Chapter 3 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

M3.3.3 Chapter 3

Multiplying and dividing fractions

What is two thirds of three quarters of a flat bread?

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

  • Find 2/3 of 3/4 and write it as 2/3 × 3/4
  • Find how many quarters are in 3, and write 3 ÷ 1/4
  • Divide one fraction by another, like 3/4 ÷ 1/8

Finding a fraction of an amount is the same as multiplying. 1/2 of 12 is 1/2 × 12 = 6. The word "of" means times.

Now take a fraction of a fraction. We can see it on a square, cut in two ways at once.

Draw it first

Find 2/3 of 3/4

  1. A square for one whole. Step 1: the square is cut into 4 columns and 3 columns are shaded with lines. Step 2: the square is also cut into 3 rows and 2 rows are shaded with dots. Step 3: the fraction 3/4 is written over the lined columns, the fraction 2/3 beside the dotted rows, and the caption says 6 of 12 squares have both lines and dots.

    1 Draw a square for 1 whole. Cut it in 4 columns. Shade 3 columns with lines. That is 3/4 of the whole.

  2. A square for one whole. Step 1: the square is cut into 4 columns and 3 columns are shaded with lines. Step 2: the square is also cut into 3 rows and 2 rows are shaded with dots. Step 3: the fraction 3/4 is written over the lined columns, the fraction 2/3 beside the dotted rows, and the caption says 6 of 12 squares have both lines and dots.

    2 Cut the square in 3 rows. Shade 2 rows with dots. That is 2/3 of the whole.

  3. A square for one whole. Step 1: the square is cut into 4 columns and 3 columns are shaded with lines. Step 2: the square is also cut into 3 rows and 2 rows are shaded with dots. Step 3: the fraction 3/4 is written over the lined columns, the fraction 2/3 beside the dotted rows, and the caption says 6 of 12 squares have both lines and dots.34236 of 12 squares

    3 The squares with lines and dots are 2/3 of 3/4. There are 6 of 12 squares, so 2/3 × 3/4 = 6/12.

Worked example

Multiply 2/3 × 3/4.

  1. Multiply the tops: 2 × 3 = 6.
  2. Multiply the bottoms: 3 × 4 = 12.
  3. So 2/3 × 3/4 = 6/12.
  4. Simplify. The HCF of 6 and 12 is 6, so 6/12 = 1/2.

Answer: 1/2

You can cancel before you multiply. In 2/3 × 3/4, a 3 is on the top and a 3 is on the bottom. Divide both by 3. You get 2/4, which is 1/2. Smaller numbers are quicker to work with.

See it three ways

With things

Take a flat bread. Use 3/4 of it. Then use 2/3 of the part you took.

As a picture

A square cut into 4 columns and 3 rows. Three columns are shaded with lines and two rows with dots. The caption says 6 of 12.6 of 12

In symbols

2/3 × 3/4 = 6/12 = 1/2. Multiply the tops, multiply the bottoms, then simplify.

Now dividing. 3 ÷ 1/4 asks: how many quarters are in 3? We can count them.

Draw it first

How many quarters are in 3?

  1. Three bars, each one whole, each cut into 4 quarters. Step 1: the 4 quarters of the first bar are counted 1 to 4. Step 2: the second bar is counted 5 to 8. Step 3: the third bar is counted 9 to 12.1234

    1 Draw 3 wholes and cut each into quarters. Count the quarters in the first whole: 4.

  2. Three bars, each one whole, each cut into 4 quarters. Step 1: the 4 quarters of the first bar are counted 1 to 4. Step 2: the second bar is counted 5 to 8. Step 3: the third bar is counted 9 to 12.12345678

    2 Count on through the second whole. Now you have 8 quarters.

  3. Three bars, each one whole, each cut into 4 quarters. Step 1: the 4 quarters of the first bar are counted 1 to 4. Step 2: the second bar is counted 5 to 8. Step 3: the third bar is counted 9 to 12.123456789101112

    3 Count on through the third whole. There are 12 quarters in 3, so 3 ÷ 1/4 = 12.

Each whole holds 4 quarters, so 3 wholes hold 3 × 4. Dividing by 1/4 is the same as multiplying by 4. The number 4, written 4/1, is the reciprocal of 1/4. You flip a fraction to find its reciprocal.

Worked example

How many quarters are in 3? Work out 3 ÷ 1/4.

  1. One whole is the same as 4 quarters.
  2. So 3 wholes are 3 × 4 quarters. 3 × 4 = 12.
  3. So 3 ÷ 1/4 = 12.

Answer: 12

Now divide one fraction by another. How many eighths are in 3/4? The two bars below have the same length. Count the eighths under the shaded quarters.

Two bars of the same length, one above the other. The top bar is cut into 4 equal parts with 3 shaded, which is 3/4. The bottom bar is cut into 8 equal parts. The first 6 parts are shaded and counted 1 to 6, and the caption says 6 eighths. The shaded parts end at the same place on both bars.1234566 eighths
3/4 is the same length as 6 eighths.

There are 6 eighths in 3/4, so 3/4 ÷ 1/8 = 6. Dividing by a fraction is the same as multiplying by its reciprocal. 3/4 × 8 = 6 too.

Worked example

Divide 3/4 ÷ 1/8.

  1. Turn the divide into a times. The reciprocal of 1/8 is 8, because 8 eighths make one whole.
  2. So 3/4 ÷ 1/8 = 3/4 × 8.
  3. The bottom stays 4. Multiply the tops: 3 × 8 = 24.
  4. So 3/4 × 8 = 24/4. Simplify: 24 ÷ 4 = 6.
  5. There are 6 eighths in 3/4.

Answer: 6

Try it

Fold a half of a third

You need: a sheet of paper, a pencil.

  1. Fold the sheet into 3 equal parts, like a letter. Open it. Shade one part with lines.
  2. Fold the sheet in half the other way. Open it.
  3. Shade the top half of the whole sheet with dots.
  4. Count the parts the folds make. Count the part with lines and dots.

What you will see: The folds make 6 equal parts, and 1 has lines and dots. So 1/2 of 1/3 is 1/6. That is 1/2 × 1/3 = 1/6.

If you cannot try it, this is what happens.
A rectangle cut into 3 columns and 2 rows. One column is shaded with lines and one row with dots. The caption says 1 of 6.1 of 6

A wrong idea many people have

Dividing always makes a number smaller.

Why it feels right: With whole numbers bigger than 1 it is true. 12 ÷ 3 = 4, and 4 is less than 12.

The test that breaks it: Count the quarters in 3. There are 12. 12 is more than 3.

What is true: For a positive number, dividing by a fraction between 0 and 1 makes the answer bigger. You count smaller parts. Multiplying by such a fraction makes it smaller.

Key words

reciprocal
a fraction turned upside down. The reciprocal of 1/4 is 4. Dividing by a fraction is multiplying by its reciprocal.

Back to your guess

The answer is B: 1/2. A part of three quarters must be less than three quarters.

Two thirds of three quarters is 1/2. Taking a part of an amount gives less than the amount. Multiplying a positive number by a fraction less than 1 makes it smaller.

Check yourself

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

  1. The columns with lines are 2/5 of the square. The rows with dots are 3/4 of the square. What fraction of the whole square has both lines and dots?

    A square cut into 5 columns and 4 rows. 2 columns are shaded with lines and 3 rows with dots. The caption says 5 columns, 4 rows.5 columns, 4 rows
    Tap your answer.

    1. Yes, that is right. The square has 5 × 4 = 20 small squares. Squares with both lines and dots are 2 columns of 3 rows, which is 6. 6/20 = 3/10.

    2. Not this one. That is the right amount, but it is not in simplest form. 6 and 20 are both in the 2 times table, so divide both by 2. Go to M3.3.3. Try another.

    3. Not this one. You added the tops and added the bottoms. A part of a part is found by multiplying, not adding. Go to M3.3.3. Try another.

    See the answer

    3/10

    The square has 5 × 4 = 20 small squares. Squares with both lines and dots are 2 columns of 3 rows, which is 6. 6/20 = 3/10.

    • If you chose B: That is the right amount, but it is not in simplest form. 6 and 20 are both in the 2 times table, so divide both by 2. Go to M3.3.3.
    • If you chose C: You added the tops and added the bottoms. A part of a part is found by multiplying, not adding. Go to M3.3.3.

  2. What is 2/3 × 3/5?

    Tap your answer.

    1. Not this one. You multiplied the tops, but you added the bottoms. Multiply the bottoms too: 3 × 5 = 15. Go to M3.3.3. Try another.

    2. Not this one. That is the right amount, but it is not in simplest form. 6 and 15 are both in the 3 times table, so divide both by 3. Go to M3.3.3. Try another.

    3. Not this one. You added the tops and added the bottoms. To multiply, multiply the tops and multiply the bottoms. Go to M3.3.3. Try another.

    4. Yes, that is right. 2 × 3 = 6 and 3 × 5 = 15. 6/15 simplifies to 2/5, because the HCF of 6 and 15 is 3.

    See the answer

    2/5

    2 × 3 = 6 and 3 × 5 = 15. 6/15 simplifies to 2/5, because the HCF of 6 and 15 is 3.

    • If you chose A: You multiplied the tops, but you added the bottoms. Multiply the bottoms too: 3 × 5 = 15. Go to M3.3.3.
    • If you chose B: That is the right amount, but it is not in simplest form. 6 and 15 are both in the 3 times table, so divide both by 3. Go to M3.3.3.
    • If you chose C: You added the tops and added the bottoms. To multiply, multiply the tops and multiply the bottoms. Go to M3.3.3.

  3. How many quarters are in 3?

    Tap your answer.

    1. Not this one. That is 3 quarters of one whole. The question asks how many quarters fit into 3 whole ones, which is many more. Go to M3.3.3. Try another.

    2. Not this one. You added 3 + 4. Each whole holds 4 quarters, so 3 wholes hold 3 × 4 quarters. Go to M3.3.3. Try another.

    3. Yes, that is right. One whole holds 4 quarters. 3 wholes hold 3 × 4 = 12 quarters.

    See the answer

    12

    One whole holds 4 quarters. 3 wholes hold 3 × 4 = 12 quarters.

    • If you chose A: That is 3 quarters of one whole. The question asks how many quarters fit into 3 whole ones, which is many more. Go to M3.3.3.
    • If you chose B: You added 3 + 4. Each whole holds 4 quarters, so 3 wholes hold 3 × 4 quarters. Go to M3.3.3.

  4. Which of these gives an answer bigger than 3?

    Tap your answer.

    1. Not this one. Multiplying by a fraction less than 1 makes the number smaller. Half of 3 is 1 1/2. Go to M3.3.3. Try another.

    2. Yes, that is right. There are 6 halves in 3, so 3 ÷ 1/2 = 6. The other two give 1 1/2 and 3.

    3. Not this one. Dividing by 1 leaves 3. It does not make it bigger. Go to M3.3.3. Try another.

    See the answer

    3 ÷ 1/2

    There are 6 halves in 3, so 3 ÷ 1/2 = 6. The other two give 1 1/2 and 3.

    • If you chose A: Multiplying by a fraction less than 1 makes the number smaller. Half of 3 is 1 1/2. Go to M3.3.3.
    • If you chose C: Dividing by 1 leaves 3. It does not make it bigger. Go to M3.3.3.

  5. What is 3/4 ÷ 1/8?

    Tap your answer.

    1. Yes, that is right. Multiply by the reciprocal. 3/4 × 8 = 24/4 = 6. There are 6 eighths in 3/4.

    2. Not this one. You multiplied the fractions. To divide, count how many eighths fit in 3/4. Or multiply by the reciprocal of 1/8, which is 8. Go to M3.3.3. Try another.

    3. Not this one. You changed 3/4 into eighths, 6/8, which is right so far. 6/8 means 6 eighths. The question asks how many eighths there are, and that is 6. Go to M3.3.3. Try another.

    See the answer

    6

    Multiply by the reciprocal. 3/4 × 8 = 24/4 = 6. There are 6 eighths in 3/4.

    • If you chose B: You multiplied the fractions. To divide, count how many eighths fit in 3/4. Or multiply by the reciprocal of 1/8, which is 8. Go to M3.3.3.
    • If you chose C: You changed 3/4 into eighths, 6/8, which is right so far. 6/8 means 6 eighths. The question asks how many eighths there are, and that is 6. Go to M3.3.3.

  6. Multiply. 3/5 × 5/6. Write the answer in simplest form.

    See the answer

    1/2

    3 × 5 = 15 and 5 × 6 = 30. 15/30 simplifies to 1/2, because the HCF of 15 and 30 is 15.

  7. How many 1/5s are in 4?

    See the answer

    20

    One whole holds 5 fifths. 4 wholes hold 4 × 5 = 20 fifths.

  8. Divide. 2/3 ÷ 1/6

    See the answer

    4

    Multiply by the reciprocal. 2/3 × 6 = 12/3 = 4.

  9. From an earlier chapter

    Add. 1/4 + 2/3

    See the answer

    11/12

    The LCM of 4 and 3 is 12. 1/4 = 3/12 and 2/3 = 8/12. 3 + 8 = 11.

Keep it

Copy this into your exercise book.

Of means times. Multiply the tops and the bottoms, then simplify. To divide by a fraction, multiply by its reciprocal.

Draw: A square cut in 4 columns and 3 rows. Shade 3 columns with lines and 2 rows with dots. Write 2/3 × 3/4 = 1/2.

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 (828 KB). It opens on any phone.