Chapter 1 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

M3.4.1 Chapter 1

Ratio: comparing parts

Ravi says there are 3 goats for every 2 sheep. I count 6 goats and 4 sheep. Who is right?

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

  • Write the ratio of 6 goats to 4 sheep as 6 : 4
  • Simplify a ratio like 6 : 4 to 3 : 2

A ratio compares two amounts of the same kind. Kofi has 6 goats and 4 sheep. The ratio of goats to sheep is 6 : 4. We read it as 6 to 4.

The order matters. The ratio of sheep to goats is 4 : 6. It is not the same as 6 : 4.

Draw it first

Draw the flock as dots

  1. Step 1: a row of 6 filled dots, labelled goats. Step 2: a row of 4 empty dots, labelled sheep. Step 3: the rows are replaced by two groups of dots. Each group has 3 filled dots over 2 empty dots, labelled each group is 3 : 2.goats

    1 Kofi has 6 goats. Draw a filled dot for each goat.

  2. Step 1: a row of 6 filled dots, labelled goats. Step 2: a row of 4 empty dots, labelled sheep. Step 3: the rows are replaced by two groups of dots. Each group has 3 filled dots over 2 empty dots, labelled each group is 3 : 2.goatssheep

    2 Kofi has 4 sheep. Draw an empty dot for each sheep.

  3. Step 1: a row of 6 filled dots, labelled goats. Step 2: a row of 4 empty dots, labelled sheep. Step 3: the rows are replaced by two groups of dots. Each group has 3 filled dots over 2 empty dots, labelled each group is 3 : 2.each group is 3 : 2

    3 Put the dots in 2 equal groups. Each group has 3 goats and 2 sheep.

Each group has 3 goats and 2 sheep. So for every 3 goats there are 2 sheep. The ratio 3 : 2 says the same as 6 : 4. We call them equal ratios.

To simplify a ratio, divide both numbers by their highest common factor. You met this in the chapter on prime factors. A ratio is in its simplest form when only 1 goes into both numbers.

Worked example

Write 12 : 18 in simplest form.

  1. The highest common factor of 12 and 18 is 6.
  2. 12 ÷ 6 = 2.
  3. 18 ÷ 6 = 3.
  4. In simplest form the ratio is 2 : 3.

Answer: 2 : 3

Both amounts must be in the same unit before you simplify. You cannot compare metres with centimetres. Change one so that both match.

Worked example

Write the ratio of 2 m to 50 cm in simplest form.

  1. Make the units match. 2 m = 200 cm.
  2. The ratio is 200 : 50.
  3. The highest common factor of 200 and 50 is 50.
  4. 200 ÷ 50 = 4 and 50 ÷ 50 = 1.
  5. In simplest form the ratio is 4 : 1.

Answer: 4 : 1

Two bars. The top bar is the rope, labelled rope: 2 m = 200 cm, and is cut into 4 equal parts of 50 cm. The bottom bar is the stick, labelled stick: 50 cm, and is one part.rope: 2 m = 200 cmstick: 50 cm
One part is 50 cm. The rope is 4 parts and the stick is 1 part, so the ratio is 4 : 1.

See it three ways

With things

Take 3 stones and 2 beans. Make the same mix again and again: 6 stones with 4 beans, then 9 stones with 6 beans.

As a picture

A table with two columns, stones and beans. The rows are 3 and 2, 6 and 4, 9 and 6, 12 and 8.stonesbeans326496128

In symbols

3 : 2 = 6 : 4 = 9 : 6 = 12 : 8. Each row is the first row times 1, 2, 3 and 4.

Try it

Make equal ratios with stones and beans

You need: 6 small stones, 4 beans or seeds.

  1. Put 6 stones in one row and 4 beans in another row.
  2. Split them into piles. Each pile must have the same number of stones and the same number of beans.
  3. Find the largest number of piles you can make with nothing left over.
  4. Count the stones and beans in one pile.

What you will see: You can make 2 piles. Each pile has 3 stones and 2 beans. The number of piles, 2, is the highest common factor of 6 and 4.

If you cannot try it, this is what happens.
Two piles. Each pile has 3 filled dots over 2 empty dots. The key says filled: stone, empty: bean.filled: stone, empty: bean

A wrong idea many people have

To make an equal ratio, add the same number to both sides. So 2 : 3 is the same as 3 : 4.

Why it feels right: Doing the same thing to both numbers sounds fair. Adding 1 to each looks like it keeps them in step.

The test that breaks it: Take 2 stones and 3 beans. Add one of each to get 3 stones and 4 beans. Now try to split both heaps into equal piles that look like 2 stones with 3 beans. You cannot. Double the first heap instead: 4 stones and 6 beans make 2 piles of 2 and 3.

What is true: Equal ratios come from multiplying or dividing both numbers by the same number. Adding the same number changes the mix.

Key words

ratio
a way to compare two amounts of the same kind, written like 6 : 4.
equal ratios
ratios that describe the same mix, such as 6 : 4 and 3 : 2.
simplest form
a ratio where only 1 goes into both numbers.

Back to your guess

The answer is B: Both are right. 6 : 4 and 3 : 2 describe the same flock.

Both are right. Divide 6 and 4 by 2 and you get 3 and 2. The flock has the same mix either way. The ratios 6 : 4 and 3 : 2 are equal.

Check yourself

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

  1. Each filled dot is a mango. Each empty dot is a lime. What is the ratio of mangoes to limes?

    A row of 8 dots. 5 are filled, for mangoes, and 3 are empty, for limes. The key says filled: mango, empty: lime.filled: mango, empty: lime
    Tap your answer.

    1. Not this one. The order is flipped. The question asks for mangoes to limes, so the mangoes come first. Go to M3.4.1. Try another.

    2. Yes, that is right. Count the filled dots: 5. Count the empty dots: 3. Mangoes come first, so the ratio is 5 : 3.

    3. Not this one. You compared the mangoes with all the fruit. A ratio compares one part with another part, so count the limes alone. Go to M3.4.1. Try another.

    See the answer

    5 : 3

    Count the filled dots: 5. Count the empty dots: 3. Mangoes come first, so the ratio is 5 : 3.

    • If you chose A: The order is flipped. The question asks for mangoes to limes, so the mangoes come first. Go to M3.4.1.
    • If you chose C: You compared the mangoes with all the fruit. A ratio compares one part with another part, so count the limes alone. Go to M3.4.1.

  2. Which is 15 : 25 in simplest form?

    Tap your answer.

    1. Yes, that is right. The highest common factor of 15 and 25 is 5. 15 ÷ 5 = 3 and 25 ÷ 5 = 5.

    2. Not this one. The numbers are right but the order is flipped. A ratio keeps its order when you simplify. Go to M3.4.1. Try another.

    3. Not this one. You divided only the first number. Divide both numbers by the same number. Go to M3.4.1. Try another.

    See the answer

    3 : 5

    The highest common factor of 15 and 25 is 5. 15 ÷ 5 = 3 and 25 ÷ 5 = 5.

    • If you chose B: The numbers are right but the order is flipped. A ratio keeps its order when you simplify. Go to M3.4.1.
    • If you chose C: You divided only the first number. Divide both numbers by the same number. Go to M3.4.1.

  3. Which ratio is equal to 4 : 6?

    Tap your answer.

    1. Not this one. You added 2 to each number. Adding does not keep the mix the same. Divide or multiply both numbers instead. Go to M3.4.1. Try another.

    2. Yes, that is right. Divide both numbers by 2: 4 ÷ 2 = 2 and 6 ÷ 2 = 3. So 4 : 6 = 2 : 3.

    3. Not this one. Only the second number changed, so the mix changed. Both numbers must change by the same factor. Go to M3.4.1. Try another.

    See the answer

    2 : 3

    Divide both numbers by 2: 4 ÷ 2 = 2 and 6 ÷ 2 = 3. So 4 : 6 = 2 : 3.

    • If you chose A: You added 2 to each number. Adding does not keep the mix the same. Divide or multiply both numbers instead. Go to M3.4.1.
    • If you chose C: Only the second number changed, so the mix changed. Both numbers must change by the same factor. Go to M3.4.1.

  4. A rope is 3 m long. A stick is 60 cm long. What is the ratio of rope to stick in simplest form?

    Two bars. The top bar is the rope, labelled rope: 3 m = 300 cm, and is cut into 5 equal parts of 60 cm. The bottom bar is the stick, labelled stick: 60 cm, and is one part.rope: 3 m = 300 cmstick: 60 cm
    Tap your answer.

    1. Not this one. The units do not match. 3 is in metres and 60 is in centimetres. Change 3 m to 300 cm first. Go to M3.4.1. Try another.

    2. Not this one. That is 3 : 60 simplified, but 3 m and 60 cm are not the same unit. Make the units match before you simplify. Go to M3.4.1. Try another.

    3. Yes, that is right. Make the units match: 3 m = 300 cm. The ratio is 300 : 60. The highest common factor is 60. 300 ÷ 60 = 5 and 60 ÷ 60 = 1.

    See the answer

    5 : 1

    Make the units match: 3 m = 300 cm. The ratio is 300 : 60. The highest common factor is 60. 300 ÷ 60 = 5 and 60 ÷ 60 = 1.

    • If you chose A: The units do not match. 3 is in metres and 60 is in centimetres. Change 3 m to 300 cm first. Go to M3.4.1.
    • If you chose B: That is 3 : 60 simplified, but 3 m and 60 cm are not the same unit. Make the units match before you simplify. Go to M3.4.1.

  5. A bag holds 12 red beads and 8 blue beads. Which ratio of red to blue is in simplest form?

    Two rows of 10 dots. 12 are filled, for red beads, and 8 are empty, for blue beads. The key says filled: red, empty: blue.filled: red, empty: blue
    Tap your answer.

    1. Not this one. This is a correct ratio, but it is not in simplest form. Both numbers can still be divided by 4. Go to M3.4.1. Try another.

    2. Not this one. This is equal to 12 : 8, but both numbers can still be divided by 2. Keep going until only 1 goes into both. Go to M3.4.1. Try another.

    3. Yes, that is right. The highest common factor of 12 and 8 is 4. 12 ÷ 4 = 3 and 8 ÷ 4 = 2. Only 1 goes into both 3 and 2.

    See the answer

    3 : 2

    The highest common factor of 12 and 8 is 4. 12 ÷ 4 = 3 and 8 ÷ 4 = 2. Only 1 goes into both 3 and 2.

    • If you chose A: This is a correct ratio, but it is not in simplest form. Both numbers can still be divided by 4. Go to M3.4.1.
    • If you chose B: This is equal to 12 : 8, but both numbers can still be divided by 2. Keep going until only 1 goes into both. Go to M3.4.1.

  6. Write 18 : 24 in simplest form.

    See the answer

    3 : 4

    The highest common factor of 18 and 24 is 6. 18 ÷ 6 = 3 and 24 ÷ 6 = 4.

  7. A rope is 4 m long. A stick is 50 cm long. Write the ratio of rope to stick in simplest form.

    See the answer

    8 : 1

    Make the units match: 4 m = 400 cm. The ratio is 400 : 50. The highest common factor is 50. 400 ÷ 50 = 8 and 50 ÷ 50 = 1.

Keep it

Copy this into your exercise book.

A ratio compares two amounts in order. 6 : 4 is equal to 3 : 2. Divide both numbers by their highest common factor to simplify. Make the units match first.

Draw: Draw 6 filled dots and 4 empty dots. Put them in 2 groups, each with 3 filled and 2 empty. Write 6 : 4 = 3 : 2.

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.