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Work & motion

Time & work: add the rates

Treat one job as a whole, translate time into work rate, and combine workers without adding their completion times.

IntermediateAbout 13 min3 worked examples
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01 / THE STARTING POINT

Before you begin

What you need: Fractions, reciprocals and LCM.

  • Convert completion time into rate
  • Combine constant work rates
  • Solve staged-work problems
02 / BUILD THE CONCEPT

Understand the idea

One job is one whole

A person who finishes one job in 6 days completes 1/6 of it per day, assuming a constant rate. Completion time and rate are reciprocals.

Rates add when workers cooperate

If A and B work independently on the same divisible job without interference, their daily rates add. Their completion times do not.

Track the work already done

For changing teams, calculate work = rate × time in each stage. Subtract completed work from one, then divide the remaining work by the rate of the remaining team.

03 / A DIFFERENT WAY TO SEE IT

A 60-unit job makes rates visible

A: 5 units/day
B: 3 units/day
Together: 8 units/day
A takes 12 days and B takes 20 days. Together, 60 ÷ 8 = 7.5 days.
04 / FROM IDEA TO ANSWER

A method you can reuse

KEEP THIS HANDY

Together time = 1/(1/a + 1/b) = ab/(a + b)

For two positive solo times a and b, constant rates, and simultaneous independent work.
  1. Choose one whole job or a convenient total number of work units.
  2. Find each worker's output per unit time.
  3. Add rates, then divide the remaining work by the combined rate.
A useful insight

The LCM of the solo times is a convenient number of total work units. It often turns fractional rates into integers.

05 / WATCH THE METHOD WORK

Worked examples

Read the question first. Try a step yourself, then compare your reasoning.

EXAMPLE 01

Work together

A takes 12 days and B takes 20 days. Find their combined time.

  1. Use a 60-unit job.
  2. A does 5 units/day; B does 3 units/day.
  3. Time = 60/(5 + 3) = 7.5 days.
Answer7.5 days
EXAMPLE 02

One worker leaves

A needs 6 days and B needs 12 days. They work together for 2 days, then A leaves. How much longer does B need?

  1. Combined daily rate = 1/6 + 1/12 = 1/4.
  2. Two days complete 1/2; 1/2 remains.
  3. B needs (1/2)/(1/12) = 6 more days.
Answer6 more days
EXAMPLE 03

Equal workers

Three identical workers finish in 8 days. How long would four need?

  1. The job takes 3 × 8 = 24 worker-days.
  2. Divide by four workers.
  3. Assume equal constant rates and unchanged conditions.
Answer6 days
06 / YOUR TURN

Check your understanding

Quick questions, with explanations. These are for self-study and do not affect your account score.

0 of 3 questions checked

QUESTION 1A finishes in 10 days. Daily rate?

See the explanation

1/10 job. Rate is one job divided by 10 days.

QUESTION 2A and B each take 8 days alone. Together?

See the explanation

4 days. 1/8 + 1/8 = 1/4 job per day, so 4 days.

QUESTION 3A does 1/3 of a job and B then completes 1/6. What remains?

See the explanation

1/2. Completed work is 1/3 + 1/6 = 1/2; the remaining half is 1/2.

07 / TAKE THE NEXT STEP

Put your understanding to work

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EXPLORE THE NUMBERS

Time and Work Calculator

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Time and Work Practice Quiz 1

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Time and Work Practice Quiz 2

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Time and Work Practice Quiz 3

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