📚 Solving Work Problems in Mathematics | 数学应用题：工程问题的解题思路
Work problems, often called “lifting problems” or “pipe problems” in school mathematics, ask us to find how long it takes for people, machines, or pipes to complete a task. The key is to treat work as a quantity and compare it with rate and time.
工程问题，也叫“工程应用题”或“水管问题”，通常要求我们计算若干人、机器或水管合作完成一项任务需要多长时间。解决这类问题的关键，是把“工作量”看作一个确定的量，并理清工作量、工作效率与工作时间三者的关系。
1. What Are Work Problems? | 什么是工程问题？
A work problem describes a task being completed by one or more workers or tools. Each worker has a fixed rate, or “efficiency.” The total work can be treated as 1 unit, an unknown quantity, or an arbitrary number chosen for convenience.
工程问题描述的是由一名或多名工人、工具共同完成一项任务的情形。每一个“做工者”都有固定的工作效率。总工作量可以设为一个单位、一个未知数，也可以为了计算方便设为任意一个整数。
The most common instructions are:
常见的表述有：
- Pipe A fills the tank in 3 hours. | A管单独注满水池需要3小时。
- Working together, they finish the job in 5 days. | 合作完成这项工作需要5天。
- One leaves halfway through. | 其中一人中途离开。
No matter how the question is worded, the underlying relationship is always the same.
无论题目怎样措辞，本质上都在使用同一个关系式。
2. The Golden Formula: Work = Rate × Time | 核心公式：工作量 = 工作效率 × 工作时间
In every work problem, we connect three quantities: total work W, rate R, and time T. The fundamental relationship is:
在每一道工程问题中，我们都涉及三个量：总工作量 W、工作效率 R 和工作时间 T。最基本的关系是：
W = R × T
From this formula, we can also write the rate and the time:
由这个公式，我们还可以写出效率和时间的表达式：
R = W / T
T = W / R
If a worker can complete 100 tasks in 4 hours, the rate is 25 tasks per hour. If the total work is unknown, we often set it equal to 1, so a worker who finishes the work in n days has rate 1/n per day.
如果一名工人4小时能完成100个工件，那么他的效率是每小时25个工件。如果总工作量未知，我们通常把总工作量设为1，那么一个在n天内完成全部工作的工人，其效率就是每天1/n。
3. One Worker: Expressing Rate | 单人工作：用分数表示效率
Suppose a worker can complete a job in 6 days. This means the worker’s rate is 1/6 of the job per day. In 2 days, the worker completes 2 × 1/6 = 1/3 of the job.
假设一名工人单独完成某项工作需要6天。这表明他的效率是每天完成整个工作的1/6。工作2天后，他完成了 2 × 1/6 = 1/3 的工作量。
In general, if a worker needs n time units to complete the whole job, then:
一般地，如果某个工人单独完成全部工作需要 n 个时间单位，那么：
Rate = 1/n
Work done in t time = t/n
This fraction representation is the heart of the “unitary method” used in work problems.
这种用分数表示效率的方法，是解决工程问题中“归一思想”的核心。
4. Working Together: Add the Rates | 合作问题：把“效率”相加
When two workers work together without interfering, their rates are added. Suppose worker A can finish a job in a days and worker B can finish it in b days. Their combined rate is:
当两名工人合作且互不干扰时，他们的效率应该相加。假设工人A单独完成工作需要a天，工人B单独完成工作需要b天，则合作效率为：
1/a + 1/b
The time needed for them to finish together is the reciprocal of this sum.
他们合作完成所需的时间是这个和的倒数。
For example, if A takes 10 days and B takes 15 days:
例如，A单独做需要10天，B单独做需要15天：
1/10 + 1/15 = (3 + 2)/30 = 5/30 = 1/6
So their combined rate is 1/6 of the work per day, and they finish in 6 days.
所以他们的合作效率是每天完成整个工作的1/6，合作完成需要6天。
A common mistake is to add the two times, 10 + 15 = 25 days. This is wrong because work rates, not times, are combined.
一个常见错误是把两人时间直接相加：10 + 15 = 25天。这是错误的，因为合作时应当把“效率”相加，而不是把“时间”相加。
5. When One Worker Joins Later | 中途加入问题：分段处理
Many problems do not involve working together from the beginning. A may work alone for some days, and then B joins him. In this situation, split the process into separate stages.
许多工程问题并不是从开始就合作。比如A先单独工作若干天，B再加入。这时应该把整段过程分成几个阶段来处理。
Example: A can finish a job in 12 days, and B can finish it in 18 days. A works alone for 4 days, and then B joins. How long will it take to finish?
例：A单独完成工作需要12天，B单独完成需要18天。A先单独工作4天，然后B加入。请问剩余工作还需要多少天完成？
First find the work done by A before B joins:
首先计算B加入前A完成的工作量：
4/12 = 1/3
Since 1/3 of the job is done, the remaining work is:
因为已经完成了1/3，剩余工作量为：
1 – 1/3 = 2/3
Now find the combined rate of A and B:
然后计算A和B的合作效率：
1/12 + 1/18 = 3/36 + 2/36 = 5/36
The extra time needed is the remaining work divided by the combined rate:
剩余时间等于剩余工作量除以合作效率：
(2/3) ÷ (5/36) = (2/3) × (36/5) = 24/5 = 4.8 days
If the question asks for the total time, add the first 4 days to this result.
如果题目问“一共需要多少天”，再把最开始的4天加上即可。
6. Pipes and Drains: Net Rate | 进水管与排水管：净效率
In pipe problems, an inlet pipe adds water to a tank, and an outlet pipe removes water. If both pipes are open, the net filling rate is the inlet rate minus the outlet rate.
在水管问题中，进水管向水池加水，排水管从水池抽水。如果两根水管同时打开，那么净注水效率等于进水管效率减去排水管效率。
If an inlet pipe can fill a tank in 3 hours, its rate is 1/3. If an outlet pipe can empty the same tank in 5 hours, its emptying rate is 1/5. With both open:
如果进水管单独注满水池需要3小时，其效率是1/3；排水管单独排空水池需要5小时，其排水效率是1/5。两根管同时打开时：
Net rate = 1/3 – 1/5 = (5 – 3)/15 = 2/15
The time required to fill the tank is:
因此注满水池所需时间为：
1 ÷ (2/15) = 15/2 = 7.5 hours
Notice that the result is longer than the inlet pipe alone would take. A drain always slows down the filling process.
注意到这个时间比进水管单独注水要更长。排水管总会减慢注水过程。
7. The LCM Multiplier Method | 设工作总量为最小公倍数
Fraction arithmetic can be improved by choosing a specific total work. A useful strategy is to set the total work equal to the least common multiple of the given times.
分数运算有时较麻烦。一个很实用的策略是：设总工作量为各个给定时间的最小公倍数，从而把分数转化为整数。
Take a job that A can do in 8 days and B can do in 12 days.
例如：A单独完成需要8天，B单独完成需要12天。
LCM(8, 12) = 24
Let the whole job be 24 units. Then:
设总工作量为24个单位，那么：
A’s rate = 24/8 = 3 units per day
B’s rate = 24/12 = 2 units per day
Together they complete 3 + 2 = 5 units per day, so the time needed is:
合作时每天完成 3 + 2 = 5 个单位，所以所需时间为：
24/5 = 4.8 days
This approach removes denominators and makes the problem look like an integer word problem.
这种设法可以去掉分母，把工程问题转化为整数应用问题。
8. Finding an Individual Rate from a Combined Rate | 由合作效率反推个人效率
Sometimes the question gives the combined time and the time of one worker, then asks for the other worker’s time. We simply subtract the known rate from the combined rate.
有时题目会给出合作完成时间以及其中一个人的单独完成时间，要求求另一个人的时间。我们只需用合作效率减去已知效率即可。
Suppose A and B together can finish a job in 6 days, and A alone can finish it in 10 days.
假设A和B合作6天可以完成工作，A单独完成需要10天。
Combined rate = 1/6
A’s rate = 1/10
Therefore B’s rate is:
因此B的效率为：
1/6 – 1/10 = (5 – 3)/30 = 2/30 = 1/15
So B alone needs 15 days.
所以B单独完成需要15天。
9. Common Pitfalls and How to Avoid Them | 常见易错点与应对策略
Work problems are full of small traps. The table below lists the most common mistakes and the correct way to think about each one.
工程问题隐藏着不少陷阱。下面这个表格列出了最常见的错误以及相应的正确思路。
| Mistake / 常见错误 | Correct Approach / 正确做法 |
| Adding times: if A takes 3 days and B takes 4 days, together they take 7 days. | Add rates, not times: 1/3 + 1/4 = 7/12, so the time is 12/7 days. |
| Thinking “A’s time is 10” means “A’s rate is 10.” | Rate is the reciprocal of time: if A takes 10 days, rate is 1/10 per day. |
| Using different units without conversion: A in hours, B in minutes. | Convert all rates into the same time unit before adding or subtracting. |
| Forgetting that a drain has negative contribution. | For pipes, use minus signs for outlet pipes and plus signs for inlet pipes. |
| Finding work done after 3 days as 3 times the total time. | Work done in t days is t × rate, not t ÷ total time. |
Always ask yourself: “What is the rate?” before doing any addition or subtraction.
做题前先问自己：“效率是多少？”然后再进行加减运算。
10. Worked Examples | 典型例题精讲
Let us study two complete examples that combine several skills.
下面通过两道完整例题，综合运用上述思路。
Example 1: A and B can finish a project in 12 days and 18 days respectively. With C’s help, all three finish it in 6 days. How long would C take alone?
例1：A、B单独完成一项工程分别需要12天和18天。如果C加入帮忙，三人合作6天可以完成。请问C单独完成需要多少天？
Calculate the combined rate of A and B:
先计算A和B的合作效率：
1/12 + 1/18 = 3/36 + 2/36 = 5/36
Find the combined rate of A, B, and C using the given total time:
再根据三人合作时间6天，求出三人总效率：
1/6 = 6/36
Subtract the combined rate of A and B to get C’s rate:
用总效率减去A、B的合作效率，得到C的效率：
6/36 – 5/36 = 1/36
So C alone needs 36 days.
所以C单独完成需要36天。
Example 2: A can dig a trench in 8 days. B can dig the same trench in 10 days. C can fill it with dirt in 20 days. If they all start at the same time, how long will it take to finish digging the trench?
例2：A挖一条沟需要8天，B挖同一条沟需要10天。C每20天能把这条沟填满。如果三个人同时开始，挖完这条沟需要多少天？
Here A and B add work to the project, while C destroys work. Using rates:
这里A和B是在增加工程量，而C在破坏工程量。用效率表示：
1/8 + 1/10 – 1/20
= 5/40 + 4/40 – 2/40 = 7/40
The net rate is 7/40 of the trench per day, so the required time is:
净效率为每天完成沟渠总量的7/40，所以所需时间为：
40/7 ≈ 5.71 days
11. Practice Problems | 练习与答案
Try the following problems on your own before reading the answers.
请先独立完成下面的练习，再对照答案。
Problem 1. Worker A can pick 600 kg of apples in 12 hours. Worker B can do the same job in 15 hours. How long will it take them to pick 600 kg of apples working together?
练习1. 工人A采摘600公斤苹果需要12小时，工人B单独完成同样工作需要15小时。如果他们合作采摘600公斤苹果，需要多少小时？
Problem 2. A pipe can fill a water tank in 6 hours. Another pipe can empty the tank in 9 hours. If the tank is empty and both pipes are opened, how many hours will it take to fill the tank?
练习2. 一根水管注满水池需要6小时，另一根水管排空水池需要9小时。如果水池原本为空，两管同时打开，需要多少小时才能注满水池？
Problem 3. A and B together can finish a task in 9 days. B alone can finish it in 18 days. How many days would A alone need?
练习3. A和B合作9天可以完成一项任务，B单独完成需要18天。请问A单独完成需要多少天？
Answers:
答案：
1. Combined rate = 1/12 + 1/15 = 3/20, so time = 20/3 hours ≈ 6.67 hours.
1. 合作效率 = 1/12 + 1/15 = 3/20，所以时间 = 20/3 小时 ≈ 6.67 小时。
2. Net rate = 1/6 – 1/9 = 1/18, so time = 18 hours.
2. 净效率 = 1/6 – 1/9 = 1/18，所以时间 = 18 小时。
3. A’s rate = 1/9 – 1/18 = 1/18, so A alone needs 18 days.
3. A的效率 = 1/9 – 1/18 = 1/18，所以A单独完成需要18天。
By always converting each worker into a rate, deciding whether a contribution is positive or negative, and carefully combining those rates, you can solve every work problem with confidence.
只要把每个参与者都转化为“效率”，判断它的贡献是正是负，再小心地对效率进行加减，你就能自信地解决所有工程问题。
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