WORK RATE PROBLEM
Work rate, cooperation solved.
Random cooperation work problems asking for combined time or remaining work, with formula-based solution steps — a classic elementary olympiad topic.
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Solution animation · work rates
t = 0 d: A did 0/9 (0%), B 0/5 (0%) — together 0%, 100% left.
What is this?
Work-rate problems treat the whole job as 1. If A takes a days alone, A's rate is 1/a per day and B's is 1/b. Combined rate = 1/a + 1/b, so time together = ab/(a+b). If A works m days first, the remaining 1 − m/a is finished by B in b(a−m)/a more days.
示例与注意事项Example & notes
一个例子A worked example
甲单独做 10 天、乙单独做 15 天：效率和为 1/10 + 1/15 = 1/6，合作需要 6 天完成。If A takes 10 days and B takes 15, the combined rate is 1/10 + 1/15 = 1/6, so they finish in 6 days.
注意事项Things to watch
- 把总工程量设为 1 是关键；设为最小公倍数可以避免分数计算。Setting the total work to 1 is the key trick — using the LCM avoids fractions entirely.
- 中途有人加入或离开必须分段计算，不能直接相加。If someone joins or leaves, compute in phases rather than adding rates.
- 效率单位是「每天完成几分之几」，别与「需要几天」混淆。Rates are fractions of the job per day — do not confuse them with days required.
相关问答More Q&A
能算先甲后乙的情况吗？Can it handle sequential work?
支持分阶段协作，可逐步输入各阶段的时间或工作量。Yes — phases can be entered one by one by time or by work done.
结果是分数怎么办？What if the answer is a fraction?
会自动约分到最简，并给出小数参考值便于判断。It reduces the fraction and adds a decimal estimate for sanity.