Time and Work ICSE Class 8: One Day's Work, Pipes and Solved Problems

What Time and Work is about

Weightage: CISCE publishes no question paper, marking scheme or chapter weightage for Class VIII Mathematics: your school sets and marks the exam. Time and Work sits in the Ratio and Proportion theme of the CISCE Curriculum for Upper Primary Classes (VI-VIII).Source: CISCE Curriculum for Upper Primary Classes (VI-VIII), RDCD 2016

Key concepts

One day's work

Take the whole job as 1.

• If A finishes it in $n$ days, A's one day's work is $\frac{1}{n}$.
• In $d$ days, A does $\frac{d}{n}$ of the job.
• If A does $\frac{1}{n}$ per day, A needs $n$ days for the whole job: time is the reciprocal of the daily rate.

The same idea works for hours, minutes or any unit of time.

Worked example

A can do a piece of work in 12 days. What part of the work does A do in 3 days?

Try it first, then show the solutionSolution

1. One day's work $=\frac{1}{12}$.
2. In 3 days: $3\times \frac{1}{12}=\frac{1}{4}$.

Answer: $\frac{1}{4}$ of the work

Working together

When people work together, their daily amounts add:
Together per day $=\frac{1}{a}+\frac{1}{b}$ (+ $\frac{1}{c}$ for a third person).

Time together $=1÷$ (together per day).

The time together is always less than the time of the faster person. Never add or average the days themselves.

Worked example

A can do a job in 12 days and B in 6 days. How long will they take together?

Try it first, then show the solutionSolution

1. Together per day $=\frac{1}{12}+\frac{1}{6}=\frac{1}{12}+\frac{2}{12}=\frac{3}{12}=\frac{1}{4}$.
2. Time $=1÷\frac{1}{4}=4$ days.

Answer: 4 days

Finding one person's time

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

Check: B's time must be more than $t$.

Worked example

A and B together can finish a job in 8 days. A alone takes 12 days. How long would B take alone?

Try it first, then show the solutionSolution

1. B per day $=\frac{1}{8}-\frac{1}{12}=\frac{3}{24}-\frac{2}{24}=\frac{1}{24}$.
2. B alone takes 24 days.

Answer: 24 days

Someone leaves (or joins) part-way

Split the job into stages:
1. Work done in stage 1 $=$ (days) × (combined daily work).
2. Work left $=1-$ work done.
3. Time for stage 2 $=$ work left ÷ daily work of those still working.

Add the stages if the question asks for the total time.

Worked example

A can do a job in 8 days and B in 24 days. They work together for 3 days, then A leaves. How long does B take to finish the job?

Try it first, then show the solutionSolution

1. Together per day $=\frac{1}{8}+\frac{1}{24}=\frac{3}{24}+\frac{1}{24}=\frac{1}{6}$; in 3 days: $\frac{3}{6}=\frac{1}{2}$ done.
2. Work left $=\frac{1}{2}$.
3. B takes $\frac{1}{2}÷\frac{1}{24}=12$ days.

Answer: 12 more days (15 days in all)

Pipes and cisterns

Filling a tank is a job too.

• An inlet pipe that fills a tank in $a$ hours adds $\frac{1}{a}$ of the tank per hour.
• An outlet (or leak) that empties a full tank in $b$ hours removes $\frac{1}{b}$ per hour: count it as negative.

Net filling per hour $=\frac{1}{a}-\frac{1}{b}$. If this is positive, the tank fills in $1÷$ (net) hours.

Worked example

A pipe fills a tank in 6 hours, and an outlet empties the full tank in 9 hours. If both are open, how long does the empty tank take to fill?

Try it first, then show the solutionSolution

1. Net per hour $=\frac{1}{6}-\frac{1}{9}=\frac{3}{18}-\frac{2}{18}=\frac{1}{18}$.
2. Time $=18$ hours.

Answer: 18 hours

Sharing wages

When people are paid jointly for a job, the money is shared in the ratio of the work each does.

If they work for the same number of days, that ratio is the ratio of their one day's work: $\frac{1}{a}:\frac{1}{b}$. Notice this is the inverse of the ratio of their times: a faster worker earns more.

Worked example

A can do a job in 10 days and B in 15 days. Working together they finish it and earn ₹2,500. How should they share it?

Try it first, then show the solutionSolution

1. Ratio of daily work $=\frac{1}{10}:\frac{1}{15}=3:2$.
2. A gets $\frac{3}{5}\times 2500=₹1500$; B gets $₹1000$.

Answer: A ₹1,500; B ₹1,000

Maths in real life: Time and Work

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.

Topics in this chapter

• Pipes and Cisterns and Sharing Wages
• Working Together and Partial Work

Time and Work formula sheet

One day's work

$\text{rate}=\frac{1}{n}$

n = days to finish alone.

Together

$\frac{1}{T}=\frac{1}{a}+\frac{1}{b}$

Add rates, not days.

Time from rate

$T=\frac{1}{\text{rate}}$

Reciprocal.

Work done

$W=\text{rate}\times \text{time}$

Whole job W = 1.

Pipes

$\text{net}=\frac{1}{a}-\frac{1}{b}$

Outlet or leak is negative.

How the school exam asks Time and Work

The question patterns to expect, and the method that works for each.

Tricks to remember Time and Work

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

• Add rates, not days

  Days never add when people work together; their daily fractions do.

  Use it when: Every 'together' problem.

• Flip at the end

  Rate → flip → time. Time → flip → rate.

  Use it when: Switching between days and daily work.

• Leaks are negative workers

  Treat an outlet pipe as a worker who undoes work.

  Use it when: Pipes and cisterns.

• Done + left = 1

  In part-way problems, always write work done and work left; they add to the whole job.

  Use it when: Someone joins or leaves.

Time and Work: exam-level solved questions

Ordered from exam level to HOTS. Cover the solution, write your own, then compare step by step.

Case-study practice (Section B style)

Read the situation, then answer every part. In a school exam set on the board pattern this is a 10-mark question split into 3 + 3 + 4 marks.

Where the marks are

How a typical answer is split into step marks. Show every step: a right final answer alone rarely earns full marks.

Where students lose marks in Time and Work

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

• ✗Mistake: Adding the days

  Working together makes the job quicker, not slower.

  Wrong working

  12 days and 6 days together: $12+6=18$ days

  Right working

  $\frac{1}{12}+\frac{1}{6}=\frac{1}{4}$, so 4 days

  ✓Do this: Add daily rates, then flip.

  Usually costs: The whole question.

• ✗Mistake: Averaging the days

  The average of 12 and 6 is 9, which is still longer than B alone.

  ✓Do this: The time together is always less than the faster person's time.

  Usually costs: The whole question.

• ✗Mistake: Forgetting to subtract a leak

  An outlet works against the inlet.

  Wrong working

  Net $=\frac{1}{10}+\frac{1}{15}$

  Right working

  Net $=\frac{1}{10}-\frac{1}{15}$

  ✓Do this: Mark outlets with a minus sign before adding.

  Usually costs: Most of the marks.

• ✗Mistake: Giving the rate as the answer

  $\frac{1}{6}$ is the work per day, not the number of days.

  ✓Do this: End with time = 1 ÷ rate.

  Usually costs: 1 mark.

• ✗Mistake: Sharing wages in the ratio of days

  The slower worker takes more days but does less work each day.

  Wrong working

  Times 10 and 15, so share 10 : 15

  Right working

  Rates $\frac{1}{10}:\frac{1}{15}=3:2$

  ✓Do this: Share by work done.

  Usually costs: 2 marks.

Practice questions on Time and Work

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

1. Q1 · 1 mark · MCQExam level

   A and B can each do a job alone in 8 days. Working together, they take:
   (a) 16 days
   (b) 8 days
   (c) 4 days
   (d) 2 days

   Hint

   $\frac{1}{8}+\frac{1}{8}$.

   Answer

   (c)

2. Q2 · 1 mark · Very short answerExam level

   A can do $\frac{1}{3}$ of a job in 5 days. How long will A take for the whole job?

   Hint

   Three times as long.

   Answer

   15 days.

3. Q3 · 2 marks · Short answerExam level

   A pipe can fill a tank in 3 hours. What part of the tank does it fill in 1 hour 30 minutes?

   Hint

   $1\frac{1}{2}\times \frac{1}{3}$.

   Answer

   $\frac{3}{2}\times \frac{1}{3}=\frac{1}{2}$ of the tank.

4. Q4 · 3 marks · Short answerHarder exam level

   A, B and C can do a job in 6, 12 and 4 days respectively. How long will they take together?

   Hint

   LCM 12.

   Answer

   $\frac{2}{12}+\frac{1}{12}+\frac{3}{12}=\frac{1}{2}$, so 2 days.

5. Q5 · 1 mark · Assertion-reasonHarder exam level

   Assertion (A): If A can do a job in 6 days and B in 3 days, they can finish it together in 2 days.
   Reason (R): Their combined one day's work is $\frac{1}{6}+\frac{1}{3}=\frac{1}{2}$.
   (a) (A) is true and (R) is false.
   (b) (A) is false and (R) is true.
   (c) Both (A) and (R) are true, and (R) is the correct explanation for (A).
   (d) Both (A) and (R) are true, but (R) is not the correct explanation for (A).

   Hint

   Flip the combined rate.

   Answer

   (c)

6. Q6 · 3 marks · Short answerHarder exam level

   A is twice as fast as B. Together they finish a job in 14 days. How long would each take alone?

   Hint

   Let B's daily work be $\frac{1}{x}$; A's is $\frac{2}{x}$.

   Answer

   $\frac{3}{x}=\frac{1}{14}$, so $x=42$. B takes 42 days and A takes 21 days.

7. Q7 · 3 marks · Short answerHarder exam level

   A can do a job in 18 days. A works for 6 days, and then B finishes the rest in 8 days. How long would B take to do the whole job alone?

   Hint

   A does $\frac{1}{3}$.

   Answer

   Work left $=\frac{2}{3}$, done by B in 8 days, so B does $\frac{1}{12}$ per day: 12 days alone.

8. Q8 · 3 marks · Short answerHarder exam level

   Two pipes can fill a tank in 20 minutes and 30 minutes. An outlet can empty it in 15 minutes. If all three are opened together on an empty tank, how long will it take to fill?

   Hint

   LCM 60.

   Answer

   $\frac{3}{60}+\frac{2}{60}-\frac{4}{60}=\frac{1}{60}$, so 60 minutes.

9. Q9 · 4 marks · Long answerHarder exam level

   A can do a job in 12 days and B in 18 days. They work together for 4 days. C then finishes the remaining work in 4 days. How long would C take to do the whole job alone?

   Hint

   Find the work left after 4 days.

   Answer

   Together per day $=\frac{5}{36}$; in 4 days $\frac{20}{36}=\frac{5}{9}$. Left $\frac{4}{9}$ in 4 days, so C does $\frac{1}{9}$ per day: 9 days alone.

10. Q10 · 2 marks · Short answerHarder exam level

    A can do a job in 6 days and B in 9 days. Working together they earn ₹1,500. Find each one's share.

    Hint

    Ratio of daily work.

    Answer

    $\frac{1}{6}:\frac{1}{9}=3:2$. A ₹900, B ₹600.

11. Q11 · 4 marks · Long answerHOTS / competency

    A can do a job in 10 days and B in 15 days. They start together, but A leaves 5 days before the job is finished. How long does the whole job take?

    Hint

    Let the total be T days; A works T − 5 days.

    Answer

    $\frac{T-5}{10}+\frac{T}{15}=1\Rightarrow 3(T-5)+2T=30\Rightarrow T=9$ days.

12. Q12 · 4 marks · Long answerHOTS / competency

    Two taps can fill a cistern in 12 minutes and 15 minutes. Both are opened; after 3 minutes the first tap is closed. How long does the second tap take to fill the rest?

    Hint

    Work done in 3 minutes together.

    Answer

    In 3 min: $3\left(\frac{1}{12}+\frac{1}{15}\right)=\frac{9}{20}$. Left $\frac{11}{20}$; time $=\frac{11}{20}\times 15=8\frac{1}{4}$ minutes (8 min 15 s).

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. 1.If A finishes a job in 8 days, what is A's one day's work?Show answer

   1/8.

2. 2.A does 1/5 of a job each day. How long for the whole job?Show answer

   5 days.

3. 3.A takes 6 days and B 3 days. How long together?Show answer

   2 days.

4. 4.How is a leak treated in pipe problems?Show answer

   Its rate is subtracted.

5. 5.In how many days does A do half a job if A does the whole job in 14 days?Show answer

   7 days.

6. 6.A and B work equal days and A is twice as fast. In what ratio are wages shared?Show answer

   2 : 1.

7. 7.What is the whole job taken as?Show answer

   1.

8. 8.Inlet fills in 4 h, outlet empties in 4 h. Does the tank fill?Show answer

   No: net rate is 0.

Quick revision checklist

The night before the exam: can you do each of these without looking?

• ☐Did you write each person's one day's work as a fraction?
• ☐Did you add rates rather than days?
• ☐Did you flip the combined rate to get the time?
• ☐Did you subtract outlet or leak rates?
• ☐In part-way problems, did you find the work left?
• ☐Did you share wages by work done?

Time and Work: frequently asked questions