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Word Problems: Work and Workers

Master word‐problems on shared work and rates. Learn how to translate a story into ‘Rate × Time = Work’, set up an equation of the form 1/T = 1/r₁ + 1/r₂ + … , and solve for the unknown time or rate. Includes clear explanations, worked examples, and practice problems that progress from basic to advanced.

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Word Problems: Work and Workers

Study Guide

Key Definition

In work problems we usually let “one job” represent the whole task. If a worker can finish that job in t hours, the worker’s rate is 1/t job per hour. When several workers cooperate, their rates add.

Important Notes

Work problems are solved with linear equations that relate rate, time, and work.
Rate is measured in jobs per unit time (e.g., fences per hour).
When several workers work together, add their individual rates to obtain a combined rate.
Use the relationship Work = Rate × Time. For an entire task we set Work = 1 job.
Solve for the unknown variable (time or rate) using the combined‐rate equation.

Mathematical Notation

$+$ addition

$-$ subtraction

$\times$ or $*$ multiplication

$\div$ or $/$ division

$\frac{1}{T} = \frac{1}{r_1} + \frac{1}{r_2} + \ldots + \frac{1}{r_n}$ (combined work equation: T = time to finish the job together, rᵢ = individual times).

Write rates as reciprocals of time to complete one job.

Why It Works

Because rate is the reciprocal of time, adding rates tells us how much of the job is finished per unit time. Taking the reciprocal of that combined rate gives the total time needed.

Remember

Always form the equation Work = Rate × Time first. Put Work = 1 job, express each rate as a reciprocal, then solve.

Quick Reference

Work Equation:$\text{Work} = \text{Rate} \times \text{Time}$

Shared Work:$\displaystyle \frac{1}{T} = \frac{1}{r_1} + \frac{1}{r_2}$

Understanding Word Problems: Work and Workers

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Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

If a worker needs t hours to finish one job, then in 1 hour the worker finishes 1/t of the job. That is the worker’s rate. Two identical workers together finish 2/t of the job per hour, so the full job takes t/2 hours.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

Two identical workers finish a job together in $4$ hours. How long would one worker, working alone, need to complete the job?

Real-World Problem

Question Exercise

Intermediate

Fence Painting

Alex can paint a fence in $5$ hours, Jordan in $7$ hours. How long will it take them if they work together?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

A machine can complete a job in $3$ hours, but it breaks down after half the job is done. A second identical machine is then used to finish the rest. How much longer will it take?

Challenge Quiz

Single Choice Quiz

Advanced

Worker A can finish a job in $4$ hours and Worker B in $6$ hours. How long do they need together?

Recap

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Review key concepts and takeaways