How to represent a question in algebra

Question

Please could someone show me how to represent this pub quiz question in algebra?

A Gym has $12$kg weights and $14$kg weights.

If it has $78$ weights in total and the total weight of the $12$kg weights is the same as the total weight of those weighing $14$kg, what is the total weight of them?

Thanks

Answer 1

If we set $t = $ number of $12$kg weights, and $f =$ number of $14$kg weights.

Then we can say that $$t+f=78$$ from the fact that there are $78$ weights in total.

We can also say that $$12t=14f$$ from the fact that the total weight of the $12$kg weights is equal to the total weight of the $14$kg weights.

We then have two simultaneous equations in two variables which we can solve easily:

\begin{align}12t&=14f\\ t&=\frac76f\\ &\Downarrow\\ \frac 76f+f&=78\\ \frac{13}{6}f&=78\\ f&=36\\ &\Downarrow\\ t+36&=78\\ t&=42\end{align}

So, we have $42$ weights weighing $12$kg and $36$ weights weighing $14$kg.

Answer 2

$12x=14y$
$x+y=78$

$x$= no. of weights of 12 kg
$y$=no. of wieghts of 14 kg