AIME 2007 II · 第 4 题

AIME 2007 II — Problem 4

专题: Contest Math
难度: L4
来源: AIME

题目详情

Problem

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 $m$ .

解析

Solutions

Suppose that it takes $x$ hours for one worker to create one widget, and $y$ hours for one worker to create one whoosit.

Therefore, we can write that (note that two hours is similar to having twice the number of workers, and so on):

100 = 300x + 200y

2(60) = 240x + 300y

3(50) = 150x + my

Solve the system of equations with the first two equations to find that $(x,y) = \left(\frac{1}{7}, \frac{2}{7}\right)$ . Substitute this into the third equation to find that $1050 = 150 + 2m$ , so $m = \boxed{450}$ .

Video Solution by OmegaLearn

https://youtu.be/00Ngozqw2d0?t=542

~ pi_is_3.14