CDS Mathematics · Time and Work

Work Rates

Someone who finishes a job in n days does 1/n of it each day; rates add when people work together.

All Time and Work notes CDS Mathematics strategy

Why this matters

Fourteen PYQs, two of them HARD. Every one is solved on RATES, never on days: find each person's fraction per day, add or subtract the fractions, and turn the total back into days at the end. Pay goes the same way — in the ratio of the rates.

Concept 1 of 2: Adding and subtracting rates

Days do not add, but daily fractions of the job do. Two people who take

6

and

10

hours do

\tfrac16 + \tfrac1{10}

of the job each hour together; the time is the reciprocal of that sum.

Definition

Alone in $n$ days: rate $\dfrac1n$ . Together: add the rates; time $= \dfrac{1}{\text{total rate}}$ .
Two people together: $\dfrac{ab}{a + b}$ days.
One partner's rate $=$ joint rate $-$ the other's.
Three pairs given: adding them gives $2(A + B + C)$ .
' $k$ times as efficient': rate $k$ times, time $\dfrac1k$ times.
Wages for a joint job: in the ratio of the rates.

Together

\dfrac{1}{T} = \dfrac1a + \dfrac1b

Worked example

P

and

Q

together finish a job in

8

days, and

Q

alone in

24

. How long does

P

take alone?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2017 · CDS (I) 2017 — Elementary Mathematics · Q28Easy

Example 1 · Time and Work · Work Rates

A and B working together can finish a piece of work in 12 days while B alone can finish it in 30 days. In how many days can A alone finish the work ?

Times do not add

Workers taking

6

and

10

hours do not finish together in

8

16

. Add the rates:

\tfrac16 + \tfrac1{10} = \tfrac4{15}

, so

3\tfrac34

hours.

Pay by rate, not by time

The faster worker earns MORE for a joint job. Times

x

1.5x

2x

give pay

6 : 4 : 3

, the reverse of

2 : 3 : 4

Drill 9 more on adding and subtracting rates

Concept 2 of 2: Joining, leaving and taking turns

Split the job into stages. Each stage does (rate

\times

time) of the work, and the stages add up to the whole job,

1

Definition

Stages: $r_1t_1 + r_2t_2 + \cdots = 1$ .
Someone leaves after $n$ days: $n\,(\text{joint rate}) + (\text{remaining time})(\text{remaining rate}) = 1$ .
Taking turns: add one full cycle, count whole cycles, then finish the last part at the next person's rate.
'One person per hour, nobody twice running': alternate the two fastest.

Stages

r_1 t_1 + r_2 t_2 = 1

Worked example

A

can do a job in

20

days and

B

30

. They start together;

A

leaves after

6

days. How long does

B

take to finish?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2022 · CDS (II) 2022 — Elementary Mathematics · Q13Moderate

Example 2 · Time and Work · Work Rates

X and Y can do a piece of work in 45 days and 40 days respectively. They begin to work together, but X leaves after n days and then Y completes the remaining work in 23 days. What is n equal to ?

The last turn may be partial

Taking turns, the job often ends part-way through a day. Count the fraction left after the whole cycles and give that to the NEXT person, not to whoever would suit the answer.

Drill 3 more on joining, leaving and taking turns

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)

Adding and subtracting rates
Together
$\dfrac{1}{T} = \dfrac1a + \dfrac1b$
Joining, leaving and taking turns
Stages
$r_1 t_1 + r_2 t_2 = 1$

Watch out for (3)

Times do not add
→ Adding and subtracting rates
Pay by rate, not by time
→ Adding and subtracting rates
The last turn may be partial
→ Joining, leaving and taking turns

Test yourself on Time and Work

15 past CDS questions from this chapter, timed at 18 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

Take the Time and Work chapter test Drill all 14 questions