题目
Working simultaneously at their respective constant rates, Machines A and B produce 800 nails in x hours. Working alone at its constant rate, Machine A produces 800 nails in y hours. In terms of x and y, how many hours does it take Machine B, working alone at its constant rate, to produce 800 nails?
解析
机器A和机器B在\(x\)小时内生产800个钉子。
效率 = 产量 ÷ 时间
因此,两台机器的**总效率**为\(\frac{800}{x}\)个/小时。
机器A单独以恒定效率工作,在\(y\)小时内生产800个钉子。
效率 = 产量 ÷ 时间
因此,机器A的效率为\(\frac{800}{y}\)个/小时。
我们已知:机器A的效率 + 机器B的效率 = 两台机器的总效率
代入得:\(\frac{800}{y} + \text{机器B的效率} = \frac{800}{x}\)
改写为:机器B的效率 = \(\frac{800}{x} - \frac{800}{y}\)
通分后:机器B的效率 = \(\frac{800y}{xy} - \frac{800x}{xy}\)
合并得:机器B的效率 = \(\frac{800y - 800x}{xy}\)
机器B单独以恒定效率工作,生产800个钉子需要多少小时?
时间 = 产量 ÷ 效率
\[
\begin{align*}
&= 800 \div \frac{800y - 800x}{xy} \\
&= 800 \times \frac{xy}{800y - 800x} \\
&= \frac{800xy}{800(y - x)} \\
&= \frac{xy}{y - x}
\end{align*}
\]
答案:E