Question

A soap factory makes $600$ units in $9$ days with the help of $20$ machines. How many units can be made in $12$ days with the help of $18$ machines?

Answer 1

The number of units increases with the number of machines. On the other hand as the number of machines increases, the number of days required decreases. Thus the number of machines is directly proportional to the number of units and inversely proportional to the number of days. The first information gives
$20=k\times 600\times \frac{1}{9}$.
This gives the constant of proportionality
$k=\frac{20\times 9}{600}$
Let $x$ be the number of units which can be manufactured using $18$ machines in $12$ days. Using the same reasoning, we obtain
$18=k\times x\times \frac{1}{12}$.
Thus we obtain
$x=\frac{18\times 12}{k}=\frac{18\times 12\times 600}{20\times 9}=720$.
Hence $720$ units can be manufactured using $18$ machines for $12$ days.