MathsICSE Class 8 · Time and Work

Time and Work: Working Together, Leaving Midway and Alternate Days (ICSE Class 8)

Updated 17 October 2026 · Fact-checked

If a worker finishes a job in $n$ days, the work done in one day is $\frac{1}{n}$. Workers together add their one-day work: $\frac{1}{a} + \frac{1}{b} = \frac{a + b}{ab}$, so they finish in $\frac{ab}{a + b}$ days. A job is always taken as 1 whole.

Each day adds A's 1/12 and B's 1/6 to make 1/4 of the job, so the whole job of 1 takes 4 days.

Illustration drawn for QuestionBuddy.

Understand Working Together and Partial Work

The CISCE curriculum asks for simple, direct word problems on time and work.

One day's work. If $A$ alone completes the job in $a$ days, then in one day $A$ does $\frac{1}{a}$ of the job. In $d$ days $A$ does $\frac{d}{a}$ of it. The whole job is $1$ .

Working together. One day's work together is the sum of the one-day works: $\frac{1}{a} + \frac{1}{b}$ . The time taken is the reciprocal of this.

Finding one worker. If $A$ and $B$ together take $t$ days and $A$ alone takes $a$ days, then $B$ 's one-day work is $\frac{1}{t} - \frac{1}{a}$ , and $B$ 's time is the reciprocal.

Someone leaves. Compute the work done before the change, subtract from $1$ , then use the remaining workers' rate for the rest. If $A$ leaves $k$ days before the end, let the total time be $t$ : $B$ works $t$ days and $A$ works $t - k$ days.

Three workers. If pair rates are given, add all three pair rates: each person appears twice, so the sum is $2 \times$ (all three together).

Alternate days. If $A$ and $B$ work on alternate days, find the work in a two-day cycle, see how many full cycles are needed, then handle the last fraction with the correct worker.

Checking. A time found for the full job must satisfy: one-day work $\times$ days $= 1$ .

Working Together and Partial Work: formulas to remember

One day's work: $work in 1 day = \frac{1}{n}$; $n$ is the days to do the whole job.
Working together: $time = \frac{ab}{a + b}$; For two workers taking $a$ and $b$ days.
Second worker: $\frac{1}{b} = \frac{1}{t} - \frac{1}{a}$; From the combined time $t$ .
Three workers from pairs: $A + B + C = \frac{1}{2} [(A + B) + (B + C) + (A + C)]$; Add the three pair rates and halve.

How to solve Working Together and Partial Work questions

Convert days to fractions of the job per day, then add, subtract and invert.

1Write each worker's one-day work as a fraction.
2Add the rates of those working at the same time.
3Take reciprocals to get time from rate.
4For a change midway, compute the work done, then the remainder.
5Check that rate times time equals the work done.

Working Together and Partial Work in real life

Each concept, met in an everyday situation. Read the situation, try the problem, then check the solution. The one-line hook is what to remember.

Painting a room

Suppose Meena paints a room in 12 days and Neha in 18 days. They start together, but Meena leaves 3 days before the work is finished.

How many days does the room take?

Try it first, then show the solutionSolution

Let the total be $t$ days: $\frac{t - 3}{12} + \frac{t}{18} = 1$ .
Multiply by 36: $3 (t - 3) + 2 t = 36$ , so $5 t = 45$ and $t = 9$ .

Answer: 9 days.

Memory hook: Each person's days worked times their daily rate, summed, equals 1.

Tricks to remember Working Together and Partial Work

Short memory aids. Say each one aloud once, then use it on a question the same day.

Reciprocal pair
Rate = 1/time. Add rates, then flip to get time.
Use it when: Everything in this topic.
Pairs add twice
When pair rates are given, the three-person rate is half the sum of the pairs.
Use it when: A+B, B+C, A+C questions.

Working Together and Partial Work: solved examples

Ordered from board level to HOTS. Try each one before you open the solution.

Example 1: Two workers · 1 mark · MCQBoard level

A can do a job in 12 days and B in 24 days. Working together they finish it in:
(a) 6 days (b) 7 days (c) 8 days (d) 18 days

Try it first, then show the solutionSolution

$\frac{1}{12} + \frac{1}{24} = \frac{3}{24} = \frac{1}{8}$ , so 8 days.

Answer: (c)

Example 2: Basic together · 3 marks · Short answerBoard level

A takes 10 days and B takes 15 days to do a job. How long will they take working together?

Try it first, then show the solutionSolution

Rate together $= \frac{1}{10} + \frac{1}{15} = \frac{3 + 2}{30} = \frac{1}{6}$ .
Time $= 6$ days.

Answer: 6 days

Example 3: Finding the other worker · 3 marks · Short answerHarder board level

A and B together can finish a job in 12 days. A alone can do it in 20 days. In how many days can B alone do it?

Try it first, then show the solutionSolution

B's rate $= \frac{1}{12} - \frac{1}{20} = \frac{5 - 3}{60} = \frac{1}{30}$ .
B takes 30 days.

Answer: 30 days

Example 4: Leaving before the end · 4 marks · Short answerHarder board level

A can do a job in 12 days and B in 18 days. They start together but A leaves 3 days before the work is completed. In how many days is the work done?

Try it first, then show the solutionSolution

Let total time be $t$ : $\frac{t - 3}{12} + \frac{t}{18} = 1$ .
Multiply by 36: $3 (t - 3) + 2 t = 36$ .
$5 t = 45$ , $t = 9$ days. Check: A does $\frac{6}{12} = \frac{1}{2}$ , B does $\frac{9}{18} = \frac{1}{2}$ .

Answer: 9 days

Example 5: Three workers from pairs · 4 marks · Long answerHOTS / competency

A and B can finish a job in 12 days, B and C in 15 days, and A and C in 20 days. Find the time taken by all three together, and by each alone.

Try it first, then show the solutionSolution

Add the pair rates: $\frac{1}{12} + \frac{1}{15} + \frac{1}{20} = \frac{5 + 4 + 3}{60} = \frac{1}{5}$ .
This is twice the rate of all three: $A + B + C = \frac{1}{10}$ , so all three take 10 days.
$A = \frac{1}{10} - \frac{1}{15} = \frac{1}{30}$ (30 days). $B = \frac{1}{10} - \frac{1}{20} = \frac{1}{20}$ (20 days). $C = \frac{1}{10} - \frac{1}{12} = \frac{1}{60}$ (60 days).
Check: $A + B = \frac{1}{30} + \frac{1}{20} = \frac{1}{12}$ .

Answer: All three 10 days; A 30, B 20, C 60 days.

Where students lose marks in Working Together and Partial Work

The slips examiners see most often, and what to write instead.

Mistake: Adding the days instead of the rates
Time does not add; work per day does.
Wrong working
A 10 days, B 15 days: together $10 + 15 = 25$ days (or $12.5$ by averaging).
Right working
$\frac{1}{10} + \frac{1}{15} = \frac{1}{6}$ , so 6 days.
Do this: Convert to work per day before combining.
Usually costs: Answer and method.
Mistake: Not letting the leaver work part of the time
A who leaves 3 days early still works $t - 3$ days.
Wrong working
A works 0 days after the first change; equation $\frac{t}{18} = 1$ .
Right working
$\frac{t - 3}{12} + \frac{t}{18} = 1$ .
Do this: Write each worker's days worked in brackets.
Usually costs: Method marks.

Exam tips

Always say "work done in 1 day" in your first line.
Check by adding the fractions of work done by each person to get 1.
Give answers in days (or days and hours when the question asks).

Practice questions on Working Together and Partial Work

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

Q1 · 1 mark · MCQBoard level
B can do a job in 8 days. The part of the job B does in 3 days is:
(a) $\frac{1}{3}$ (b) $\frac{3}{8}$ (c) $\frac{5}{8}$ (d) $\frac{8}{3}$
Hint
One day is 1/8.
Answer
(b)
Q2 · 2 marks · Short answerBoard level
A can do a piece of work in 9 days and B in 18 days. Find the time they take together.
Hint
$\frac{1}{9} + \frac{1}{18}$ .
Answer
$\frac{2 + 1}{18} = \frac{1}{6}$ ; 6 days.
Q3 · 3 marks · Short answerHarder board level
A can do a job in 20 days and B in 30 days. They work together for 6 days and then A leaves. In how many more days does B finish the job?
Hint
Work done together in 6 days first.
Answer
Together rate $= \frac{1}{12}$ ; in 6 days $\frac{1}{2}$ done. Remaining $\frac{1}{2}$ at B's rate $\frac{1}{30}$ : $15$ days.
Q4 · 3 marks · Short answerHarder board level
A and B together can do a job in 8 days; B alone can do it in 24 days. How long does A alone take?
Hint
Subtract B's rate.
Answer
A's rate $= \frac{1}{8} - \frac{1}{24} = \frac{3 - 1}{24} = \frac{1}{12}$ ; A takes 12 days.
Q5 · 4 marks · Long answerHOTS / competency
A can do a job in 12 days and B in 18 days. They work on alternate days, A starting first. In how many days is the job finished?
Hint
Two-day cycle rate is 5/36.
Answer
In a 2-day cycle: $\frac{1}{12} + \frac{1}{18} = \frac{5}{36}$ . After 7 cycles (14 days): $\frac{35}{36}$ . Remaining $\frac{1}{36}$ on day 15 is A's turn: $\frac{1/36}{1/12} = \frac{1}{3}$ day. Total $14 \frac{1}{3}$ days.

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1.A does a job in 15 days. Work in 1 day?Show answer
1/15.
2.Time for A (10 days) and B (15 days) together?Show answer
6 days.
3.How do you find B alone from A and A+B?Show answer
1/b = 1/t - 1/a.
4.What is the whole job taken as?Show answer
1.

Working Together and Partial Work: frequently asked questions

Why is rate added but not time?

Because work done per day is additive when both work at the same time.

What if the workers have different efficiencies?

Their one-day work fractions already reflect that.