Formula

The work-equivalence chain rule $M_{1} D_{1} H_{1} = M_{2} D_{2} H_{2}$

Arithmetic • Time and Work

Formula

M_{1} D_{1} H_{1} = M_{2} D_{2} H_{2}

M

workers of equal efficiency take

D

days of

H

hours each to finish a job, the product

M D H

measures the job and is the same for any other workforce doing the same job:

M_{1} D_{1} H_{1} = M_{2} D_{2} H_{2}

. With efficiencies

E

differing between groups, the invariant is

M D H E

How to use it

1Compute the job once as a total in man-hours, then divide by the new daily output to get the new number of days.
2When two kinds of worker are mixed, first convert one into the other using the stated exchange rate (e.g. $3$ women $= 2$ men, so one woman is $\frac{2}{3}$ of a man).
3Drop $H$ when the working day is not mentioned; the rule then reads $M_{1} D_{1} = M_{2} D_{2}$ .

Worked example

10

men take

12

days of

7

hours each to build a wall, how many days of

6

hours each will

14

men take?

Step-by-step

1
$10 \times 12 \times 7 = 840$ man-hours
Why: Measure the whole job in man-hours.
2
$14 \times D \times 6 = 840$
Why: The same job, done by the new workforce.
3
$D = \frac{840}{84} = 10$

Answer

10

days

Watch out

It assumes every worker in a group is equally efficient and that output scales linearly. Convert mixed workforces to a single unit BEFORE multiplying, and never add men to women directly.

Builds on

Unitary method: find one unit, then scale

Also known as

men days hours, work equivalence, chain rule of work, man-hours, M1D1H1 = M2D2H2, men and women work ratio

Appears in past papers

Formula

Worked example

Appears in past papers

2020 (II)

2020 (I)

2019 (II)

2019 (I)

2018 (II)

2018 (I)

2017 (II)

2017 (I)