Question

The workers in a factory produce widgets and whoosits. For each product, production time is constant and identical for all workers, but not necessarily equal for the two products. In one hour, $100$ workers can produce $300$ widgets and $200$ whoosits. In two hours, $60$ workers can produce $240$ widgets and $300$ whoosits. In three hours, $50$ workers can produce $150$ widgets and $m$ whoosits. Find $m.$

Answer

Solution:

Let $a$ and $b$ be the worker-hours required to make one widget and one whoosit. The three scenarios supply $100,$ $120,$ and $150$ worker-hours, so $300a + 200b = 100, \qquad 240a + 300b = 120, \qquad 150a + mb = 150.$

The first two equations simplify to $3a + 2b = 1$ and $4a + 5b = 2,$ giving $a = \frac{1}{7}$ and $b = \frac{2}{7}.$ Substituting into the third, $\frac{150}{7} + \frac{2m}{7} = 150,$ so $150 + 2m = 1050$ and $m = 450.$

2007 AIME II Problem 4

Solution:

Problem 4 in Other Years