AFCAT Numerical Ability · Time and Work

Work as a Rate, and Efficiency

Turn every work statement into a rate, the share of the job done in one day or hour, and read efficiency as the opposite of time.

All Time and Work notes

Why this matters

Most of this page is EASY, and two AFCAT past-year questions sit here, from 2018 and 2019. Marks are lost on the direction of a change: 25% more efficient does not mean 25% less time, and a part of the work done in some days has to be scaled up to the whole job before anything else.

Concept 1 of 4: Part of the work in some days

Treat the whole job as 1, or as 100%. If

\tfrac 27

of a job takes 6 days, then

\tfrac 17

takes 3 days and the whole job takes 21 days. Every later method in this chapter starts from this step: how long the whole job takes, or how much of it is done in one day.

Definition

A fraction $\tfrac pq$ of the work in $d$ days means the whole work takes $d \times \tfrac qp$ days.
The rate is the share done in one day: the whole job in $T$ days means $\tfrac 1T$ of it a day, or $\tfrac{100}{T}\%$ a day.
Work done is rate × time, and the work left is 1 minus the work done.
To compare people, change each one to the time for the whole job, then compare.

Whole job from a part

\tfrac pq \text{ of the job in } d \text{ days} \;\Rightarrow\; T = d \times \dfrac qp

Worked example

A painter finishes

\tfrac 27

of a wall in 6 days. How long does the whole wall take, and how many more days does he need?

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionModerate

Example 1 · Time and Work · Work and Efficiency

A contractor estimates that

30\%

of work should be completed in

12

days, but only

20\%

gets completed. How many more days are needed to complete the remaining

80\%

work at the current rate?

Scale the part the right way

\tfrac 35

of a job takes 9 days, the whole job takes

9 \times \tfrac 53 = 15

days, not

9 \times \tfrac 35

. The whole job is bigger than the part, so the days must go up.

'More days' is what is left

When the question asks how many more days, take away the days already worked. A 15-day job that is 9 days in needs 6 more days, not 15.

Drill 16 more on part of the work in some days

Concept 2 of 4: Work done in stages, in percentages

When a job is done in stages, keep one running total of what is left. The catch is the phrase 'of the remaining work': a percentage of what is left is a smaller share of the whole job.

Definition

Stages that are shares of the WHOLE job just add: 25%, then 35%, then 15% leaves 25%.
'k% of the remaining work' is k% of what is left at that moment. Change it into a share of the whole job before adding.
A rate in percent a day works stage by stage: days = percentage still to do ÷ percentage a day.

A share of the remainder

\text{done} = \tfrac{k}{100} \times (\text{work left})

Worked example

A clears 40% of a backlog. B then clears 75% of what is left, and C finishes the rest. What share of the whole backlog did C clear?

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionModerate

Example 2 · Time and Work · Work and Efficiency

A completes

20\%

of a work in

5

days and B completes

30\%

of the remaining work in

6

days. What percentage of work is still remaining?

Of the remaining work

If X does 75% of a job and Y then does 40% of the remaining work, Y has done only 10% of the whole job. 15% is still left, not none.

Drill 7 more on work done in stages, in percentages

Concept 3 of 4: Efficiency and time

Efficiency is the rate at which someone works. For the same job, twice the efficiency means half the time. So time is efficiency turned upside down: efficiencies

3 : 4

give times

4 : 3

Definition

Time × efficiency is fixed for a job, so time is inversely proportional to efficiency.
x% more efficient: the rate is $\tfrac{100 + x}{100}$ times as much, so the time is $\tfrac{100}{100 + x}$ times as much.
x% less efficient: the rate is $\tfrac{100 - x}{100}$ times, so the time is $\tfrac{100}{100 - x}$ times.
Working at a fraction $f$ of normal efficiency stretches the time to $\tfrac 1f$ times the normal time.
Efficiencies $a : b : c$ give times $\tfrac 1a : \tfrac 1b : \tfrac 1c$ . Clear the fractions with the LCM.
'A is k times as efficient as B and takes d days fewer': B takes $k$ parts of time and A 1 part, so the $k - 1$ parts between them are the d days.

Time and efficiency

\dfrac{T_A}{T_B} = \dfrac{E_B}{E_A}

Worked example

B is 25% more efficient than A, and A takes 30 days for a job. How long does B take?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

AFCAT · 2019 · AFCAT (I) 2019 — Numerical Ability · 03-203Easy

Example 3 · Time and Work · Work and Efficiency

A

is 5 times as efficient as

B

A

completes a piece of work in 60 days less than

B

. How much time will they take individually?

More efficient is not the same percentage less time

25% more efficient cuts the time to

\tfrac{100}{125}

of it, which is 20% less, not 25% less. 50% more efficient cuts the time by a third.

The ratio of times turns round

Efficiencies

2 : 3

mean times

3 : 2

. Writing the times as

2 : 3

gives the slower worker the shorter time.

Drill 17 more on efficiency and time

Concept 4 of 4: Output per hour: machines and shifts

Some questions measure work in units made instead of jobs finished. The rate is then units per hour, and everything above still holds: rates add, efficiency scales the rate, and time is output divided by rate.

Definition

Rate = units ÷ hours. Output = rate × hours. Time = units ÷ rate.
An efficiency of $p\%$ means the rate is $p\%$ of the standard rate.
Shifts or machines working at the same time add their outputs.
When efficiency falls by the same step every day, the daily outputs are evenly spaced: add them as (first + last) ÷ 2 × number of days.

Output

\text{output} = \text{rate} \times \text{time}

Worked example

A machine makes 90 bolts in 6 hours. A second machine is 40% more efficient. How long does the second machine take to make 252 bolts?

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionEasy

Example 4 · Time and Work · Work and Efficiency

A machine can produce

100

units in

8

hours. Another machine is

60

% more efficient. How long will the second machine take to produce

200

units?

Efficiency scales the rate

A machine 25% less efficient than one making 80 units an hour makes 60 an hour. It needs

\tfrac 43

of the time, not 25% more time.

Drill 7 more on output per hour: machines and shifts

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 (4)

Part of the work in some days
Whole job from a part
$\tfrac pq \text{ of the job in } d \text{ days} \;\Rightarrow\; T = d \times \dfrac qp$
Work done in stages, in percentages
A share of the remainder
$\text{done} = \tfrac{k}{100} \times (\text{work left})$
Efficiency and time
Time and efficiency
$\dfrac{T_A}{T_B} = \dfrac{E_B}{E_A}$
Output per hour: machines and shifts
Output
$\text{output} = \text{rate} \times \text{time}$

Watch out for (6)

Scale the part the right way
→ Part of the work in some days
'More days' is what is left
→ Part of the work in some days
Of the remaining work
→ Work done in stages, in percentages
More efficient is not the same percentage less time
→ Efficiency and time
The ratio of times turns round
→ Efficiency and time
Efficiency scales the rate
→ Output per hour: machines and shifts

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