how much did either worker work?

Question

The first worker earned 32.4€ and the second worker 28.8€, however, the first worker worked 2 hours longer than the second worker. The workers paycheck was different. If the first worker had worked as much as the second, and the second as much as the first, the second worker wouldve earned twice as much money as the first. How many hours did either worker work?

Answer 1

Let p1 and t1 and p2 and t2 represent the rates of pay (€/hr) and time worked (hours) of the two workers. p1t1=32.4 and p2t2=28.8. We also know that t1=t2+2, because worker 1 worked 2 hours more than worker 2, and p2t1=2p1t2, because if the workers' hours are swapped, worker 2 earns twice as much as worker 1. So, dividing two equations, p2t2/p2t1=28.8/2p1t2=14.4/p1t2. p2 cancels out: t2/t1=14.4/p1t2. Substitute p1=32.4/t1, t2/t1=14.4t1/32.4t2. Therefore: (t2/t1)^2=14.4/32.4=4/9 and t2/t1=2/3. t2=2/3t1 can be substituted in t1=t2+2 to give t1=2/3t1+2, so t1/3=2 and t1=6, making t2=4. Worker 1 worked 6 hours and worker 2 4 hours.