Chapter 2 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

M3.3.2 Chapter 2

Adding and taking away fractions

How can I add a third and a quarter when the parts are different sizes?

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

  • Add 1/3 + 1/4 by finding a common bottom number
  • Take away mixed numbers, like 2 1/2 − 1 3/4

Fractions with the same bottom number are quick to add. 2/7 + 3/7 = 5/7. The parts are the same size, so you add the tops and keep the bottom.

Parts of different sizes cannot be counted together, like 3 stones and 4 beans. First change both fractions so they have the same bottom number. That bottom number is a common denominator.

The best common denominator is the lowest common multiple (LCM) of the two bottom numbers. The LCM of 3 and 4 is 12. Both 3 and 4 go into 12 exactly.

Draw it first

Add 1/3 and 1/4

  1. Bars of the same length. In step 1 the top bar is cut into 3 parts with 1 shaded, labelled 1/3, and the second bar is cut into 4 parts with 1 shaded, labelled 1/4. In step 2 both bars are cut into 12 parts. The top bar has 4 shaded, labelled 4/12. The second bar has 3 shaded, labelled 3/12. In step 3 a third bar is cut into 12 parts with 7 shaded, labelled 7/12.1314

    1 Draw two bars of the same length. Cut one in 3 and shade 1 part. Cut the other in 4 and shade 1 part. The parts are not the same size.

  2. Bars of the same length. In step 1 the top bar is cut into 3 parts with 1 shaded, labelled 1/3, and the second bar is cut into 4 parts with 1 shaded, labelled 1/4. In step 2 both bars are cut into 12 parts. The top bar has 4 shaded, labelled 4/12. The second bar has 3 shaded, labelled 3/12. In step 3 a third bar is cut into 12 parts with 7 shaded, labelled 7/12.412312

    2 Cut both bars into 12 parts. Now 1/3 is 4/12 and 1/4 is 3/12. The parts are the same size.

  3. Bars of the same length. In step 1 the top bar is cut into 3 parts with 1 shaded, labelled 1/3, and the second bar is cut into 4 parts with 1 shaded, labelled 1/4. In step 2 both bars are cut into 12 parts. The top bar has 4 shaded, labelled 4/12. The second bar has 3 shaded, labelled 3/12. In step 3 a third bar is cut into 12 parts with 7 shaded, labelled 7/12.412312712

    3 Count the shaded twelfths together. 4 + 3 = 7. So 1/3 + 1/4 = 7/12.

Worked example

Add 1/3 + 1/4.

  1. The bottom numbers are 3 and 4. The lowest common multiple of 3 and 4 is 12.
  2. Change 1/3. 12 ÷ 3 = 4, so 1/3 = 4/12.
  3. Change 1/4. 12 ÷ 4 = 3, so 1/4 = 3/12.
  4. The bottom stays 12. Add the tops: 4 + 3 = 7.
  5. So 1/3 + 1/4 = 7/12.

Answer: 7/12

Worked example

Add 3/4 + 5/6.

  1. The lowest common multiple of 4 and 6 is 12.
  2. Change 3/4. 3 × 3 = 9 and 4 × 3 = 12, so 3/4 = 9/12.
  3. Change 5/6. 5 × 2 = 10 and 6 × 2 = 12, so 5/6 = 10/12.
  4. Add the tops. The bottom stays 12. 9 + 10 = 19.
  5. The sum is 19/12, which is more than 1. 19 − 12 = 7, so the answer is 1 7/12.

Answer: 1 7/12

Three bars of the same length. The top bar is cut into 3 parts with 1 shaded, labelled 1/3. The middle bar is cut into 4 parts with 1 shaded, labelled 1/4. The bottom bar is cut into 12 parts with 7 shaded, labelled 7/12. The bottom bar shows the two top bars joined.1314712
1/3 + 1/4 = 7/12. It is a little more than half, and less than 1.

Taking away works the same way. Change both fractions to the same bottom number. Then take away the tops.

With mixed numbers, take away the fraction parts first. Sometimes you cannot, because the top is too small. Then change one whole into parts, as in this example.

Worked example

Take away. 2 1/2 − 1 3/4

  1. Change to a common bottom number. The lowest common multiple of 2 and 4 is 4.
  2. 1/2 = 2/4, so 2 1/2 = 2 2/4.
  3. You cannot take 3 quarters from 2 quarters. Change 1 whole into 4 quarters. Now 2 quarters + 4 quarters = 6 quarters.
  4. So 2 2/4 is 1 6/4. Take away: 1 − 1 = 0 wholes, and 6 − 3 = 3 quarters.
  5. So 2 1/2 − 1 3/4 = 3/4.

Answer: 3/4

A number line from 0 to 3 with a mark at every quarter. A point labelled start is at 2 1/2. A first hop of 1 goes left to 1 1/2. A second hop of 3/4 goes left to a point labelled answer, at 3/4.0123134startanswer
Start at 2 1/2. Hop left 1, then hop left 3/4. You land on 3/4.

See it three ways

With things

Ravi eats 1/3 of a flat bread. Lucía eats 1/4 of a bread of the same size. How much bread did they eat in all?

As a picture

Three bars of the same length. The top bar is cut into 3 parts with 1 shaded, labelled one third. The middle bar is cut into 4 parts with 1 shaded, labelled one quarter. The bottom bar is cut into 12 parts with 7 shaded, labelled seven twelfths.one thirdone quarterseven twelfths

In symbols

1/3 + 1/4 = 4/12 + 3/12 = 7/12.

Try it

Add on squared paper

You need: squared paper, a pencil.

  1. Draw a bar 12 squares long and 1 square high.
  2. Shade 1/3 of the bar. That is 4 squares.
  3. Shade 1/4 of the bar next to it. That is 3 squares.
  4. Count all the shaded squares.

What you will see: 4 + 3 = 7 squares are shaded out of 12. So 1/3 + 1/4 = 7/12. Twelve squares let you cut the bar into both thirds and quarters.

If you cannot try it, this is what happens.
A bar cut into 12 equal parts with 7 shaded, labelled 7 of 12 squares.7 of 12 squares

A wrong idea many people have

1/3 + 1/4 = 2/7, because 1 + 1 = 2 and 3 + 4 = 7.

Why it feels right: You add the tops and you add the bottoms. It looks like adding.

The test that breaks it: 2/7 is smaller than 1/3. Adding a quarter to a third cannot give less than the third you began with.

What is true: The bottom number is the size of the parts, so it is not added. Change both fractions to the same bottom number. Then add only the tops.

Key words

common denominator
a bottom number that two fractions can both be changed to.
lowest common multiple
the smallest number that is in both times tables. It is the best common denominator.

Back to your guess

The answer is A: 1/3 + 1/4 = 7/12. Cut both into twelfths, then add.

Parts of different sizes cannot be counted together. Cut both fractions into twelfths first. 1/3 = 4/12 and 1/4 = 3/12. Then 4 + 3 = 7, so the answer is 7/12.

Check yourself

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

  1. The bars show one half and one quarter. How much is shaded in all?

    Two bars of the same length. The top bar is cut into 2 parts with 1 shaded, labelled one half. The bottom bar is cut into 4 parts with 1 shaded, labelled one quarter.one halfone quarter
    Tap your answer.

    1. Not this one. You multiplied the fractions. 1/2 × 1/4 = 1/8, but here you are putting the parts together, so you add. Go to M3.3.2. Try another.

    2. Not this one. You added the tops and added the bottoms. The parts are different sizes, so that does not work. One half is 2 quarters. Go to M3.3.2. Try another.

    3. Not this one. You added the tops, 1 + 1, but kept the bottom 4 without changing 1/2. One half is 2/4, so add 2 + 1. Go to M3.3.2. Try another.

    4. Yes, that is right. One half is 2/4. Then 2/4 + 1/4 = 3/4.

    See the answer

    3/4

    One half is 2/4. Then 2/4 + 1/4 = 3/4.

    • If you chose A: You multiplied the fractions. 1/2 × 1/4 = 1/8, but here you are putting the parts together, so you add. Go to M3.3.2.
    • If you chose B: You added the tops and added the bottoms. The parts are different sizes, so that does not work. One half is 2 quarters. Go to M3.3.2.
    • If you chose C: You added the tops, 1 + 1, but kept the bottom 4 without changing 1/2. One half is 2/4, so add 2 + 1. Go to M3.3.2.

  2. What is the lowest common bottom number for 1/6 and 1/9?

    Tap your answer.

    1. Not this one. 3 is a factor of 6 and 9. A common bottom number must be a multiple of both. 6 and 9 must both go into it. Go to M3.3.2. Try another.

    2. Not this one. 54 is a common multiple of 6 and 9, so it works. But it is not the lowest one, and it gives bigger numbers to work with. Go to M3.3.2. Try another.

    3. Yes, that is right. Multiples of 6: 6, 12, 18. Multiples of 9: 9, 18. The lowest one in both lists is 18.

    4. Not this one. You added 6 + 9. 6 does not go into 15 exactly. Use the lowest common multiple instead. Go to M3.3.2. Try another.

    See the answer

    18

    Multiples of 6: 6, 12, 18. Multiples of 9: 9, 18. The lowest one in both lists is 18.

    • If you chose A: 3 is a factor of 6 and 9. A common bottom number must be a multiple of both. 6 and 9 must both go into it. Go to M3.3.2.
    • If you chose B: 54 is a common multiple of 6 and 9, so it works. But it is not the lowest one, and it gives bigger numbers to work with. Go to M3.3.2.
    • If you chose D: You added 6 + 9. 6 does not go into 15 exactly. Use the lowest common multiple instead. Go to M3.3.2.

  3. What is 2/3 + 1/6?

    Tap your answer.

    1. Not this one. You added the tops and added the bottoms. The parts are different sizes, so that does not work. Change 2/3 into sixths first. Go to M3.3.2. Try another.

    2. Yes, that is right. 2/3 = 4/6. Then 4/6 + 1/6 = 5/6.

    3. Not this one. You changed the bottom to 6 but not the top. 2/3 is 4/6, not 2/6. Go to M3.3.2. Try another.

    See the answer

    5/6

    2/3 = 4/6. Then 4/6 + 1/6 = 5/6.

    • If you chose A: You added the tops and added the bottoms. The parts are different sizes, so that does not work. Change 2/3 into sixths first. Go to M3.3.2.
    • If you chose C: You changed the bottom to 6 but not the top. 2/3 is 4/6, not 2/6. Go to M3.3.2.

  4. Take away. 3 1/2 − 1 3/4

    Tap your answer.

    1. Yes, that is right. 3 1/2 = 3 2/4 = 2 6/4. Take away 1 3/4: 2 − 1 = 1 whole and 6 − 3 = 3 quarters.

    2. Not this one. You took the smaller quarters from the bigger ones. 1/2 is 2/4, and that is less than 3/4. Change 1 whole into 4 quarters first. Go to M3.3.2. Try another.

    3. Not this one. You changed 1 whole into 4 quarters but did not take 1 from the wholes. 3 2/4 becomes 2 6/4. Go to M3.3.2. Try another.

    See the answer

    1 3/4

    3 1/2 = 3 2/4 = 2 6/4. Take away 1 3/4: 2 − 1 = 1 whole and 6 − 3 = 3 quarters.

    • If you chose B: You took the smaller quarters from the bigger ones. 1/2 is 2/4, and that is less than 3/4. Change 1 whole into 4 quarters first. Go to M3.3.2.
    • If you chose C: You changed 1 whole into 4 quarters but did not take 1 from the wholes. 3 2/4 becomes 2 6/4. Go to M3.3.2.

  5. Joseph has a rope 3/4 m long. He cuts off a piece 1/3 m long. How long is the rope that is left?

    Tap your answer.

    1. Not this one. You changed the bottoms to 12 but did not change the tops. 3/4 = 9/12 and 1/3 = 4/12. Go to M3.3.2. Try another.

    2. Not this one. You took away the tops and took away the bottoms. That gives 2 m, which is more than the rope he started with. Go to M3.3.2. Try another.

    3. Yes, that is right. The LCM of 4 and 3 is 12. 3/4 = 9/12 and 1/3 = 4/12. 9 − 4 = 5, so 5/12 m is left.

    See the answer

    5/12 m

    The LCM of 4 and 3 is 12. 3/4 = 9/12 and 1/3 = 4/12. 9 − 4 = 5, so 5/12 m is left.

    • If you chose A: You changed the bottoms to 12 but did not change the tops. 3/4 = 9/12 and 1/3 = 4/12. Go to M3.3.2.
    • If you chose B: You took away the tops and took away the bottoms. That gives 2 m, which is more than the rope he started with. Go to M3.3.2.

  6. Add 1/2 + 1/6. Write the answer in simplest form.

    See the answer

    2/3

    The LCM of 2 and 6 is 6. 1/2 = 3/6. Then 3 + 1 = 4, so the sum is 4/6. In simplest form that is 2/3.

  7. Take away. 3/4 − 1/6

    See the answer

    7/12

    The LCM of 4 and 6 is 12. 3/4 = 9/12 and 1/6 = 2/12. 9 − 2 = 7.

  8. Take away. 4 1/4 − 2 1/2. Write the answer in simplest form.

    See the answer

    1 3/4

    1/2 = 2/4. You cannot take 2 quarters from 1 quarter, so change 1 whole into 4 quarters. 4 1/4 = 3 5/4. Then 3 − 2 = 1 whole and 5 − 2 = 3 quarters.

  9. From an earlier chapter

    Write 16/24 in simplest form.

    See the answer

    2/3

    The HCF of 16 and 24 is 8. 16 ÷ 8 = 2 and 24 ÷ 8 = 3.

Keep it

Copy this into your exercise book.

Parts must be the same size before you add. Find the LCM of the bottoms, change both fractions, then add the tops. The bottom stays.

Draw: Two bars of the same length, in thirds and quarters. Below them, a bar in twelfths with 7 shaded.

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.