AFCAT Numerical Ability · Time and Work

Working on Alternate Days, and Other Work Patterns

When people take turns, or work and rest in a fixed pattern, count whole cycles first, then finish the last part day by day, keeping track of who works next.

All Time and Work notes

Why this matters

One AFCAT past-year question is here, from 2015, and over a quarter of the page is HARD. Almost every slip is in the last cycle: running the whole cycles too far, or forgetting that the person who starts also works first in the final, partial cycle.

Concept 1 of 2: Two people on alternate days

A day of A and then a day of B make one cycle. Add their rates to get the work per cycle, run as many whole cycles as fit, then let the starter work the next day. If something is still left, the other person finishes it, perhaps in part of a day.

Definition

Work per 2-day cycle = $\tfrac 1a + \tfrac 1b$ . Take the largest number of whole cycles that does not overshoot the job.
After the whole cycles, the starter works first. If the starter's day covers what is left, the job ends on that day.
A partial last day counts as a fraction: work left ÷ that person's rate.
Who starts matters: when the faster person starts, the job can end sooner.
Working backwards: if the total time is given, count each person's days (the starter has the extra or partial day), then solve for the unknown rate.

One cycle

\text{work per cycle} = \dfrac 1a + \dfrac 1b

Worked example

A can do a job in 6 days and B in 12 days. They work on alternate days, A first. How long does the job take?

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionModerate

Example 1 · Time and Work · Working on Alternate Days

A

and

B

working separately can do a piece of work in 9 and 15 days respectively. If they work for a day alternately, beginning with

A

, then the work will be completed in

Do not run the cycles past the job

If each cycle does

\tfrac 27

, three cycles do

\tfrac 67

and a fourth would overshoot. Stop at three and finish day by day.

The starter works first in the last cycle

After the whole cycles, the next day belongs to whoever started. Giving it to the other person changes the answer whenever their rates differ.

Drill 16 more on two people on alternate days

Concept 2 of 2: Turns, helpers and rest days

Any repeating pattern works like the alternate-day cycle. Find the length of one repeat and the work it does, fit in whole repeats, and finish by hand. The repeat might be three people in turn, helpers who join every few days, or work days followed by rest days.

Definition

Helpers every k-th day: a block of $k$ days does $k$ days of the main worker plus one day of each helper.
Three in turn: a 3-day block does $\tfrac 1a + \tfrac 1b + \tfrac 1c$ .
Work and rest: with $w$ days on and $r$ days off, count the working days needed, then place the last one on the calendar. No rest day comes after the last working day.
Together, then in turns: do the joint stage first, then run the cycles on what is left.

Repeating blocks

\text{blocks} = \dfrac{\text{work}}{\text{work per block}}

Worked example

A can do a job in 10 days. B and C, who would each take 40 days, join him every second day. How long does the job take?

Practice this conceptself-check · 3 quick reps

The same idea in a real exam question:

AFCAT · 2015 · AFCAT (I) 2015 — Numerical Ability · 03-217Moderate

Example 2 · Time and Work · Working on Alternate Days

A

B

and

C

can do a piece of work in 20 days, 30 days and 60 days, respectively. In how many days can

A

do the work if he is assisted by

B

and

C

on every third day?

Rest after the end does not count

Someone working 6 days a week who needs 16 working days finishes on day 18, not day 19: the rest days after the last working day are never taken.

A helper's day is one day per block

If helpers join every fourth day, a 4-day block has four days of the main worker but only one day of each helper.

Drill 7 more on turns, helpers and rest days

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

Two people on alternate days
One cycle
$\text{work per cycle} = \dfrac 1a + \dfrac 1b$
Turns, helpers and rest days
Repeating blocks
$\text{blocks} = \dfrac{\text{work}}{\text{work per block}}$

Watch out for (4)

Do not run the cycles past the job
→ Two people on alternate days
The starter works first in the last cycle
→ Two people on alternate days
Rest after the end does not count
→ Turns, helpers and rest days
A helper's day is one day per block
→ Turns, helpers and rest days

Drill the questions on this topic

25 questions from the bank, with cart and Word export.

Drill all 25 questions