Question

6 men working 9 hours a day can do a piece of work in 4 weeks. How many men are required working 8 hours a day for 3 weeks to complete the same work?

Answer 1

Here, the number of men is inversely proportional to the number of hours and inversely proportional to the number of days.

Number of men	Number of hours	Number of days
6	9	$4\times 7=28$
$x$	8	$3\times 7=21$
6:$x$	9:8	28:21 = 4:3

$\text{Number of men}\propto \frac{1}{\text{Number of hours}}$
$\Rightarrow 6:x=8:9$ ...(i)

$\text{Number of men}\propto \frac{1}{\text{Number of days}}$
$\Rightarrow 6:x=3:4$ ...(ii)

From (i) and (ii)
6 : $x$
$=\text{compound ratio of}8:9\text{and}3:4$
$\Rightarrow 6:x=(8:9)\times (3:4)$
$\Rightarrow 6:x=(8\times 3):(9\times 4)$
$\Rightarrow 6:x=(24:36)=(2:3)$

$\because$ Product of means = Product of extremes
$\Rightarrow x\times 2=6\times 3$
$\Rightarrow x=9$

Hence, 9 men are required to finish that work in 3 weeks working 8 hours a day.