Time Work Efficiency related problem

Question

A bakery bakes 60 cakes in 9 hours every morning. They make the first batch of P cakes in Q hours and then increase their efficiency by 20% to make the remaining cakes, finishing just on time. How much time could they save if they worked throughout at increased efficiency?

Answer 1

We can try solving this using equations.
The bakery has to bake 6o cakes in total,which is P cakes plus the remaining.We will name theese R.
$$60=P+R$$ Total time is 9 hours,Q plus remaining which we will call T. $$9=Q+T$$ We have 2 baking speeds.The first one is P cakes/Q time. $$S_1=\frac{P}{Q}$$ The second speed is 20% bigger or $\frac{1}{5}$ bigger.So it's $\frac{6}{5}$ of speed 1. $$S_2=\frac{6}{5} * \frac{P}{Q}$$ Speed 2 is also R cakes/T time. $$S_2=\frac{R}{T}$$ Now we know $$\frac{R}{T}=\frac{6}{5} * \frac{P}{Q}$$ R must than be $$R=T*\frac{6}{5} * \frac{P}{Q}$$ Combine $$R=T*\frac{6}{5} * \frac{P}{Q}$$ and $$60=P+R$$ You get $$60=P+T*\frac{6}{5} * \frac{P}{Q}$$ Now taking $$9=Q+T$$we know $$Q=9-T$$If you combine this with $$60=P+T*\frac{6}{5} * \frac{P}{Q}$$you get $$60=P+T*\frac{6}{5} * \frac{P}{9-T}$$
We successfully eliminated 2 unknown factors,but I get stuck here.I think it is because there just isn't enough information to solve this problem.We would need at least one number more.Make sure you included all the information you have about ths problem in your question and that it is clear.

It is also possible that this problem is not meant to have only one solution,but several.You could try solving it with a graphic calculator and choosing only the positive numbers as solutions.