AFCAT Numerical Ability · Time and Work

Working Together: Adding Rates

People working together add their daily rates. Subtract a known rate to find an unknown one, and use pair totals or efficiency ratios when single times are not given.

All Time and Work notes

Why this matters

More AFCAT past-year questions sit on this page than on any other, ten of them, from every year from 2015 to 2019. Rates add but times never do, and most wrong answers come from adding or averaging the days themselves.

Concept 1 of 5: Adding rates

In one day A does

\tfrac 1a

of the job and B does

\tfrac 1b

. Working side by side they do both shares every day, so the rates add. The time together is 1 divided by the total rate.

Definition

Together: $\tfrac 1T = \tfrac 1a + \tfrac 1b$ , so $T = \dfrac{ab}{a + b}$ for two people.
Three people: add three rates. The product over sum shortcut is for two people only.
LCM units: call the job the LCM of the times. With times 9 and 18 the job is 18 units; A does 2 a day and B 1, so together 3 a day and 6 days.
The time together is always less than the fastest person's time alone. An answer above it is wrong.

Two people together

T = \dfrac{ab}{a + b}

Worked example

A can do a job in 10 days, B in 15 days and C in 30 days. How long do they take together?

Practice this conceptself-check · 3 quick reps

The same idea in a real exam question:

AFCAT · 2015 · AFCAT (II) 2015 — Numerical Ability · 03-215Easy

Example 1 · Time and Work · Combined Work

A

can do

\frac{1}{2}

of a piece of work in 8 days while

B

can do

\frac{1}{3}

of the same work in 8 days. In how many days can both do it together?

Never add or average the days

A in 10 days and B in 15 days do not take 25 days together, or 12.5. They take

\tfrac{10 \times 15}{25} = 6

days, less than either alone.

Drill 9 more on adding rates

Concept 2 of 5: One person's time from the joint time

If A and B together do

\tfrac 16

of a job each day and A alone does

\tfrac 19

, what is left over is B's share:

\tfrac 16 - \tfrac 19 = \tfrac 1{18}

. Subtracting rates undoes adding them.

Definition

B's rate = joint rate minus A's rate: $\tfrac 1b = \tfrac 1T - \tfrac 1a$ , so $b = \dfrac{aT}{a - T}$ .
Three people together take $T$ and two of them take $S$ : the third alone does $\tfrac 1T - \tfrac 1S$ a day.
Joint work given as a percentage: 'together they do 45% in 9 days' is 5% a day. Subtract the known person's percentage a day.
The ratio of the two rates is the efficiency ratio.

The other person's time

b = \dfrac{aT}{a - T}

Worked example

A and B together finish a job in 12 days, and A alone takes 18 days. How long does B take alone?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

AFCAT · 2019 · AFCAT (I) 2019 — Numerical Ability · 03-205Easy

Example 2 · Time and Work · Combined Work

1 man and 1 woman together can complete a piece of work in 8 days. A man alone can complete the work in 10 days. In how many days can one woman alone complete the work?

Subtract rates, not days

Together 9 days and A alone 12 days does not make B 3 days. B's rate is

\tfrac 19 - \tfrac 1{12} = \tfrac 1{36}

, so B takes 36 days.

Drill 19 more on one person's time from the joint time

Concept 3 of 5: Pairs working together

When only pairs are timed, add the three pair rates. Each person appears in two pairs, so the total is twice the rate of all three together. Halve it, then take away a pair to leave one person.

Definition

$2(A + B + C) = (A + B) + (B + C) + (C + A)$ , where each letter stands for a daily rate.
One person alone: $A = (A + B + C) - (B + C)$ .
Two pair rates with a shared member compare the other two: $(A + B) - (B + C) = A - C$ .

All three from the pairs

A + B + C = \tfrac 12\left[(A + B) + (B + C) + (C + A)\right]

Worked example

A and B take 8 days for a job, B and C take 12 days, and C and A take 8 days. How long do all three take together, and how long does A take alone?

Practice this conceptself-check · 3 quick reps

The same idea in a real exam question:

AFCAT · 2016 · AFCAT (I) 2016 — Numerical Ability · 03-214Moderate

Example 3 · Time and Work · Combined Work

A

and

B

can do a piece of work in 72 days,

B

and

C

in 120 days, and

A

and

C

in 90 days. In what time can

A

alone do it?

Halve the total

Adding the three pair rates counts everyone twice. For pairs taking 8, 12 and 8 days the sum is

\tfrac 13

, so all three together take 6 days, not 3.

Drill 6 more on pairs working together

Concept 4 of 5: Efficiency ratios with a joint time

When you know how much faster one person is and how long they take together, work in units of efficiency. If A is twice as fast as B, give A 2 units a day and B 1. Together that is 3 units a day, and the job is 3 times the joint time in units.

Definition

Efficiencies $a : b$ and a joint time $T$ : the job is $(a + b)T$ units, and A alone takes $\dfrac{(a + b)T}{a}$ days.
A does $\tfrac{a}{a + b}$ of the joint rate.
'A takes k times as long as B' means B is k times as efficient as A.
If a gap in days is given instead of a joint time, use the parts rule from Work and Efficiency first, then add the rates.

One person from a ratio

T_A = \dfrac{(a + b)\,T}{a}

Worked example

A is twice as efficient as B, and together they finish a job in 16 days. How long does each take alone?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

AFCAT · 2017 · AFCAT (II) 2017 — Numerical Ability · 03-209Moderate

Example 4 · Time and Work · Combined Work

A

is thrice as efficient as

B

A

takes 40 days less than

B

to complete a work. If they both work together, how many days will it take to finish the work?

Which way the ratio runs

'A takes twice as long as B' makes B the faster worker, with efficiencies A : B = 1 : 2. Giving A the 2 units answers for the wrong person.

Drill 19 more on efficiency ratios with a joint time

Concept 5 of 5: Work done in a given time, and each share

Not every question asks for a finishing time. Some ask how much is done after a few days together, or what share of the work, or of the pay, each person earns. Work done is rate × time, and each person's share of the work is their share of the joint rate.

Definition

Work done in $d$ days together is $d\left(\tfrac 1a + \tfrac 1b\right)$ , and the work left is 1 minus that.
Time for part of the job = that part ÷ the joint rate. Two thirds of a job at $\tfrac 19$ a day takes 6 days.
When two people do a whole job together, their shares of the work are in the ratio of their rates: times 6 and 12 give shares $2 : 1$ .
Wages follow the work done. With different daily wages, the pay is in the ratio of days × daily wage.

Work done together

W = d\left(\dfrac 1a + \dfrac 1b\right)

Worked example

A can finish a job in 9 days and B in 18 days. They work together for 4 days. What fraction of the job is left?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

AFCAT · 2018 · AFCAT (I) 2018 — Numerical Ability · 03-208Easy

Example 5 · Time and Work · Combined Work

A

can do a work in 20 days and

B

in 40 days. If they work on it together for 5 days, then the fraction of the work that is left is

Done, or left?

After people work for a while, read whether the question asks for the work done or the work left. With

\tfrac 23

done,

\tfrac 13

is left, and both are usually among the options.

Pay follows the work, not the days

Two people who work the same number of days at different speeds are not paid equally. Split the money in the ratio of the work each one did.

Drill 9 more on work done in a given time, and each share

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 (5)

Adding rates
Two people together
$T = \dfrac{ab}{a + b}$
One person's time from the joint time
The other person's time
$b = \dfrac{aT}{a - T}$
Pairs working together
All three from the pairs
$A + B + C = \tfrac 12\left[(A + B) + (B + C) + (C + A)\right]$
Efficiency ratios with a joint time
One person from a ratio
$T_A = \dfrac{(a + b)\,T}{a}$
Work done in a given time, and each share
Work done together
$W = d\left(\dfrac 1a + \dfrac 1b\right)$

Watch out for (6)

Never add or average the days
→ Adding rates
Subtract rates, not days
→ One person's time from the joint time
Halve the total
→ Pairs working together
Which way the ratio runs
→ Efficiency ratios with a joint time
Done, or left?
→ Work done in a given time, and each share
Pay follows the work, not the days
→ Work done in a given time, and each share

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