节: 连接: 正在完成任务 | 9.10 连接: 完成任务

章节大纲

We can determine the amount of a task that is completed by a worker by multiplying the rate at which the task is completed by the time the task has been worked on. For example, if a secretary types at a rate of 50 words per minute and has been typing for 15 minutes, the secretary will have typed 750 words.
::我们可以通过将任务完成速度乘以任务完成时间来决定工人完成的任务数量。比如,如果秘书按每分钟50个字的速度打字并打15分钟,秘书将打750个字。

lesson content

In general, if a job can be completed in t units of time, the worker's rate is 1 job in $\begin{aligned} t \end{aligned}$ units of time—that is, the rate is $\begin{aligned} \frac{1}{t} \end{aligned}$ . We can use this to help us determine how long it will take a group of workers to complete a task.
::一般而言,如果一项工作能够在t时间单位内完成,那么工人的比率是t时间单位的1份工作,也就是说,这个比率是1吨。我们可以利用这个比率来帮助我们确定一组工人完成一项任务需要多长时间。

Let's consider an example. If it takes Chrissy 2 hours to build a dog house, and it takes Hannah 5 hours to build a dog house, how long will it take them to complete a dog house if they build it together?
::让我们举个例子。如果克里斯西需要两个小时来建造狗屋, 而汉娜需要五个小时来建造狗屋, 如果他们一起建造狗屋,他们需要多长时间才能完成狗屋?

Since we're trying to determine the amount of time it will take them to build the house together, let's assign that a variable—say, x . A table may help us organize our information as well.
::由于我们正试图确定他们一起建造房子需要多少时间,让我们指定一个变量, x。一个表格也可以帮助我们组织我们的信息。

	Rate	Time Worked	Fractional Part of Job Done
Chrissy	$\begin{aligned} \frac{1}{2} \end{aligned}$	$\begin{aligned} x \end{aligned}$	$\begin{aligned} \frac{1}{2} x \end{aligned}$
Hannah	$\begin{aligned} \frac{1}{5} \end{aligned}$	$\begin{aligned} x \end{aligned}$	$\begin{aligned} \frac{1}{5} x \end{aligned}$

Together they complete 1 job, so
::他们一起完成了一个工作,所以

\begin{aligned} \frac{1}{2} x + \frac{1}{5} x = 1. \end{aligned}

::12x+15x=1。

We can multiply both sides by the LCD, 10, to get
::我们可以把两边乘以LCD, 10,获得

\begin{aligned} 5 x + 2 x & = 10 \\ 7 x & = 10 \\ x & = \frac{10}{7} = 1 \frac{3}{7} . \end{aligned}

::5x+2x=107x=10x=107=137。

Together it will take them a little less than an hour an a half to complete the job.
::他们一起需要不到一个半小时的时间来完成这项工作。

Solve the following:
::解决以下问题:

1. A teenager can mow all the fields on his family's farm 3 times faster than his father can. Together they can do the job in 4 hours. How long would it take each of them, working alone, to mow the fields?
::1. 青少年在家庭农场上耕种所有田地的速度比父亲快3倍,他们可以在4小时内一起耕种,每人单独耕种田地需要多长时间?

2. A pool can be filled by one pipe in 5 hours and by a different pipe in 7 hours. If both pipes are used, how long will it take to fill the pool?
::2. 池水可在5小时内用1根管子填充,在7小时内用1根不同的管子填充,如果使用两根管子,填充池水需要多长时间?

3. For the past 10 years, it has taken John exactly 3 hours to fill his family's swimming pool. This year, when it took him 8 hours to fill his family's pool, John knew something was wrong. He discovered he had left the drain valve open. How long will it take John to drain his family's pool once it is completely filled, and only the drain valve is open?
::3. 在过去10年中,John花了整整3小时才填满家人的游泳池,今年,他花了8小时填满家人的游泳池,John知道有问题,发现排水阀开着,John要花多长时间才能排干家人的游泳池,只有排水阀开着?

4. Jane can paint her living room in 4 1/2 hours. It takes Jolene only 3 hours to complete the same task. How long would it take them to paint the living room if they worked together?
::4. Jane可以在4个半小时半的时间里油漆起居室,Jolene只需3小时完成同样的任务,如果他们一起工作,他们需要多久才能油漆起居室?

5. Andrea can wash the windows on the house in 30 minutes, and Jorge can wash the windows on the house in 40 minutes. How long will it take them to wash the windows together?
::5. Andrea可以在30分钟内洗家门窗,Jorge可以在40分钟内洗家门窗。

by martensmath provides examples of combined word problems involving rational equations.
::Martensmath 提供了涉及合理方程式的合并单词问题的例子。