Q work together on the same job.

Concept / Approach:
If P is 30 percent more efficient than Q, then P's rate is 1.3 times Q's rate. Instead of working directly with fractions, a convenient approach is to assume a simple base rate for Q and scale P accordingly. We then adjust the total work so that P alone finishes it in 23 days. From this, we get the combined daily work of P and Q together and finally find the total number of days needed when they both work on the job.

Step-by-Step Solution:
Step 1: Assume Q's efficiency is 10 units of work per day. Then P's efficiency is 30 percent more, that is 13 units per day. Step 2: Let the total work be such that P alone finishes it in 23 days. Then total work equals 23 times P's daily work which is 23 times 13 = 299 units. Step 3: Combined daily work of P and Q is 13 + 10 = 23 units per day. Step 4: Time taken when both work together is total work divided by combined rate, that is 299 divided by 23 days. Step 5: 299 divided by 23 equals 13 days exactly.

Verification / Alternative check:
We can also use rate equations. Let Q's rate be r. Then P's rate is 1.3r. Since P alone takes 23 days, total work W equals 23 times 1.3r which equals 29.9r. Combined rate of P and Q is 1.3r + r = 2.3r. Time taken together is W divided by 2.3r which is 29.9r divided by 2.3r = 13 days. Both the unit method and algebraic method converge to the same result, confirming that 13 days is correct.

Why Other Options Are Wrong:

• 16 days and 15 days are longer than necessary and would imply a lower combined rate than P working alone, which is impossible.
• 12 days would mean a slightly higher combined rate than 23 units per day in the unit method, which does not match the given efficiency relation.
• Only 13 days fits all the given conditions and the relative efficiency relationship.

Common Pitfalls:
Many learners confuse 30 percent more efficient with taking 30 percent less time, which is not correct. Efficiency refers to work per day, while time is the inverse of this rate. Another pitfall is failing to set up a consistent hypothetical unit system, leading to messy fractions. Using a simple assumption like 10 units for Q helps keep calculations clean and intuitive.

Final Answer:
P and Q together will complete the job in 13 days.