Chapter 5 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

M3.4.5 Chapter 5

Direct proportion

When one amount doubles, does the other amount always double too?

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 two amounts grow at the same rate
  • Use a table of values to solve a proportion problem

Two amounts are in direct proportion when they grow in step. If one doubles, so does the other. If one is 3 times as big, so is the other.

A table of values lists pairs of amounts. Divide the second amount by the first in every row. If you always get the same number, the amounts are in direct proportion.

Draw it first

Draw rice and water as a graph

  1. Step 1: a graph with cups of rice across and cups of water up. Dots are at 1 and 2, 2 and 4, 3 and 6, and 4 and 8. Step 2: a straight line runs from the origin through every dot. Step 3: dashed guides show that 3 cups of rice go with 6 cups of water.0246801234cups of watercups of rice

    1 Rice needs twice as much water. Mark the pairs 1 and 2, 2 and 4, 3 and 6, 4 and 8.

  2. Step 1: a graph with cups of rice across and cups of water up. Dots are at 1 and 2, 2 and 4, 3 and 6, and 4 and 8. Step 2: a straight line runs from the origin through every dot. Step 3: dashed guides show that 3 cups of rice go with 6 cups of water.0246801234cups of watercups of rice

    2 Join the dots. They lie on a straight line. It starts at the origin, the point 0 and 0.

  3. Step 1: a graph with cups of rice across and cups of water up. Dots are at 1 and 2, 2 and 4, 3 and 6, and 4 and 8. Step 2: a straight line runs from the origin through every dot. Step 3: dashed guides show that 3 cups of rice go with 6 cups of water.0246801234cups of watercups of rice

    3 Read across from 3 cups of rice. The line gives 6 cups of water.

On a graph, direct proportion gives a straight line through the origin. The origin is the point where both amounts are 0. No rice needs no water, so the line must start there.

Worked example

Beans are weighed by the cup. 2 cups weigh 300 g, 3 cups weigh 450 g and 5 cups weigh 750 g. How much do 8 cups weigh?

  1. Divide each weight by its cups: 300 ÷ 2 = 150, 450 ÷ 3 = 150 and 750 ÷ 5 = 150.
  2. The answers are all equal, so the weight is in direct proportion to the cups. 1 cup weighs 150 g.
  3. 8 cups weigh 8 × 150 = 1 200 g.

Answer: 1 200 g

A table with two columns, cups and weight (g). The rows are 2 and 300, 3 and 450, 5 and 750.cupsweight (g)230034505750
Weight ÷ cups is 150 g in every row.

See it three ways

With things

Ravi cooks rice with twice as much water. 1 cup of rice goes with 2 cups of water. 2 cups of rice go with 4 cups of water.

As a picture

A table with two columns, rice (cups) and water (cups). The rows are 1 and 2, 2 and 4, 3 and 6, and 5 and 10.rice (cups)water (cups)122436510

In symbols

Water ÷ rice is 2 each time: 2 ÷ 1 = 2, 4 ÷ 2 = 2 and 6 ÷ 3 = 2. The ratio of rice to water stays 1 : 2.

Now look at a case that is not direct proportion. Ana plants a seedling that is 4 cm tall. It grows a steady 2 cm each week, but it started at 4 cm, not 0. So when the weeks double, the height does not.

A graph with weeks across and height (cm) up. Dots are at 0 and 4, 1 and 6, 2 and 8, and 4 and 12. The straight line starts at 4 on the height axis, not at the origin.02468101201234height (cm)weeks
The line starts at 4, not at 0. It does not go through the origin.

Worked example

Ana's seedling is 4 cm tall. It is 6 cm after 1 week. It is 8 cm after 2 weeks. It is 12 cm after 4 weeks. Is the height in direct proportion to the weeks?

  1. Week 1: 6 ÷ 1 = 6.
  2. Week 2: 8 ÷ 2 = 4.
  3. Week 4: 12 ÷ 4 = 3.
  4. The answers 6, 4 and 3 are not equal. So the height is not in direct proportion to the weeks.

Answer: No. The answers 6, 4 and 3 are not equal.

Try it

Test a square for direct proportion

You need: squared paper, a pencil.

  1. Draw 3 squares on squared paper. Their sides are 1 cm, 2 cm and 3 cm.
  2. For each square, count the squares along its edge all the way round. This is the distance round.
  3. For each square, count all the small squares inside it. This is the area.
  4. Divide the distance round by the side, then the area by the side.

What you will see: Distance round ÷ side is 4 every time, so the distance round is in direct proportion to the side. Area ÷ side is 1, then 2, then 3, so the area is not.

If you cannot try it, this is what happens.
A table. The top row says side (cm), 1, 2, 3. The next row says round (cm), 4, 8, 12. The last row says area (cm²), 1, 4, 9.side (cm)123round (cm)4812area (cm²)149

A wrong idea many people have

If one amount grows when the other grows, they are in direct proportion.

Why it feels right: Often more of one thing goes with more of another. So the idea seems to hold.

The test that breaks it: Look at the seedling. It grew from 8 cm to 12 cm while the weeks went from 2 to 4. The weeks doubled, but the height did not.

What is true: Direct proportion needs the second amount divided by the first to stay the same. The graph must pass through the origin.

Key words

direct proportion
two amounts that grow in step, so one divided by the other never changes.
table of values
a table that lists pairs of amounts, one pair in each row.
origin
the point on a graph where both amounts are 0.

Back to your guess

The answer is B: No. It only happens when the two amounts are in direct proportion.

Rice and water double together when the water is always twice the rice. That is direct proportion. A plant that starts 4 cm tall does not double its height when its weeks double. Always test with a table.

Check yourself

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

  1. A builder lays bricks at an even speed. Are the bricks laid in direct proportion to the hours?

    A table with two columns, hours and bricks. The rows are 1 and 30, 2 and 60, 3 and 90, 4 and 120.hoursbricks1302603904120
    Tap your answer.

    1. Yes, that is right. 30 ÷ 1 = 30, 60 ÷ 2 = 30, 90 ÷ 3 = 30 and 120 ÷ 4 = 30. The answers are all equal.

    2. Not this one. Growing together is not enough. The seedling grew too. Divide the bricks by the hours in each row. Is the answer the same? Go to M3.4.5. Try another.

    3. Not this one. The numbers do change, but look at the ratio. Divide each row: 30 ÷ 1, 60 ÷ 2, 90 ÷ 3. What do you get? Go to M3.4.5. Try another.

    See the answer

    Yes. Bricks ÷ hours is 30 every time.

    30 ÷ 1 = 30, 60 ÷ 2 = 30, 90 ÷ 3 = 30 and 120 ÷ 4 = 30. The answers are all equal.

    • If you chose B: Growing together is not enough. The seedling grew too. Divide the bricks by the hours in each row. Is the answer the same? Go to M3.4.5.
    • If you chose C: The numbers do change, but look at the ratio. Divide each row: 30 ÷ 1, 60 ÷ 2, 90 ÷ 3. What do you get? Go to M3.4.5.

  2. A tank has 4 litres in it at the start. A tap adds 1 litre each minute. The litres after 1, 2 and 4 minutes are 5, 6 and 8. Are the litres in direct proportion to the minutes?

    A table with two columns, minutes and litres. The rows are 1 and 5, 2 and 6, 4 and 8.minuteslitres152648
    Tap your answer.

    1. Not this one. Check by dividing: 5 ÷ 1 = 5, but 6 ÷ 2 = 3. They are not the same. The tank started with 4 litres, not 0. Go to M3.4.5. Try another.

    2. Yes, that is right. 5 ÷ 1 = 5, 6 ÷ 2 = 3 and 8 ÷ 4 = 2. The answers are not equal. A graph would start at 4 litres, not at the origin.

    3. Not this one. Going up by the same amount each time is not the same as direct proportion. Direct proportion needs litres ÷ minutes to stay the same. Go to M3.4.5. Try another.

    See the answer

    No. Litres ÷ minutes gives 5, 3 and 2.

    5 ÷ 1 = 5, 6 ÷ 2 = 3 and 8 ÷ 4 = 2. The answers are not equal. A graph would start at 4 litres, not at the origin.

    • If you chose A: Check by dividing: 5 ÷ 1 = 5, but 6 ÷ 2 = 3. They are not the same. The tank started with 4 litres, not 0. Go to M3.4.5.
    • If you chose C: Going up by the same amount each time is not the same as direct proportion. Direct proportion needs litres ÷ minutes to stay the same. Go to M3.4.5.

  3. Paint is poured at a steady rate. The table shows the litres poured out. How many litres pour out in 9 minutes?

    A table with two columns, minutes and litres. The rows are 2 and 14, 3 and 21, and 9 and ? (a question mark).minuteslitres2143219?
    Tap your answer.

    1. Not this one. You added 9 to 21. But the paint adds 7 litres every minute, so 6 more minutes add far more than 9 litres. Go to M3.4.5. Try another.

    2. Not this one. That is the litres each minute. The question asks about 9 minutes, so multiply 7 by 9. Go to M3.4.5. Try another.

    3. Yes, that is right. 14 ÷ 2 = 7 and 21 ÷ 3 = 7, so there are 7 litres a minute. 9 × 7 = 63 litres.

    See the answer

    63 litres

    14 ÷ 2 = 7 and 21 ÷ 3 = 7, so there are 7 litres a minute. 9 × 7 = 63 litres.

    • If you chose A: You added 9 to 21. But the paint adds 7 litres every minute, so 6 more minutes add far more than 9 litres. Go to M3.4.5.
    • If you chose B: That is the litres each minute. The question asks about 9 minutes, so multiply 7 by 9. Go to M3.4.5.

  4. Leila walks at an even pace. The graph shows how far she walks. How far has she walked after 2.5 hours?

    A graph with hours across and km up. Dots are at 1 and 4, 2 and 8, 3 and 12, and 4 and 16. A straight line runs from the origin through every dot.048121601234kmhours
    Tap your answer.

    1. Not this one. That is the distance at 2 hours. 2.5 hours is half an hour further on. Go to M3.4.5. Try another.

    2. Not this one. You added 1 km for the half hour. But Leila walks 4 km each hour, so half an hour is 2 km. Go to M3.4.5. Try another.

    3. Yes, that is right. The line gives 4 km for every hour. 2.5 hours is 2.5 × 4 = 10 km. It lies halfway between 8 km and 12 km.

    4. Not this one. That is the distance at 3 hours. After 2.5 hours she has not walked as far. Go to M3.4.5. Try another.

    See the answer

    10 km

    The line gives 4 km for every hour. 2.5 hours is 2.5 × 4 = 10 km. It lies halfway between 8 km and 12 km.

    • If you chose A: That is the distance at 2 hours. 2.5 hours is half an hour further on. Go to M3.4.5.
    • If you chose B: You added 1 km for the half hour. But Leila walks 4 km each hour, so half an hour is 2 km. Go to M3.4.5.
    • If you chose D: That is the distance at 3 hours. After 2.5 hours she has not walked as far. Go to M3.4.5.

  5. Which graph shows two amounts in direct proportion?

    Two graphs side by side. Graph A has hours across and km (graph A) up. Its straight line starts at the origin. Graph B has weeks across and cm (graph B) up. Its straight line starts at 2 on the up axis, above the origin.0246801234km (graph A)hours0246801234cm (graph B)weeks
    Tap your answer.

    1. Yes, that is right. Graph A is a straight line through the origin. Graph B is a straight line too. But it starts at 2, so its second amount divided by its first is not always the same.

    2. Not this one. Graph B is a straight line, but it starts at 2, above the origin. For direct proportion the line must start at 0 and 0. Go to M3.4.5. Try another.

    3. Not this one. A straight line is not enough. It must also start at the origin, where both amounts are 0. Go to M3.4.5. Try another.

    See the answer

    Graph A

    Graph A is a straight line through the origin. Graph B is a straight line too. But it starts at 2, so its second amount divided by its first is not always the same.

    • If you chose B: Graph B is a straight line, but it starts at 2, above the origin. For direct proportion the line must start at 0 and 0. Go to M3.4.5.
    • If you chose C: A straight line is not enough. It must also start at the origin, where both amounts are 0. Go to M3.4.5.

  6. Water pours at a steady rate. In 4 minutes 20 litres pour out, and in 6 minutes 30 litres. How many litres pour out in 9 minutes?

    See the answer

    45 litres

    20 ÷ 4 = 5 and 30 ÷ 6 = 5, so there are 5 litres a minute. 9 × 5 = 45 litres.

  7. Two amounts are in direct proportion. When the first is 4, the second is 12. What is the second when the first is 7?

    See the answer

    21

    12 ÷ 4 = 3, so the second is always 3 times the first. 7 × 3 = 21.

  8. From an earlier chapter

    A soup for 8 people uses 24 carrots. How many carrots does the same soup need for 5 people?

    See the answer

    15 carrots

    1 person needs 24 ÷ 8 = 3 carrots. 5 people need 5 × 3 = 15 carrots.

Keep it

Copy this into your exercise book.

In direct proportion, the second amount divided by the first is the same in every row. The graph is a straight line through the origin. Test every pair, not just one.

Draw: Draw a graph with axes from 0. Plot 1 and 2, 2 and 4, 3 and 6. Join them with a line through the origin.

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.