Chapter 6 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

M3.3.6 Chapter 6

Multiplying and dividing decimals

Why is 0.3 × 0.2 equal to 0.06 and not 0.6?

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

  • Multiply a decimal by a whole number, like 2.4 × 3
  • Multiply two decimals, like 0.3 × 0.2
  • Divide a decimal by a whole number, like 7.5 ÷ 3

Start with a decimal times a whole number. 2.4 × 3 means three groups of 2.4. You can add: 2.4 + 2.4 + 2.4.

Draw it first

Work out 2.4 × 3 in a chart

  1. A place value chart with the headings ones and tenths. Step 1: one row, 2.4. Step 2: three rows of 2.4. Step 3: plus signs are written beside the second and third rows, a double line is drawn, and a fourth row shows the total, 7.2.onestenths24

    1 Write 2.4. It is 2 ones and 4 tenths.

  2. A place value chart with the headings ones and tenths. Step 1: one row, 2.4. Step 2: three rows of 2.4. Step 3: plus signs are written beside the second and third rows, a double line is drawn, and a fourth row shows the total, 7.2.onestenths242424

    2 Write it 3 times, because × 3 means three groups of 2.4.

  3. A place value chart with the headings ones and tenths. Step 1: one row, 2.4. Step 2: three rows of 2.4. Step 3: plus signs are written beside the second and third rows, a double line is drawn, and a fourth row shows the total, 7.2.onestenths24242472++

    3 Add the columns. 3 × 4 tenths = 12 tenths, which is 1 one and 2 tenths. 3 × 2 ones = 6 ones, and 1 more makes 7 ones. The total is 7.2.

Worked example

Work out 2.4 × 3.

  1. Ignore the point for now. 24 × 3 = 72.
  2. 2.4 has 1 decimal place, so the answer needs 1 decimal place.
  3. Put the point back. 72 becomes 7.2.
  4. Check with an estimate. 2.4 is about 2.5, and 2.5 × 3 = 7.5. That is close to 7.2.

Answer: 7.2

An estimate is a quick answer that is close to the real one. Make one before you calculate. If your answer is far from your estimate, look for a mistake.

Now two decimals. 0.3 × 0.2 means 3 tenths of 2 tenths. A hundred grid shows it. The whole grid is 1. One column is 0.1.

Draw it first

Find 0.3 of 0.2 on a hundred grid

  1. A square grid of 10 columns and 10 rows, one whole. Step 1: 3 columns are shaded with lines, 3/10 of the whole. Step 2: 2 rows are shaded with dots, 2/10 of the whole. Step 3: the fractions 3/10 and 2/10 are written beside the shaded parts, and the caption says 6 of 100 squares.

    1 Shade 3 columns of the grid with lines. That is 3 tenths, 0.3.

  2. A square grid of 10 columns and 10 rows, one whole. Step 1: 3 columns are shaded with lines, 3/10 of the whole. Step 2: 2 rows are shaded with dots, 2/10 of the whole. Step 3: the fractions 3/10 and 2/10 are written beside the shaded parts, and the caption says 6 of 100 squares.

    2 Shade 2 rows with dots. That is 2 tenths, 0.2.

  3. A square grid of 10 columns and 10 rows, one whole. Step 1: 3 columns are shaded with lines, 3/10 of the whole. Step 2: 2 rows are shaded with dots, 2/10 of the whole. Step 3: the fractions 3/10 and 2/10 are written beside the shaded parts, and the caption says 6 of 100 squares.3102106 of 100 squares

    3 The squares with lines and dots are 6 of the 100 squares. That is 6 hundredths, 0.06.

Worked example

Work out 0.3 × 0.2.

  1. Ignore the points. 3 × 2 = 6.
  2. Count the decimal places. 0.3 has 1 and 0.2 has 1. That is 2 in all.
  3. The answer needs 2 decimal places. Write 6 with 2 places after the point: 0.06.
  4. So 0.3 × 0.2 = 0.06.

Answer: 0.06

A square grid of 100 small squares with 6 squares shaded with dots in the top row. The caption says 0.06 is 6 squares.0.06 is 6 squares
The answer, 0.06, is 6 hundredths. It is less than 0.2, because it is only a part of 0.2.

Here is the rule. Multiply as if there were no points. Count the decimal places in both numbers. Write the whole number answer with that many decimal places. Then you may drop zeros at the end: 0.20 = 0.2.

Now dividing a decimal by a whole number. Divide from the left, the biggest place first, as with whole numbers. When you reach the point, put a point in the answer in the same place. Then carry on with the tenths.

See it three ways

With things

Ana shares 7.5 kg of rice between 3 bags. Each bean in the picture stands for 0.5 kg. She puts the same number of beans in each bag.

As a picture

Three bags, each with 5 beans. Each bean stands for 0.5 kg. Every bag is labelled 2.5 kg.2.5 kg2.5 kg2.5 kg

In symbols

7.5 ÷ 3 = 2.5. Each bag gets 2.5 kg.

Worked example

Work out 7.5 ÷ 3.

  1. Divide the ones (there are no tens here). 7 ÷ 3 = 2, and 1 one is left over.
  2. Write the 2 in the ones place of the answer. Put the decimal point after it.
  3. The 1 left over is 10 tenths. Add the 5 tenths: 10 + 5 = 15 tenths.
  4. Divide the tenths. 15 ÷ 3 = 5.
  5. So 7.5 ÷ 3 = 2.5.

Answer: 2.5

Try it

Find 0.4 of 0.5 on squared paper

You need: squared paper, a pencil, a ruler.

  1. Draw a square 10 squares wide and 10 squares high. It is one whole.
  2. Shade 4 columns with lines. That is 0.4.
  3. Shade 5 rows with dots. That is 0.5.
  4. Count the squares that have both lines and dots.

What you will see: There are 20 squares with both. 20 hundredths is 0.2. So 0.4 × 0.5 = 0.2.

If you cannot try it, this is what happens.
A square grid of 10 columns and 10 rows. 4 columns are shaded with lines and 5 rows with dots. The caption says 20 of 100 squares.20 of 100 squares

A wrong idea many people have

0.3 × 0.2 = 0.6, because 3 × 2 = 6.

Why it feels right: The digits are right. And you may think the answer must have one decimal place, like the numbers.

The test that breaks it: Find 0.3 of 0.2. That is a part of 0.2, so it must be less than 0.2. But 0.6 is more than 0.2. On the hundred grid you count only 6 squares of 100.

What is true: Write the whole number answer with as many decimal places as both numbers together. 1 + 1 = 2 places, so 6 becomes 0.06. Then you may drop zeros at the end: 0.20 = 0.2.

Key words

estimate
a quick answer that is close to the real one. You use it to check your work.
decimal places
the digits after the decimal point. 0.06 has 2 decimal places.

Back to your guess

The answer is A: 0.06. 3 × 2 = 6, and the two numbers have 2 decimal places in all.

0.3 × 0.2 = 0.06. A tenth of a tenth is a hundredth. 3 tenths of 2 tenths is 6 hundredths. The answer must be smaller than 0.2, because it is only a part of 0.2.

Check yourself

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

  1. The grid has 10 columns and 10 rows. The columns with lines are 0.6 of the whole. The rows with dots are 0.7 of the whole. Which sentence does the grid show?

    A square grid of 10 columns and 10 rows. 6 columns are shaded with lines and 7 rows with dots. The caption says 10 columns, 10 rows.10 columns, 10 rows
    Tap your answer.

    1. Not this one. The squares with lines and dots are 42 of the 100. Each square is one hundredth, so 42 hundredths is 0.42, not 4.2. Go to M3.3.6. Try another.

    2. Yes, that is right. 6 columns and 7 rows cross in 6 × 7 = 42 squares. 42 of 100 squares is 42 hundredths, which is 0.42.

    3. Not this one. You added 0.6 and 0.7. The picture shows a part of a part, which is found by multiplying. Go to M3.3.6. Try another.

    See the answer

    0.6 × 0.7 = 0.42

    6 columns and 7 rows cross in 6 × 7 = 42 squares. 42 of 100 squares is 42 hundredths, which is 0.42.

    • If you chose A: The squares with lines and dots are 42 of the 100. Each square is one hundredth, so 42 hundredths is 0.42, not 4.2. Go to M3.3.6.
    • If you chose C: You added 0.6 and 0.7. The picture shows a part of a part, which is found by multiplying. Go to M3.3.6.

  2. What is 2.4 × 3?

    Tap your answer.

    1. Not this one. The digits are right, but you forgot to put the decimal point back. 2.4 has 1 decimal place, so the answer has 1. Go to M3.3.6. Try another.

    2. Not this one. You wrote 2 × 3 = 6 and 4 × 3 = 12 side by side. 12 tenths is 1 one and 2 tenths, so the 1 moves into the ones. 6 + 1 = 7. Go to M3.3.6. Try another.

    3. Yes, that is right. 24 × 3 = 72. 2.4 has 1 decimal place, so the answer is 7.2.

    See the answer

    7.2

    24 × 3 = 72. 2.4 has 1 decimal place, so the answer is 7.2.

    • If you chose A: The digits are right, but you forgot to put the decimal point back. 2.4 has 1 decimal place, so the answer has 1. Go to M3.3.6.
    • If you chose B: You wrote 2 × 3 = 6 and 4 × 3 = 12 side by side. 12 tenths is 1 one and 2 tenths, so the 1 moves into the ones. 6 + 1 = 7. Go to M3.3.6.

  3. What is 0.4 × 0.6?

    Tap your answer.

    1. Yes, that is right. 4 × 6 = 24. There are 2 decimal places in all, so the answer is 0.24.

    2. Not this one. You worked out 4 × 6 = 24 but did not count decimal places. Each number has 1, so the answer has 2. Go to M3.3.6. Try another.

    3. Not this one. That has 3 decimal places. 0.4 and 0.6 have 1 each, so the answer has 2 in all. Go to M3.3.6. Try another.

    See the answer

    0.24

    4 × 6 = 24. There are 2 decimal places in all, so the answer is 0.24.

    • If you chose B: You worked out 4 × 6 = 24 but did not count decimal places. Each number has 1, so the answer has 2. Go to M3.3.6.
    • If you chose C: That has 3 decimal places. 0.4 and 0.6 have 1 each, so the answer has 2 in all. Go to M3.3.6.

  4. What is 7.2 ÷ 3?

    Tap your answer.

    1. Not this one. The digits are right, but you forgot the decimal point. 7.2 is about 7, and 7 ÷ 3 is about 2, so the answer is near 2. Go to M3.3.6. Try another.

    2. Yes, that is right. Divide the ones: 7 ÷ 3 = 2, and 1 is left. 1 one and 2 tenths is 12 tenths. 12 ÷ 3 = 4 tenths. So 7.2 ÷ 3 = 2.4.

    3. Not this one. The point is in the wrong place. 7.2 is about 7, and 7 ÷ 3 is about 2, not a quarter. Go to M3.3.6. Try another.

    See the answer

    2.4

    Divide the ones: 7 ÷ 3 = 2, and 1 is left. 1 one and 2 tenths is 12 tenths. 12 ÷ 3 = 4 tenths. So 7.2 ÷ 3 = 2.4.

    • If you chose A: The digits are right, but you forgot the decimal point. 7.2 is about 7, and 7 ÷ 3 is about 2, so the answer is near 2. Go to M3.3.6.
    • If you chose C: The point is in the wrong place. 7.2 is about 7, and 7 ÷ 3 is about 2, not a quarter. Go to M3.3.6.

  5. Make an estimate first. Which answer to 4.1 × 2.9 is sensible?

    Tap your answer.

    1. Not this one. Estimate: 4 × 3 = 12. 118.9 is far too big. The point has moved too far to the right. Go to M3.3.6. Try another.

    2. Not this one. Estimate: 4 × 3 = 12. 1.189 is far too small. Count the decimal places: 1 + 1 = 2, so the answer has 2 places. Go to M3.3.6. Try another.

    3. Yes, that is right. 4.1 is about 4 and 2.9 is about 3. 4 × 3 = 12, so the answer should be near 12. 41 × 29 = 1189, and 2 decimal places gives 11.89.

    See the answer

    11.89

    4.1 is about 4 and 2.9 is about 3. 4 × 3 = 12, so the answer should be near 12. 41 × 29 = 1189, and 2 decimal places gives 11.89.

    • If you chose A: Estimate: 4 × 3 = 12. 118.9 is far too big. The point has moved too far to the right. Go to M3.3.6.
    • If you chose B: Estimate: 4 × 3 = 12. 1.189 is far too small. Count the decimal places: 1 + 1 = 2, so the answer has 2 places. Go to M3.3.6.

  6. Work out 3.6 × 4.

    See the answer

    14.4

    36 × 4 = 144. 3.6 has 1 decimal place, so the answer is 14.4.

  7. Work out 0.7 × 0.4.

    See the answer

    0.28

    7 × 4 = 28. There are 2 decimal places in all, so the answer is 0.28.

  8. Work out 9.6 ÷ 4.

    See the answer

    2.4

    Divide the ones: 9 ÷ 4 = 2, and 1 is left. 1 one and 6 tenths is 16 tenths. 16 ÷ 4 = 4 tenths. So the answer is 2.4.

  9. From an earlier chapter

    Take away. 6.3 − 2.45

    See the answer

    3.85

    Write 6.3 as 6.30. Hundredths: 10 − 5 = 5. Tenths: 2 − 4 cannot be done, so borrow: 12 − 4 = 8. Ones: 5 − 2 = 3.

Keep it

Copy this into your exercise book.

Multiply as whole numbers, then count the decimal places in both numbers. The answer has that many. Make an estimate to check. When you divide, keep the point in line.

Draw: A hundred grid. Shade 3 columns with lines and 2 rows with dots. Write 0.3 × 0.2 = 0.06.

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.