Quick answer: C alone would take 28.8 days. That answer is not listed among 32, 36, 40, and 48 days, so none of the supplied options is correct as written. The calculation below shows exactly why and gives you a repeatable method for time-and-work questions.

The question

A can complete a piece of work in 18 days, while B can complete the same work in 24 days. They work together for 4 days, after which A leaves. C joins B, and the remaining work is completed in 8 days. In how many days can C alone complete the entire work?

A) 32 days   B) 36 days   C) 40 days   D) 48 days

First understand the basic idea

Time-and-work questions become easier when you stop adding the number of days and instead compare the amount of work completed in one day. A worker who finishes a job in fewer days has a higher daily rate. If the whole job is called one unit, A's rate is 1/18 of the job per day and B's rate is 1/24 per day.

When people work at the same time, their daily rates are added. When a worker leaves, remove that worker's rate from the team. Keep the units consistent from start to finish, and do not confuse a worker's time with the group's time.

Quick solving guide

For a team that changes during a job, keep one total-work unit throughout and calculate each phase separately:

  • Choose total work divisible by each known completion time.
  • Add the daily rates of workers who are together.
  • Subtract completed work to find what remains.
  • Use the second phase to find the team rate, then subtract known workers to isolate the requested rate.

Method 1: use 72 convenient work units

The least common multiple of 18 and 24 is 72. Imagine that the whole job contains 72 small units. This changes fractions into easy whole-number daily rates.

  • A completes 72 ÷ 18 = 4 units per day.
  • B completes 72 ÷ 24 = 3 units per day.
  • Working together, A and B complete 4 + 3 = 7 units per day.

Step 1: Find the work completed in the first 4 days

For each of those four days, A and B together finish 7 units. So they finish 7 × 4 = 28 units.

Step 2: Find the unfinished work

The entire job is 72 units. After 28 units are done, 72 − 28 = 44 units remain. This is the amount B and C complete during the next 8 days.

Step 3: Find B and C's combined daily rate

If B and C finish 44 units in 8 days, their combined rate is 44 ÷ 8 = 5.5 units per day.

Step 4: Subtract B's rate to get C's rate

B can do 3 units per day. Therefore C's rate is 5.5 − 3 = 2.5 units per day.

Step 5: Find C's time for the whole job

C alone would need 72 ÷ 2.5 = 28.8 days. The answer is 28.8 days, not 32, 36, 40, or 48.

Method 2: solve using fractions

You can solve the same question by treating the complete job as 1. A's one-day work is 1/18 and B's one-day work is 1/24. Together they do:

1/18 + 1/24 = 4/72 + 3/72 = 7/72 of the job per day.

During four days, they complete 4 × 7/72 = 28/72 = 7/18. The remaining work is 1 − 7/18 = 11/18.

B and C finish that 11/18 in eight days, so their combined rate is (11/18) ÷ 8 = 11/144 of the job per day. B's rate is 1/24 = 6/144. Subtracting gives C's rate:

11/144 − 6/144 = 5/144 of the job per day.

So C's time is the reciprocal of that rate: 1 ÷ (5/144) = 144/5 = 28.8 days. Both methods lead to the same result, which is a useful accuracy check.

Why none of the answer choices works

An exam option should agree with the rates given in the question. Here, B alone contributes 3 units each day, and the remaining work requires B and C together to contribute 5.5 units each day. That leaves exactly 2.5 units per day for C. Since the job has 72 units, C needs 28.8 days. Each printed option would imply a different rate for C:

  • 32 days: C would do 72/32 = 2.25 units per day. B and C would then need 5.25 units per day and would take 44/5.25, or about 8.38 days, not 8.
  • 36 days: C would do 2 units per day. Together with B, that is 5 units per day, so 44 units would take 8.8 days.
  • 40 days: C would do 1.8 units per day. Together with B, that is 4.8 units per day, so 44 units would take about 9.17 days.
  • 48 days: C would do 1.5 units per day. Together with B, that is 4.5 units per day, so 44 units would take about 9.78 days.

None of those implied completion times is 8 days. The question or its answer choices likely contains a typo. In a real test, do not force a wrong option to fit: recheck the numbers, and if the question is exactly as printed, report that no listed choice is correct.

Five common traps in time-and-work questions

  1. Adding days instead of rates. “18 days” and “24 days” describe completion times, not daily output. Convert them to rates, 1/18 and 1/24, before combining workers.
  2. Using the wrong remaining work. The first team completes 28 of 72 units, so 44 remain. A quick check is 28 + 44 = 72. Do not accidentally subtract from a different total.
  3. Forgetting that B stays on the job. The 5.5-unit rate belongs to B and C together. C's rate is not 5.5; subtract B's 3 units per day.
  4. Mixing up a part-job time and a whole-job time. Eight days is how long B and C take to finish the remaining 44 units. The question asks how long C needs for all 72 units.
  5. Rounding before the final step or trusting the options blindly. Keep 5.5 exact, calculate 72 ÷ 2.5, and compare the result with every option. A neat-looking choice is not automatically correct.

A quick exam method you can remember

For a question where workers change partway through, write a small timeline: who is working, for how long, and at what combined rate. Then follow this sequence: (1) choose total work units divisible by the given completion times; (2) calculate the first team's rate; (3) calculate work completed during the first phase; (4) subtract from the total; (5) divide the remaining work by the second phase's duration; (6) subtract the rate of any worker whose individual rate is known; and (7) divide the total work by the remaining worker's rate.

Before submitting, ask two questions: does the second team's rate exceed the known worker's rate, and does multiplying each phase's rate by its duration reproduce the stated amount of work? These checks catch negative rates, arithmetic slips, and inconsistent options.

Four more time-and-work practice questions

Practice 1: A worker joins after the first phase

A finishes a job in 12 days and B in 24 days. They work together for 6 days. A then leaves, and C joins B. B and C finish the remaining work in 3 days. How long would C take alone?

Solution: A's rate is 1/12 and B's is 1/24, so their combined rate is 1/8. In 6 days they finish 6/8 = 3/4, leaving 1/4. B and C complete 1/4 in 3 days, so their rate is 1/12. Subtract B's 1/24: C's rate is 1/24. C alone takes 24 days.

Practice 2: Find the rate after a team changes

A can complete a job in 15 days and B in 20 days. They work together for 5 days. A leaves, and B and C finish the remaining work in 5 days. Find C's solo time.

Solution: Their combined rate is 1/15 + 1/20 = 7/60. In 5 days, they finish 35/60 = 7/12, leaving 5/12. B and C's rate is (5/12) ÷ 5 = 1/12. Subtract B's 1/20: 1/12 − 1/20 = 2/60 = 1/30. C alone takes 30 days.

Practice 3: One worker starts first

A completes a job in 15 days and B in 20 days. A works alone for 3 days, then A and B work together until the job is complete. How many days does the entire job take?

Solution: A completes 3/15 = 1/5, leaving 4/5. The combined rate is 1/15 + 1/20 = 7/60. The time for the remaining work is (4/5) ÷ (7/60) = 48/7 days, or 6 6/7 days. Add A's first 3 days: 9 6/7 days in total.

Practice 4: Use an efficiency ratio

A is three times as efficient as B. Working together, they finish a job in 12 days. How long would each take alone?

Solution: Their efficiency ratio is A:B = 3:1. Think of their combined output as 4 equal rate parts per day. If four parts complete one job in 12 days, one part is 1/48 of the job per day. B is one part, so B takes 48 days. A is three parts, so A takes 48/3 = 16 days. Check: 1/16 + 1/48 = 3/48 + 1/48 = 1/12.

Takeaway

The reliable habit is to work with daily efficiency, track the unfinished work carefully, and subtract the rate of workers who remain on a team before finding an individual worker's time. For this question, that process gives 28.8 days. Because the choices are 32, 36, 40, and 48, the correct conclusion is that none of the supplied options is correct. For more timed practice, explore the practice tests and review each solution by checking the work done in every phase.

Exam-smart tips

  • Draw a short timeline when workers join or leave.
  • Check that the remaining-work amount is positive and less than the total.
  • Substitute your final rates back into both phases before choosing an option.
  • If no option matches the stated numbers, report the mismatch instead of forcing an answer.

Frequently asked questions

What is the correct answer to this time-and-work question?

Using the numbers exactly as stated, C alone takes 28.8 days. No listed option matches, so the multiple-choice item is inconsistent as written.

Why do we subtract B's work from the combined rate?

The 8-day phase is completed by B and C together. Their rate includes both workers. Since B's individual rate is already known, subtract it from the team's rate to isolate C's rate.

Can C's answer be a decimal number of days?

Yes. A worker can finish part of a day's work on the final day. The rate calculation gives 28.8 working days; it does not have to be a whole number unless the question specifically imposes that condition.

How do I choose total work units?

Use a convenient common multiple of the completion times. For 18 and 24 days, 72 units works because both 72/18 and 72/24 are whole numbers. Fractions are equally valid, but work units can make arithmetic easier.

What should I do when no option matches?

Recalculate each phase and check the wording once. If the data still produce an answer absent from the choices, do not claim a false match; note the inconsistency. In a live exam, follow the official marking instructions and use the time available wisely.