Chapter 8 of 8

All 8 chapters
  1. 1Ratio: comparing parts, readunderstood
  2. 2Ratios and fractions, readunderstood
  3. 3Sharing in a ratio, readunderstood
  4. 4Finding one first, readunderstood
  5. 5Direct proportion, readunderstood
  6. 6Rates and speed, readunderstood
  7. 7Maps and scale drawings, readunderstood
  8. 8Inverse proportion, readunderstood

M3.4.8 Chapter 8

Inverse proportion

3 people dig a ditch in 8 days. How long would 6 people take?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

  • You picked A. Read on. At the end, the chapter comes back to your guess.

  • You picked B. Read on. At the end, the chapter comes back to your guess.

  • You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

  • Spot when one amount goes down as the other goes up
  • Find how long 6 people take if 3 people take 8 days

Sometimes more of one amount means less of another. When more people share a job, it takes fewer days. If the two amounts always multiply to the same number, this is inverse proportion. Then when one amount doubles, the other halves.

Measure the work in person-days. One person working for one day does 1 person-day. 3 people working for 8 days do 3 × 8 = 24 person-days.

Draw it first

Draw the same job two ways

  1. Step 1: a rectangle 3 squares across, labelled 3 people, and 8 squares down, labelled 8 days. It has 24 squares. Step 2: a second rectangle 6 squares across, labelled 6 people, and 4 squares down, labelled 4 days. It also has 24 squares. Step 3: the label says 3 × 8 = 24 and 6 × 4 = 24.3 people8 days

    1 Each square is 1 person-day. 3 people for 8 days make a rectangle of 3 by 8 squares.

  2. Step 1: a rectangle 3 squares across, labelled 3 people, and 8 squares down, labelled 8 days. It has 24 squares. Step 2: a second rectangle 6 squares across, labelled 6 people, and 4 squares down, labelled 4 days. It also has 24 squares. Step 3: the label says 3 × 8 = 24 and 6 × 4 = 24.3 people8 days6 people4 days

    2 6 people for 4 days make a rectangle of 6 by 4 squares. It is a different shape.

  3. Step 1: a rectangle 3 squares across, labelled 3 people, and 8 squares down, labelled 8 days. It has 24 squares. Step 2: a second rectangle 6 squares across, labelled 6 people, and 4 squares down, labelled 4 days. It also has 24 squares. Step 3: the label says 3 × 8 = 24 and 6 × 4 = 24.3 people8 days6 people4 days3 × 8 = 24 and 6 × 4 = 24

    3 Count the squares in each. Both rectangles have 24 squares, so both are the same job.

Worked example

3 people dig a ditch in 8 days. How long would 6 people take?

  1. 3 people for 8 days is 3 × 8 = 24 person-days.
  2. Everyone works at the same speed, so the job is always 24 person-days.
  3. With 6 people, each person works 24 ÷ 6 = 4 days.

Answer: 4 days

The job stays the same size, so the person-days stay the same. In inverse proportion the two amounts multiply to the same number every time.

This works only when every person works at the same speed, and the job can be shared out.

Going down is not enough. A candle burns for more hours and gets shorter, but hours × height does not stay the same.

Worked example

3 painters paint a long wall in 10 days. How long would 5 painters take?

  1. 3 painters for 10 days is 3 × 10 = 30 person-days.
  2. Everyone works at the same speed, so the wall is always 30 person-days of work.
  3. With 5 painters, each works 30 ÷ 5 = 6 days.

Answer: 6 days

Two rectangles of squares. The left one is 3 squares across, labelled 3 people, and 10 squares down, labelled 10 days. The right one is 5 squares across, labelled 5 people, and 6 squares down, labelled 6 days. Each has 30 squares.3 people10 days5 people6 days
Each square is 1 person-day. Both rectangles have 30 squares. Here the people are painters.

See it three ways

With things

Take 24 stones. Each stone is 1 person-day. Make equal rows. Try 2 rows of 12, then 3 of 8, 4 of 6 and 6 of 4.

As a picture

A table with two columns, people and days. The rows are 1 and 24, 2 and 12, 3 and 8, 4 and 6, 6 and 4, 8 and 3.peopledays12421238466483

In symbols

people × days = 24 every time. 1 × 24 = 2 × 12 = 3 × 8 = 4 × 6 = 6 × 4 = 8 × 3 = 24.

Try it

Share 24 stones into equal groups

You need: 24 small stones, or beans.

  1. Make 2 equal groups. Count the stones in one group.
  2. Mix them up. Make 3 equal groups, then 4, then 6, and count each time.
  3. Write each pair in a table: the number of groups and the stones in each group.
  4. Multiply the two numbers in each row.

What you will see: More groups means fewer stones in each group. The two numbers multiply to 24 every time, so they are in inverse proportion.

If you cannot try it, this is what happens.
Four groups of 6 stones. There are 24 stones in all.

A wrong idea many people have

If 3 people take 8 days, then 6 people take 16 days, because 6 is double 3.

Why it feels right: When one amount doubles, the other doubles too in many problems, such as rice and water.

The test that breaks it: 6 people working for 16 days do 6 × 16 = 96 person-days. The job is only 24 person-days. That is four times too much work.

What is true: More people do the job faster. When the people double, the days halve.

Key words

inverse proportion
two amounts that always multiply to the same number, so one goes down as the other goes up.
person-days
the size of a job: the number of people times the number of days.

Back to your guess

The answer is B: 4 days, because twice as many people finish in half the time.

More people share the digging, so each one works fewer days. 3 people for 8 days is 24 person-days of work. 6 people share 24 person-days, so each works 24 ÷ 6 = 4 days.

Check yourself

Try each one before you look. A wrong answer tells you where to go next.

  1. A job takes 6 people 10 days. How many days would 4 people take?

    A table with two columns, people and days. The rows are 6 and 10, and 4 and ? (a question mark).peopledays6104?
    Tap your answer.

    1. Not this one. You took away 2 because 4 is 2 fewer than 6. But fewer people need more days, not fewer. Go to M3.4.8. Try another.

    2. Not this one. You multiplied 4 by 10. But 6 × 10 = 60 person-days is the whole job, so divide 60 by 4. Go to M3.4.8. Try another.

    3. Yes, that is right. 6 people for 10 days is 6 × 10 = 60 person-days. 4 people share 60 person-days: 60 ÷ 4 = 15 days.

    See the answer

    15 days

    6 people for 10 days is 6 × 10 = 60 person-days. 4 people share 60 person-days: 60 ÷ 4 = 15 days.

    • If you chose A: You took away 2 because 4 is 2 fewer than 6. But fewer people need more days, not fewer. Go to M3.4.8.
    • If you chose B: You multiplied 4 by 10. But 6 × 10 = 60 person-days is the whole job, so divide 60 by 4. Go to M3.4.8.

  2. A trip takes 3 hours at 60 km per hour. How long does the same trip take at 90 km per hour?

    A table with two columns, km per hour and hours. The rows are 60 and 3, and 90 and ? (a question mark).km per hourhours60390?
    Tap your answer.

    1. Not this one. A faster speed should take less time, not more. You scaled the time up by the same factor as the speed. Go to M3.4.8. Try another.

    2. Not this one. You halved the time, but 90 is not double 60. First find the distance: 60 × 3 = 180 km. Go to M3.4.8. Try another.

    3. Yes, that is right. The trip is 60 × 3 = 180 km. At 90 km per hour it takes 180 ÷ 90 = 2 hours.

    See the answer

    2 hours

    The trip is 60 × 3 = 180 km. At 90 km per hour it takes 180 ÷ 90 = 2 hours.

    • If you chose A: A faster speed should take less time, not more. You scaled the time up by the same factor as the speed. Go to M3.4.8.
    • If you chose B: You halved the time, but 90 is not double 60. First find the distance: 60 × 3 = 180 km. Go to M3.4.8.

  3. Which pair of amounts is in inverse proportion?

    Tap your answer.

    1. Not this one. If every notebook weighs the same, the total weight is the number of notebooks times one weight. That is direct proportion, not inverse. Go to M3.4.5. Try another.

    2. Yes, that is right. More people sharing the same cake means a smaller slice for each. People × size of each slice is the whole cake, which stays the same.

    3. Not this one. The candle gets shorter as the hours go up, but hours × height does not stay the same. Inverse proportion needs the two amounts to multiply to the same number. Go to M3.4.8. Try another.

    See the answer

    The number of people sharing a cake and the size of each slice

    More people sharing the same cake means a smaller slice for each. People × size of each slice is the whole cake, which stays the same.

    • If you chose A: If every notebook weighs the same, the total weight is the number of notebooks times one weight. That is direct proportion, not inverse. Go to M3.4.5.
    • If you chose C: The candle gets shorter as the hours go up, but hours × height does not stay the same. Inverse proportion needs the two amounts to multiply to the same number. Go to M3.4.8.

  4. The same job is done by different numbers of people. Find the missing number of people.

    A table with two columns, people and days. The rows are 1 and 24, 2 and 12, 4 and 6, ? and 3.peopledays12421246?3
    Tap your answer.

    1. Not this one. You counted on by 2 from 4. But the days go down 24, 12, 6, 3, so the people go up in a different way. Use people × days = 24. Go to M3.4.8. Try another.

    2. Yes, that is right. The job is 1 × 24 = 24 person-days. For 3 days you need 24 ÷ 3 = 8 people.

    3. Not this one. You copied a number from the days column. The people go with the days, so use people × days = 24. Go to M3.4.8. Try another.

    See the answer

    8 people

    The job is 1 × 24 = 24 person-days. For 3 days you need 24 ÷ 3 = 8 people.

    • If you chose A: You counted on by 2 from 4. But the days go down 24, 12, 6, 3, so the people go up in a different way. Use people × days = 24. Go to M3.4.8.
    • If you chose C: You copied a number from the days column. The people go with the days, so use people × days = 24. Go to M3.4.8.

  5. A job takes 2 people 12 days. How many days would 8 people take?

    A rectangle 2 squares across, labelled 2 people, and 12 squares down, labelled 12 days. The caption says one whole job.2 people12 daysone whole job
    Tap your answer.

    1. Not this one. You multiplied 12 by 4 because 8 is 4 times 2. More people finish sooner, so the days must go down, not up. Go to M3.4.8. Try another.

    2. Not this one. You took 6 away from 12 because 8 is 6 more than 2. But the days go down by dividing, not by taking away. Go to M3.4.8. Try another.

    3. Not this one. You multiplied 8 by 12. That is the work of 8 people for 12 days, not the days they need. Go to M3.4.8. Try another.

    4. Yes, that is right. 2 × 12 = 24 person-days. 8 people share it: 24 ÷ 8 = 3 days.

    See the answer

    3 days

    2 × 12 = 24 person-days. 8 people share it: 24 ÷ 8 = 3 days.

    • If you chose A: You multiplied 12 by 4 because 8 is 4 times 2. More people finish sooner, so the days must go down, not up. Go to M3.4.8.
    • If you chose B: You took 6 away from 12 because 8 is 6 more than 2. But the days go down by dividing, not by taking away. Go to M3.4.8.
    • If you chose C: You multiplied 8 by 12. That is the work of 8 people for 12 days, not the days they need. Go to M3.4.8.

  6. 4 people build a wall in 9 days. How many days would 6 people take?

    See the answer

    6 days

    4 × 9 = 36 person-days. 6 people share it: 36 ÷ 6 = 6 days.

  7. A tank is emptied by 5 pumps in 6 hours. How many hours would 10 pumps take?

    See the answer

    3 hours

    5 × 6 = 30 pump-hours. 10 pumps share it: 30 ÷ 10 = 3 hours.

  8. From an earlier chapter

    A map has the scale 1 : 50 000. Two towns are 8 cm apart on the map. How many km apart are they?

    See the answer

    4 km

    8 × 50 000 = 400 000 cm. That is 4 000 m, which is 4 km.

Keep it

Copy this into your exercise book.

In inverse proportion one amount goes down as the other goes up. Their product stays the same. Find the job in person-days, then divide by the new number of people. Everyone must work at the same speed.

Draw: Draw a rectangle 3 squares across and 8 down. Draw another 6 across and 4 down. Write 24 in each.

Reading with a child? A short page for the adult: what this chapter teaches and three questions to ask.

No internet tomorrow? Save the whole book as one file (710 KB). It opens on any phone.