Question

A factory makes tennis rackets and cricket bats. A tennis racket takes $1.5$ hours of machine time and $3$ hours of craftsman's time in its making while a cricket bat takes $3$ hour of machine time and $1$ hour of craftsman's time. In a day, the factory has the availability of not more than $42$ hours of machine time and $24$ hours of craftsman's time.
(i) What number of rackets and bats must be made if the factory is to work at full capacity?

Answer 1

Let the number of rackets and the number of bats to be made be $x$ and $y$ respectively.
The machine time is not available for more than $42$ hours.

$\therefore$ $1.5x+3y\le \mathrm{42.........}\left(1\right)$
The craftsman's time is not available for more than $42$ hours.

$\therefore$ $3x+y\le \mathrm{24.........}\left(2\right)$
The factory is to work at full capacity. Therefore,

$1.5x+3y=42$

$3x+y=24$

On solving these equations, we obtain
$x=4,$ $y=12$

Thus, $4$ rackets and $12$ bats must be made.