Solve workers, pipe & cistern, and individual rate problems with step-by-step working.
Enter the time (in the same unit) each worker takes to complete the job alone. Leave blank to exclude.
Worker Adays/hours
Worker Bdays/hours
Worker Cdays/hours
Worker Ddays/hours
Worker Edays/hours
Enter the time each pipe takes to fill or drain the cistern alone.
Pipe 1hrs to
Pipe 2hrs to
Pipe 3hrs to
Pipe 4hrs to
Two workers (A and B) together finish a job in T hours. Worker A alone takes TA hours. Find how long Worker B takes alone.
What Is a Work Rate Problem?
A work rate problem is a classic category of arithmetic problems that asks: if different workers (or pipes) can complete a task at different speeds, how long will it take them working together? These problems appear in everyday life — how many painters to finish a house by a deadline, how many pumps to drain a reservoir, how long a shared project takes with multiple contributors.
The key insight is that rates are additive. If Worker A does 1/6 of a job per day and Worker B does 1/4 per day, together they do 1/6 + 1/4 = 5/12 of the job per day. The total time is simply 12/5 = 2.4 days.
The Work Rate Formula
The fundamental formula is based on rates (fraction of job per unit time):
Rate of Worker = 1 / (Time to complete alone)
Combined Rate = 1/T₠+ 1/T₂ + 1/T₃ + ...
Time Together (T) = 1 / Combined Rate
The LCM method is an efficient alternative: take the LCM of all individual times as the "total work units." Each worker's output per day equals (LCM ÷ their time). Sum all outputs for the combined output per day, then divide total work by combined output for the answer.
Solving Pipe and Cistern Problems
Pipe and cistern problems are identical in structure but include both filling and draining pipes. Filling pipes contribute a positive rate; draining pipes contribute a negative rate. The net rate determines whether the cistern fills or empties, and how quickly.
For example: Pipe A fills in 3 hours (+1/3 per hour), Pipe B fills in 6 hours (+1/6 per hour), Pipe C drains in 4 hours (−1/4 per hour). Net rate = 1/3 + 1/6 − 1/4 = 4/12 + 2/12 − 3/12 = 3/12 = 1/4 per hour. The cistern fills in 4 hours.
Step-by-Step Worked Example
Problem: A completes a job in 6 days; B completes it in 4 days. How long together?
Rate A = 1/6 per day
Rate B = 1/4 per day
Combined = 1/6 + 1/4 = 2/12 + 3/12 = 5/12 per day
Time = 1 ÷ (5/12) = 12/5 = 2.4 days = 2 days 9 hrs 36 min
Common Variations
Partial work: A works for 3 days then B joins. Calculate work done by A, then apply combined rate for remaining work.
Worker leaves early: A and B start together; A leaves after t hours. Use combined rate for t hours, then solo rate for remaining work.
Reverse problem: Given combined time and one worker's time, find the other worker's time using 1/TB = 1/T − 1/TA.
Efficiency ratio: Workers with given efficiency ratio can be converted to times: if A is twice as fast as B and B takes T, then A takes T/2.
Tips for Competitive Exams
Work and time problems are standard in CAT, GRE, GMAT, and government competitive exams (SSC, IBPS). Here are exam-proven shortcuts:
Use LCM as total work — avoids fraction arithmetic and speeds up calculation.
For two workers: T = (T₠× T₂) / (T₠+ T₂) — memorize this shortcut.
Express all units consistently (hours, days, or minutes) before calculating.
For pipe problems, list rates with signs: filling = positive, draining = negative.
Always sanity-check: combined time must always be less than the fastest individual time.
Frequently Asked Questions
A work rate problem asks how long it takes multiple workers or pipes to complete a task together, given each one's individual completion time. The key principle is that rates (fractions of work per unit time) are additive.
If each worker completes the job alone in Tâ‚, Tâ‚‚, ..., Tâ‚™ units of time, their combined rate is 1/Tâ‚ + 1/Tâ‚‚ + ... + 1/Tâ‚™. The time for all to finish together is 1 ÷ (combined rate).
Combined rate = 1/6 + 1/4 = 5/12 per day. Time together = 12/5 = 2.4 days = 2 days and 9 hours 36 minutes. Shortcut: T = (6×4)/(6+4) = 24/10 = 2.4 days.
Structurally they are identical — but in pipe problems some agents reduce the total (draining pipes). Their rates are subtracted from filling rates rather than added. The net rate tells you if the cistern fills or empties overall.
Assign draining pipes a negative rate. Net rate = (sum of filling rates) − (sum of draining rates). If net rate is positive, the cistern fills; negative means it empties. Time = 1 / |net rate|.
Use: 1/TB = 1/T − 1/TA, so TB = TA × T / (TA − T). For this to be valid, T must be less than TA (you can't work slower together than the slowest worker alone).
CAT, GRE, GMAT, and banking exams regularly test work and time. Use the LCM method for speed: set total work = LCM of all times, compute each worker's daily output, sum for combined output, divide total work by combined output for the answer.