Time & Work • Topic 4 of 5

Work Equivalence

Work equivalence is the "man-days" idea: a fixed job needs a fixed amount of total effort, so men × days is constant for the same job. If 10 workers finish a wall in 6 days, that wall is 60 man-days of work; 15 workers will finish it in 60/15 = 4 days. The same constant lets you solve "some workers leave or join" problems: compute the man-days already spent, subtract from the total, then divide the remainder by the new workforce. When hours per day also vary, extend it to man-day-hours: 10 men × 6 days × 8 hours = 480 man-hours. The crucial CAT caveat is that this assumes every worker is equally efficient; if efficiencies differ, convert everyone to a common unit first (e.g. in terms of "one man’s daily output"). Watch for inverse proportion: more workers ⇒ fewer days, so the relationship is a product staying constant, never a simple ratio.

✅ Solved examples

1. 12 men build a wall in 10 days. How many days for 15 men (same wall)?
Man-days = 12 × 10 = 120. With 15 men: 120/15 = 8 days.
2. 20 workers can finish a road in 30 days. After 6 days, 5 workers leave. How many more days?
Total = 20 × 30 = 600 man-days. Done = 20 × 6 = 120 ⇒ 480 left. Remaining workers = 15 ⇒ 480/15 = 32 days.
3. 8 men working 9 hours/day finish a job in 10 days. How many days will 12 men working 6 hours/day take?
Total man-hours = 8 × 9 × 10 = 720. New rate = 12 × 6 = 72/day ⇒ 720/72 = 10 days.
4. A contractor has 24 men to finish a job in 20 days. After 8 days only ¼ is done. How many extra men are needed to finish on time?
Planned = 24 × 20 = 480 man-days. In 8 days, ¼ done = 120 man-days of output; ¾ left = 360 man-days in 12 remaining days ⇒ need 30 men ⇒ 6 extra.

✏️ Practice — try these, take hints as needed

1. 15 men finish a job in 16 days. How many men finish it in 12 days?
Man-days = 15 × 16 = 240.
Men = 240 ÷ 12.
More work in fewer days ⇒ more men.
20 men
2. 18 workers do a task in 20 days. After 5 days, 6 leave. Days to finish the rest?
Total = 360 man-days; done = 18 × 5 = 90.
Remaining = 270 man-days, workers = 12.
270 ÷ 12.
22.5 days
3. 10 men, 8 hrs/day, finish in 12 days. How many days for 8 men at 10 hrs/day?
Total man-hours = 10 × 8 × 12 = 960.
New rate = 8 × 10 = 80/day.
960 ÷ 80.
12 days
4. 6 men or 9 women finish a job in 10 days. In how many days will 4 men and 6 women finish it?
6 men = 9 women ⇒ 1 man = 1.5 women; total work in women-days = 9 × 10 = 90.
4 men = 6 women, so 4 men + 6 women = 12 women.
90 ÷ 12.
7.5 days
5. 40 men finish a project in 60 days. After 30 days, half the work is done. How many men should leave so it still ends on time?
Total = 2400 man-days; done = 40 × 30 = 1200 (half).
Half left = 1200 man-days in 30 days ⇒ need 40 men.
So 0 leave (right on pace).
0 men (they are exactly on schedule)

📝 Topic test — 8 questions

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