How to Solve "Work Done in a Fraction of the Total Days" Problems
Solve “fraction of work done in some days” aptitude problems using rate-fraction-days relationships, with worked examples.
Expected Interview Answer
Multiply the worker's (or group's) daily rate by the number of days actually worked to get the fraction of the job completed, then subtract that fraction from 1 to find the remaining work, and divide the remaining work by whichever rate applies next to find the remaining time.
If a worker completes a job alone in n days, their daily rate is 1/n, so working for k days completes k/n of the job, leaving (1 − k/n) as the remaining fraction. This remaining fraction is then divided by the rate of whoever finishes the job — the same worker, a different worker, or a combined group — to get the additional time required. The phrase “did 2/5 of the work in some days” should immediately be read as a fraction-of-job-completed statement, convertible into either a rate (fraction ÷ days) or a time (fraction ÷ rate), whichever the question needs. Keeping “fraction of job” and “number of days” as clearly separate quantities, related only through the rate, prevents the most common setup errors.
- One relationship (fraction = rate × days) answers both “how much done” and “how long left” questions
- Cleanly separates work completed from work remaining
- Extends directly to problems where a different worker finishes the remainder
AI Mentor Explanation
A team chasing a target has scored 3/5 of the required runs after using 30 of the innings' 50 overs, so the fraction of runs still needed is 2/5, and dividing that fraction by the required run rate for the remaining 20 overs tells you exactly what scoring rate is needed to finish. This “fraction done, fraction remaining, divide by the applicable rate” structure is exactly how “work done in a fraction of the days” problems are solved, whether the chase is completed by the same batter or a different one coming in.
Worked example
A's progress (8 days)
- 8 × 1/20 = 2/5 done
Remaining work
- 1 − 2/5 = 3/5
B finishes the rest
- (3/5) ÷ (1/15) = 9 days
Step-by-Step Explanation
Step 1
Find the daily rate
Rate = 1 ÷ solo completion time for the worker in question.
Step 2
Compute fraction completed
Fraction done = rate × days actually worked.
Step 3
Find the remaining fraction
Remaining = 1 − fraction done.
Step 4
Solve for remaining time
Time left = remaining fraction ÷ rate of whoever finishes it.
What Interviewer Expects
- Correctly converts “fraction done in k days” into a rate
- Correctly computes the remaining fraction as 1 minus the completed fraction
- Applies the correct rate (same or different worker) to the remaining fraction
- Keeps fraction-of-job and number-of-days as distinct, clearly related quantities
Common Mistakes
- Confusing “days worked” with “days remaining” and using the wrong one in the rate formula
- Forgetting to subtract from 1 to get the remaining fraction
- Applying the first worker's rate to the remainder when a different worker takes over
- Treating the fraction given as a rate directly instead of dividing by days worked first
Best Answer (HR Friendly)
“I turn “worked for some days” into a rate by dividing the fraction of the job completed by the number of days spent, or the reverse when the rate is already known. Then I subtract from 1 to see what fraction is left, and divide that remaining fraction by whichever rate finishes the job to get the extra time needed.”
Follow-up Questions
- How would this change if the worker's rate itself changes partway through?
- How do you handle a problem where two different fractions are given for two different workers?
- What if the problem asks for the fraction completed rather than the time remaining?
- How would you verify your fraction and rate calculations are consistent?
MCQ Practice
1. A can complete a job in 25 days. After working for 10 days, what fraction of the job remains?
Fraction done = 10/25 = 2/5, so remaining = 1 − 2/5 = 3/5.
2. A completes 3/8 of a job in 6 days. At the same rate, how many more days to finish the rest?
Rate = (3/8)/6 = 1/16 per day. Remaining = 5/8. Days = (5/8) ÷ (1/16) = (5/8)×16 = 10 days.
3. A works 5 days completing 1/4 of a job, then B (who alone finishes in 12 days) takes over. Days for B to finish?
Remaining work = 3/4. B's rate = 1/12. Time = (3/4)/(1/12) = 9 days.
Flash Cards
Formula linking fraction, rate, and days? — Fraction completed = rate × days worked.
How do you find remaining work? — Remaining fraction = 1 − fraction already completed.
How do you find time to finish the remainder? — Remaining fraction ÷ the rate of whoever completes it.
Key pitfall in fraction-of-days problems? — Mixing up days worked with days remaining when computing the rate.