A can finish a work in 18 days and B can finish the same work in half the time taken by A. Working together at their usual rates, what fraction of the total work can A and B complete in one day?

Introduction / Context:
This is a simple time and work question focusing on the fraction of work completed per day by two workers A and B when they work together. You are given A’s total time to complete the job and told that B takes half that time. The goal is to compute the combined daily work as a fraction of the whole job.

Given Data / Assumptions:
A alone can complete the work in 18 days. B alone can complete the work in half the time taken by A, which means in 9 days. Both work at constant rates and, when working together, their rates simply add. The total work is considered to be one complete unit.

Concept / Approach:
The key concept is that the rate of work of a person is the reciprocal of the time they take to complete the job alone. To find the fraction of work done per day when they work together, we compute A’s rate and B’s rate separately and then add them. The result is the fraction of the job completed in one day by A and B working jointly.

Step-by-Step Solution:
Step 1: Let the total work be 1 unit. Step 2: A alone takes 18 days to complete the work, so A’s rate is 1 / 18 of the work per day. Step 3: B alone takes half of 18 days, which is 9 days, so B’s rate is 1 / 9 of the work per day. Step 4: The combined daily rate when A and B work together is 1 / 18 + 1 / 9. Step 5: Convert to a common denominator of 18: 1 / 18 remains 1 / 18, and 1 / 9 = 2 / 18. Step 6: Add the fractions: 1 / 18 + 2 / 18 = 3 / 18. Step 7: Simplify 3 / 18 by dividing numerator and denominator by 3 to get 1 / 6. Step 8: Therefore, A and B together can complete 1 / 6 of the work in one day.

Verification / Alternative check:
We can verify by computing the total time they would take together. If they do 1 / 6 of the work each day, then they would need 6 days to complete the entire job. As a quick consistency check, note that A alone needs 18 days and B alone needs 9 days. It is reasonable for the joint time to be less than 9 days, and 6 days fits this expectation. This confirms that 1 / 6 of the work per day is a sensible and correct result.

Why Other Options Are Wrong:
One-fourth, one-third, half, and two-thirds of the work per day are all too large given the individual times of 18 and 9 days. For example, if they did half the work in a day, they would finish in 2 days, which is unrealistically fast compared to their individual times. Similarly, one-fourth or one-third would lead to total times of 4 or 3 days, which do not match the work rates implied by 18 and 9 days. Only one-sixth of the work per day is consistent with their individual capacities.

Common Pitfalls:
Some learners mistakenly average the times 18 and 9 (getting 13.5 days) instead of working with rates. Others may forget to simplify the combined fraction 3 / 18 to 1 / 6. Always remember that when dealing with time and work, you should add rates (1 / time) rather than adding or averaging the times themselves.

Final Answer:
A and B together can complete one-sixth of the work in one day.