Working 9 hours a day, 12 persons complete a work in 25 days. Working 6 hours…

Concept

When one and the same job is finished under two different working arrangements, the total amount of labour the job demands does not change. Measuring that labour in man-hours turns this into a single invariant: persons × hours worked per day × days = total man-hours.

Hence M1H1D1 = M2H2D2. The number of days is inversely proportional to the product of the number of persons and the hours each of them works per day.

Application

1. First arrangement: 12 persons work 9 hours a day for 25 days.

2. Total work = 12 × 9 × 25 = 2700 man-hours. This total is fixed by the job itself, so it must also hold for the second arrangement.

3. Second arrangement: 15 persons work 6 hours a day, so together they supply 15 × 6 = 90 man-hours each day.

4. Let D be the number of days required. Equating the labour supplied to the labour demanded: 90 × D = 2700.

5. D = 2700 ÷ 90 = 30.

Cross-check

The same figure follows from proportionality alone, without ever computing the total.

• Persons rise from 12 to 15, a factor of 15/12 = 5/4. More workers shorten the time, so the days get multiplied by the reciprocal 4/5.

• Hours per day fall from 9 to 6, a factor of 6/9 = 2/3. Shorter shifts lengthen the time, so the days get multiplied by the reciprocal 3/2.

• Days = 25 × 4/5 × 3/2 = 30, which agrees with the man-hour computation.

The same work is therefore completed in 30 days.