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Work and Time

Competitive Mathematics · Time, Work and Distance

7 Part3 worked examples19 practice questions

A painter takes 10 days to paint a whole wall. How much of the wall does the painter finish in one day? One-tenth. That is the foundation of this chapter: a job done in days has a **one-day work of . When two people work together their one-day works add**, when someone leaves they drop, and in the end the work done must equal (the whole job).

The first formula
1

Working together

Add the rates, not the days

Finding the time for two or three workers together, and one worker’s time when the pair is known.

Two or three workers together

Add each worker’s one-day work to get the time together.

Time taken together
  1. Write each person’s one-day work: .
  2. Add them — this is the one-day work together.
  3. The reciprocal (the fraction turned upside down) is the time together.
  4. Check: the time together must be less than either worker’s time alone.
Example 1 · Basic
A can do a job in 12 days and B in 4 days. How long will they take together?
Solution, step by step
  1. 1One-day work .
  2. 2So the time is days — less than both 4 and 12 ✓
Answer 3 days
Example 2 · Standard
A, B and C alone do a job in 10, 15 and 30 days. How long will all three take together?
Solution, step by step
  1. 1.
  2. 2Time days
Answer 5 days
BasicTry it yourself first
A alone can finish a job in 6 days and B alone in 3 days. In how many days will they finish it working together?
BasicTry it yourself first
A alone can finish a job in 20 days and B alone in 5 days. In how many days will they finish it working together?
StandardTry it yourself first
A alone can finish a job in 9 days and B alone in 18 days. In how many days will they finish it working together?
StandardTry it yourself first
A alone can finish a job in 14 days and B alone in 21 days. In how many days will they finish it working together?
StandardTry it yourself first
A alone can finish a job in 8 days and B alone in 12 days. In how many days will they finish it working together?
AdvancedTry it yourself first
A alone can finish a job in 7 days and B alone in 11 days. In how many days will they finish it working together?
AdvancedTry it yourself first
A alone can finish a job in 16 days and B alone in 24 days. In how many days will they finish it working together?
BasicTry it yourself first
A, B and C can do a job alone in 6, 12 and 12 days. In how many days will they do it together?
StandardTry it yourself first
A, B and C can do a job alone in 12, 18 and 36 days. In how many days will they do it together?
StandardTry it yourself first
A, B and C can do a job alone in 10, 20 and 25 days. In how many days will they do it together?
AdvancedTry it yourself first
A, B and C can do a job alone in 8, 12 and 24 days. In how many days will they do it together?
AdvancedTry it yourself first
A, B and C can do a job alone in 15, 20 and 30 days. In how many days will they do it together?

Pair known, find one worker

Subtract

When the pair’s one-day work and one worker’s are given, find the other’s.

The other worker’s time

If the pair takes days and A takes days alone, then B’s one-day work = the pair’s work minus A’s. Notice that is always larger than — a single worker is always slower than the pair.

Example 3 · Standard
A and B together finish a job in 6 days. A alone takes 15 days. How long does B alone take?
Solution, step by step
  1. 1B’s one-day work .
  2. 2So B alone takes days.
Answer 10 days
BasicTry it yourself first
A and B together do a job in 4 days. A alone can do it in 12 days. In how many days can B alone do it?
BasicTry it yourself first
A and B together do a job in 6 days. A alone can do it in 10 days. In how many days can B alone do it?
StandardTry it yourself first
A and B together do a job in 12 days. A alone can do it in 20 days. In how many days can B alone do it?
StandardTry it yourself first
A and B together do a job in 8 days. A alone can do it in 24 days. In how many days can B alone do it?
StandardTry it yourself first
A and B together do a job in 15 days. A alone can do it in 40 days. In how many days can B alone do it?
AdvancedTry it yourself first
A and B together do a job in 9 days. A alone can do it in 21 days. In how many days can B alone do it?
AdvancedTry it yourself first
A and B together do a job in 14 days. A alone can do it in 35 days. In how many days can B alone do it?
2

Efficiency and ratios

Faster worker, fewer days

Efficiency ratios, “ times as efficient”, “ more efficient” and pair questions.

Efficiency and time are inverse

The time ratio flips

A worker times as fast takes of the time.

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From pairs to singles

Add the three pairs

Given the times of A+B, B+C and C+A, find all three together.

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3

Leaving, joining and alternate days

Split the job into stages

Someone leaves or joins midway, workers alternate days, men and women are combined.

Leaving and joining

Add the work of each stage

Find the work done in each stage; the stages must add up to .

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Working on alternate days

The two-day cycle

Questions where workers take turns, one day each.

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Men, women and hours

The man-day unit

and “ men or women” questions.

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4

The LCM method

Make the total work a whole number

Take the total work as the LCM of the days and avoid fractions.

Total work = LCM

Think in units

Find each worker’s daily units; adding them gives the time directly.

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5

Traps and fast tricks

Slips · tricks

Where marks go in work-and-time questions.

The commonest mistakes

Six traps

The slips that cost marks.

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Fast tricks

LCM and man-days

Habits for speed.

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6

The practice ladder

Basic → standard → advanced

New questions of every type, in three levels.

Level 1 — Basic

Direct rules

One-rule questions.

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Level 2 — Standard

Two ideas combined

Exam-level questions.

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Level 3 — Advanced

Many steps and clever thinking

Questions at the hard end of competitive exams.

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7

Revision — formula cards

Flip each card and test yourself

Read the front, think, then flip.

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