If person X can complete a piece of work in p days and person Y can complete the same piece of work in q days (with p and q positive), then in how many days will X and Y working together be able to complete that work?

Introduction / Context:
This is a conceptual time and work question that asks for the general formula for the combined time when two people, X and Y, work together on a task. Instead of numerical values, the question uses variables p and q to represent the individual times. The goal is to derive a formula that expresses the number of days required when both work simultaneously.

Given Data / Assumptions:

• X alone finishes the work in p days.
• Y alone finishes the work in q days.
• Both X and Y work together from the beginning.
• Total work is considered as 1 unit.
• p and q are positive real numbers, and p + q is not zero.

Concept / Approach:
The key idea is that time and work questions are easier when we work with rates rather than times directly. If X finishes the work in p days, then X's work rate is 1/p units per day. Similarly, Y's rate is 1/q units per day. When they work together, their combined rate is the sum of these rates, that is, 1/p + 1/q. The time taken together equals total work divided by combined rate. Simplifying this expression gives the required formula in terms of p and q.

Step-by-Step Solution:
Step 1: Assume total work is 1 unit. Step 2: Since X completes 1 unit of work in p days, X's rate = 1/p units per day. Step 3: Since Y completes 1 unit of work in q days, Y's rate = 1/q units per day. Step 4: When X and Y work together, their combined rate = 1/p + 1/q. Step 5: Combine the fractions: 1/p + 1/q = (q + p) / (p*q). Step 6: Time taken when working together = total work / combined rate = 1 / ( (p + q) / (p*q) ). Step 7: This simplifies to (p*q) / (p + q) days.

Verification / Alternative check:
We can test the formula with simple numeric values. Suppose p = 8 days and q = 12 days. Then the formula gives (8*12)/(8+12) = 96/20 = 4.8 days. Checking with rates: 1/8 + 1/12 = (3/24 + 2/24) = 5/24. Time together should be 1 / (5/24) = 24/5 = 4.8 days, which matches the formula. This confirms the correctness of the expression (p*q)/(p+q).

Why Other Options Are Wrong:

• (p+q)/2 days: This is a simple arithmetic average of the times, which does not correctly model combined work rates.
• p+q days: This is even larger and represents the sum of individual times, which is meaningless in the context of simultaneous work.
• (p*q)/(p-q) days: This form appears similar to the correct one but uses a difference instead of a sum in the denominator. It generally gives incorrect and sometimes negative values when p is less than q.

Common Pitfalls:
Many students incorrectly average or add times instead of using rates. It is crucial to remember that work rates add for workers operating together, not their times. Another common mistake is failing to simplify 1 / (1/p + 1/q) correctly, leading to algebraic errors. Working through the algebra step by step helps avoid such mistakes.

Final Answer:
The time taken by X and Y working together to complete the work is (p*q)/(p+q) days.