Chapter 1 of 8

All 8 chapters
  1. 1Equal fractions and simplest form, readunderstood
  2. 2Adding and taking away fractions, readunderstood
  3. 3Multiplying and dividing fractions, readunderstood
  4. 4Tenths, hundredths and thousandths, readunderstood
  5. 5Adding and taking away decimals, readunderstood
  6. 6Multiplying and dividing decimals, readunderstood
  7. 7Fractions and decimals, readunderstood
  8. 8Percentages, readunderstood

M3.3.1 Chapter 1

Equal fractions and simplest form

My answer is 12/18 and the book says 2/3. 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

  • Simplify 12/18 by dividing the top and bottom by 6
  • Change 7/3 into 2 1/3 and back again

Two fractions can look different and still name the same amount. 1/2 and 2/4 are like that. We call them equal fractions.

The top number of a fraction is the numerator. It counts the parts you have. The bottom number is the denominator. It says how many equal parts make one whole.

To make an equal fraction, multiply the top and the bottom by the same number. Every part is cut into smaller parts. The shaded amount does not change.

Draw it first

Cut every part into smaller parts

  1. Three bars of the same length, stacked. The top bar is cut into 3 parts with 2 shaded, labelled 2/3. The middle bar is cut into 6 parts with 4 shaded, labelled 4/6. The bottom bar is cut into 12 parts with 8 shaded, labelled 8/12. Every shaded part ends at the same place.23

    1 Draw a bar in 3 equal parts. Shade 2 parts. That is 2/3.

  2. Three bars of the same length, stacked. The top bar is cut into 3 parts with 2 shaded, labelled 2/3. The middle bar is cut into 6 parts with 4 shaded, labelled 4/6. The bottom bar is cut into 12 parts with 8 shaded, labelled 8/12. Every shaded part ends at the same place.2346

    2 Cut every part in 2. Now there are 6 parts and 4 are shaded. That is 4/6.

  3. Three bars of the same length, stacked. The top bar is cut into 3 parts with 2 shaded, labelled 2/3. The middle bar is cut into 6 parts with 4 shaded, labelled 4/6. The bottom bar is cut into 12 parts with 8 shaded, labelled 8/12. Every shaded part ends at the same place.2346812

    3 Cut every part of the first bar in 4 instead. There are 12 parts and 8 are shaded. That is 8/12.

The bars have the same length, and the shaded parts end in the same place. So 2/3 = 4/6 = 8/12. Each time the top and the bottom were multiplied by the same number.

Worked example

Write 3/5 as a fraction with 20 on the bottom.

  1. The bottom changes from 5 to 20. 20 ÷ 5 = 4, so multiply by 4.
  2. Multiply the top by 4 too. 3 × 4 = 12.
  3. The new bottom is 5 × 4 = 20. So 3/5 = 12/20.

Answer: 3/5 = 12/20

You can go the other way too. Divide the top and the bottom by the same number. This works only when both numbers are in that number's times table.

A fraction is in simplest form when only 1 divides both its top and its bottom. To reach it, divide both by their highest common factor (HCF).

Worked example

Write 12/18 in simplest form.

  1. Find the highest common factor of 12 and 18. They share 2 and 3, so the HCF is 2 × 3 = 6.
  2. Divide the top by 6. 12 ÷ 6 = 2.
  3. Divide the bottom by 6. 18 ÷ 6 = 3.
  4. So 12/18 = 2/3. Only 1 divides both 2 and 3, so this is the simplest form.

Answer: 12/18 = 2/3

Three bars of the same length, stacked. The top bar is cut into 18 parts with 12 shaded, labelled 12/18. The middle bar is cut into 9 parts with 6 shaded, labelled 6/9. The bottom bar is cut into 3 parts with 2 shaded, labelled 2/3. The shaded parts end at the same place.12186923
12/18 = 6/9 = 2/3. You can divide in small steps. Dividing by the HCF does it in one step.

See it three ways

With things

Hana has 18 beans. 12 of them are red. She puts the beans in groups of 6. There are 3 groups, and 2 of them are red.

As a picture

Eighteen beans in 3 groups of 6. Two groups are filled in and labelled red, red. They stand for the 12 red beans. The third group is labelled not red.redrednot red

In symbols

12/18 = 2/3. Divide the top and the bottom by 6.

A fraction with a top bigger than its bottom is an improper fraction. 7/3 is one. It is more than 1 whole. A mixed number has whole numbers and a fraction, like 2 1/3. Both write the same amount.

Draw it first

Change 7/3 into a mixed number

  1. Three bars, each cut into 3 equal parts. The top bar is all shaded and labelled 1 whole. The middle bar is all shaded and labelled 1 whole. The bottom bar has 1 part shaded and is labelled 1/3. Together the shaded parts are 7 thirds.1 whole

    1 Shade a whole bar. It is cut in 3 equal parts, so it holds 3 thirds.

  2. Three bars, each cut into 3 equal parts. The top bar is all shaded and labelled 1 whole. The middle bar is all shaded and labelled 1 whole. The bottom bar has 1 part shaded and is labelled 1/3. Together the shaded parts are 7 thirds.1 whole1 whole

    2 Shade a second whole bar. That is 3 more thirds, 6 thirds in all.

  3. Three bars, each cut into 3 equal parts. The top bar is all shaded and labelled 1 whole. The middle bar is all shaded and labelled 1 whole. The bottom bar has 1 part shaded and is labelled 1/3. Together the shaded parts are 7 thirds.1 whole1 whole13

    3 Shade 1 part of a third bar. That makes 7 thirds: 2 wholes and 1/3.

Worked example

Change 17/5 into a mixed number.

  1. Count how many fives fit in 17. 5 × 3 = 15, so 3 whole bars are full.
  2. Take away what the full bars use. 17 − 15 = 2.
  3. The 2 left over are fifths. They make the fraction 2/5.
  4. So 17/5 = 3 2/5.

Answer: 3 2/5

To go back, multiply the whole number by the bottom number. Then add the top number. For 3 2/5: 3 × 5 = 15, and 15 + 2 = 17. The bottom stays 5. So 3 2/5 = 17/5.

Try it

Two strips, one amount

You need: two strips of paper of the same length, a pencil.

  1. Fold the first strip into 3 equal parts. Open it and shade 2 parts.
  2. Fold the second strip into 3 parts, then fold each part in half. Open it. It has 6 parts.
  3. Shade 4 parts of the second strip.
  4. Put the strips side by side, with the left ends together.

What you will see: Both shaded parts end at the same place. So 2/3 and 4/6 are the same amount. You multiplied the top and the bottom by 2.

If you cannot try it, this is what happens.
Two strips of the same length side by side. The top strip has 3 parts with 2 shaded, labelled two thirds. The bottom strip has 6 parts with 4 shaded, labelled four sixths. The shaded parts have the same length.two thirdsfour sixths

A wrong idea many people have

To make an equal fraction, add the same number to the top and the bottom.

Why it feels right: Doing the same thing to both numbers looks fair.

The test that breaks it: Add 1 to the top and the bottom of 1/2. You get 2/3. Shade a bar for each. 1/2 is 1 part of 2. 2/3 is 2 parts of 3. The shaded parts do not end in the same place.

What is true: Multiply or divide the top and the bottom by the same number. Adding changes the amount.

Key words

equal fractions
fractions that name the same amount, like 1/2 and 2/4.
numerator
the top number of a fraction. It counts the parts you have.
denominator
the bottom number of a fraction. It says how many equal parts make one whole.
simplest form
a fraction whose top and bottom have no common factor except 1.
improper fraction
a fraction whose top is bigger than its bottom, like 7/3.
mixed number
a whole number and a fraction together, like 2 1/3.

Back to your guess

The answer is B: We are both right. The two fractions name the same amount.

12/18 and 2/3 are the same amount. The book gives 2/3 because it is in simplest form. The top and the bottom are as small as they can be.

Check yourself

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

  1. Which of these is NOT equal to 2/3?

    Tap your answer.

    1. Yes, that is right. To get from 3 to 10 you cannot multiply by a whole number. 8/10 = 4/5, which is not 2/3. The other two come from multiplying 2/3 by 2 and by 3.

    2. Not this one. Multiply the top and the bottom of 2/3 by 2. You get 4/6, so it is equal to 2/3. Go to M3.3.1. Try another.

    3. Not this one. Multiply the top and the bottom of 2/3 by 3. You get 6/9, so it is equal to 2/3. Go to M3.3.1. Try another.

    See the answer

    8/10

    To get from 3 to 10 you cannot multiply by a whole number. 8/10 = 4/5, which is not 2/3. The other two come from multiplying 2/3 by 2 and by 3.

    • If you chose B: Multiply the top and the bottom of 2/3 by 2. You get 4/6, so it is equal to 2/3. Go to M3.3.1.
    • If you chose C: Multiply the top and the bottom of 2/3 by 3. You get 6/9, so it is equal to 2/3. Go to M3.3.1.

  2. What fraction of this bar is shaded? Choose the answer in simplest form.

    A bar cut into 8 equal parts with 6 shaded. The caption says 8 equal parts.8 equal parts
    Tap your answer.

    1. Not this one. That is the part that is not shaded. 2 parts of 8 are not shaded, and 2/8 = 1/4. Count the shaded parts. Go to M3.3.1. Try another.

    2. Not this one. That is the right amount, but it is not in simplest form. 6 and 8 are both in the 2 times table, so divide both by 2. Go to M3.3.1. Try another.

    3. Yes, that is right. 6 of 8 parts are shaded, so the fraction is 6/8. The HCF of 6 and 8 is 2. 6 ÷ 2 = 3 and 8 ÷ 2 = 4.

    4. Not this one. You divided the top by 2 but not the bottom. To keep the amount, divide both numbers by the same number. Go to M3.3.1. Try another.

    See the answer

    3/4

    6 of 8 parts are shaded, so the fraction is 6/8. The HCF of 6 and 8 is 2. 6 ÷ 2 = 3 and 8 ÷ 2 = 4.

    • If you chose A: That is the part that is not shaded. 2 parts of 8 are not shaded, and 2/8 = 1/4. Count the shaded parts. Go to M3.3.1.
    • If you chose B: That is the right amount, but it is not in simplest form. 6 and 8 are both in the 2 times table, so divide both by 2. Go to M3.3.1.
    • If you chose D: You divided the top by 2 but not the bottom. To keep the amount, divide both numbers by the same number. Go to M3.3.1.

  3. Which mixed number is the same as 11/4?

    Tap your answer.

    1. Not this one. 3 whole bars need 3 × 4 = 12 quarters, and there are only 11. 2 wholes use 8 quarters, and 3 are left. Go to M3.3.1. Try another.

    2. Yes, that is right. 4 × 2 = 8, and 11 − 8 = 3. So there are 2 wholes and 3 quarters.

    3. Not this one. 2 wholes use 8 quarters. 11 − 8 = 3 quarters are left over, not 1. Go to M3.3.1. Try another.

    See the answer

    2 3/4

    4 × 2 = 8, and 11 − 8 = 3. So there are 2 wholes and 3 quarters.

    • If you chose A: 3 whole bars need 3 × 4 = 12 quarters, and there are only 11. 2 wholes use 8 quarters, and 3 are left. Go to M3.3.1.
    • If you chose C: 2 wholes use 8 quarters. 11 − 8 = 3 quarters are left over, not 1. Go to M3.3.1.

  4. Which improper fraction is the same as 2 3/5?

    Tap your answer.

    1. Yes, that is right. 2 × 5 = 10, and 10 + 3 = 13. The bottom stays 5.

    2. Not this one. That puts the 2 and the 3 side by side. 2 wholes are 2 × 5 = 10 fifths. Then add the 3 fifths. Go to M3.3.1. Try another.

    3. Not this one. You multiplied 2 × 3. Multiply the whole number by the bottom number, 5. Then add the top number, 3. Go to M3.3.1. Try another.

    See the answer

    13/5

    2 × 5 = 10, and 10 + 3 = 13. The bottom stays 5.

    • If you chose B: That puts the 2 and the 3 side by side. 2 wholes are 2 × 5 = 10 fifths. Then add the 3 fifths. Go to M3.3.1.
    • If you chose C: You multiplied 2 × 3. Multiply the whole number by the bottom number, 5. Then add the top number, 3. Go to M3.3.1.

  5. How do you change 2/3 into twelfths?

    Tap your answer.

    1. Not this one. That gives 2/12, which is 1/6. That is a much smaller amount. The top must change too. Go to M3.3.1. Try another.

    2. Not this one. Adding 9 gives 11/12, which is not 2/3. Multiply both numbers instead. Go to M3.3.1. Try another.

    3. Yes, that is right. 12 ÷ 3 = 4, so multiply by 4. 2 × 4 = 8 and 3 × 4 = 12. So 2/3 = 8/12.

    See the answer

    Multiply the top and the bottom by 4

    12 ÷ 3 = 4, so multiply by 4. 2 × 4 = 8 and 3 × 4 = 12. So 2/3 = 8/12.

    • If you chose A: That gives 2/12, which is 1/6. That is a much smaller amount. The top must change too. Go to M3.3.1.
    • If you chose B: Adding 9 gives 11/12, which is not 2/3. Multiply both numbers instead. Go to M3.3.1.

  6. Write a fraction equal to 3/4 with 12 on the bottom.

    See the answer

    9/12

    12 ÷ 4 = 3, so multiply the top and the bottom by 3. 3 × 3 = 9 and 4 × 3 = 12.

  7. Write 18/24 in simplest form.

    See the answer

    3/4

    The HCF of 18 and 24 is 6. 18 ÷ 6 = 3 and 24 ÷ 6 = 4. Only 1 divides both 3 and 4.

  8. Change 23/6 into a mixed number.

    See the answer

    3 5/6

    6 × 3 = 18, and 23 − 18 = 5. So there are 3 wholes and 5 sixths left over.

Keep it

Copy this into your exercise book.

Equal fractions name the same amount. Multiply or divide the top and the bottom by the same number. Simplest form: divide by the HCF.

Draw: Two bars of the same length. Cut one in 3 and shade 2. Cut the other in 6 and shade 4. Write 2/3 = 4/6.

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 (828 KB). It opens on any phone.