Chapter 5 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

M3.3.5 Chapter 5

Adding and taking away decimals

What is 2.5 + 0.75, and why must the decimal points line up?

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 and take away decimals by lining up the decimal points
  • Multiply and divide decimals by 10, 100 and 1 000
  • Round a decimal to the nearest whole or one decimal place

You add decimals the way you add whole numbers, in columns. Each digit must sit in a column with digits of the same worth. That is why the decimal points go in a line.

Draw it first

Add 2.5 and 0.75 in a chart

  1. A place value chart with the headings ones, tenths and hundredths. Step 1: the numbers 2.5 and 0.75 are written with their points in a line. Step 2: a 0 is added after the 5 in a dashed box, so 2.5 reads 2.50. Step 3: a plus sign is written beside 0.75, a double line is drawn, and a third row shows the sum, 3.25.onestenthshundredths25075

    1 Write 2.5 and 0.75 with the decimal points in a line. Tenths sit under tenths.

  2. A place value chart with the headings ones, tenths and hundredths. Step 1: the numbers 2.5 and 0.75 are written with their points in a line. Step 2: a 0 is added after the 5 in a dashed box, so 2.5 reads 2.50. Step 3: a plus sign is written beside 0.75, a double line is drawn, and a third row shows the sum, 3.25.onestenthshundredths250075

    2 Fill the empty place with a 0, so both numbers have two decimal places. 2.5 = 2.50.

  3. A place value chart with the headings ones, tenths and hundredths. Step 1: the numbers 2.5 and 0.75 are written with their points in a line. Step 2: a 0 is added after the 5 in a dashed box, so 2.5 reads 2.50. Step 3: a plus sign is written beside 0.75, a double line is drawn, and a third row shows the sum, 3.25.onestenthshundredths250075325+

    3 Add one column at a time, from the right. The sum is 3.25.

Worked example

Add 2.5 + 0.75.

  1. Line up the points. Write 2.5 as 2.50.
  2. Add the hundredths: 0 + 5 = 5.
  3. Add the tenths: 5 + 7 = 12, which is 1 one and 2 tenths.
  4. Add the ones: 2 + 0 + 1 = 3.
  5. So 2.5 + 0.75 = 3.25.

Answer: 3.25

Taking away works the same way. Line up the points and fill empty places with 0. If a digit is too small to take from, borrow 1 from the column on its left. That 1 is worth 10 in the column you are working in.

Worked example

Take away. 5.2 − 1.75

  1. Line up the points. Write 5.2 as 5.20.
  2. Hundredths: 0 − 5 cannot be done. Borrow 1 tenth, which is 10 hundredths. 10 − 5 = 5.
  3. Tenths: 1 tenth is left after borrowing. 1 − 7 cannot be done. Borrow 1 one, which is 10 tenths. 11 − 7 = 4.
  4. Ones: 4 are left after borrowing. 4 − 1 = 3.
  5. So 5.2 − 1.75 = 3.45.

Answer: 3.45

A place value chart with the headings ones, tenths and hundredths. The top row is 5.20. The second row is 1.75, the number taken away, with a minus sign beside it. A double line is drawn under it. The third row is the answer, 3.45.onestenthshundredths520175345−
Top row: 5.20. Second row: take away 1.75. Under the double line: the answer, 3.45.

See it three ways

With things

Mei shades 35 squares of a hundred grid. Kofi shades 28 squares of another one. How much is shaded in all?

As a picture

Two hundred grids side by side with a plus sign between them. The left grid has 35 squares shaded and is labelled 0.35. The right grid has 28 squares shaded and is labelled 0.28.0.35+0.28

In symbols

0.35 + 0.28 = 0.63. 35 + 28 = 63 hundredths.

Now multiplying by 10. Every digit moves one place to the left, so it becomes 10 times bigger. The decimal point stays where it is. The digits move past it.

A place value chart with the headings tens, ones, tenths and hundredths. The top row is 3.47. The second row is 34.7. Each digit of 3.47 has moved one column to the left in the second row.tensonestenthshundredths347347
3.47 × 10 = 34.7. Every digit moves one place to the left.

Multiplying by 100 moves the digits 2 places left. Multiplying by 1 000 moves them 3 places left. Dividing goes the other way. Dividing by 10, 100 or 1 000 moves the digits 1, 2 or 3 places to the right.

Worked example

Work out 0.047 × 1 000.

  1. 1 000 has 3 zeros, so move every digit 3 places to the left.
  2. Do it one place at a time. First × 10 once: 0.047 × 10 = 0.47.
  3. × 10 again: 0.47 × 10 = 4.7.
  4. × 10 a third time: 4.7 × 10 = 47.
  5. So 0.047 × 1 000 = 47.

Answer: 47

The digits after the decimal point are the decimal places. 3.47 has 2 decimal places. To round a decimal means to change it to a nearby number with fewer decimal places. Look at the digit just after the place you keep.

If that digit is 5 or more, round up. If it is 4 or less, leave the kept digit as it is. To round to a whole number, look at the tenths digit.

Draw it first

Round 3.46 to one decimal place

  1. A number line from 3.40 to 3.50 with a tick for each hundredth. Step 1: the line is drawn, with the halfway mark 3.45. Step 2: a point labelled 3.46 is marked one tick after 3.45. Step 3: a note says it rounds up to 3.5.3.403.453.50

    1 Draw the line from 3.40 to 3.50. The halfway mark is 3.45.

  2. A number line from 3.40 to 3.50 with a tick for each hundredth. Step 1: the line is drawn, with the halfway mark 3.45. Step 2: a point labelled 3.46 is marked one tick after 3.45. Step 3: a note says it rounds up to 3.5.3.403.453.503.46

    2 Mark 3.46. It is one step after the halfway mark.

  3. A number line from 3.40 to 3.50 with a tick for each hundredth. Step 1: the line is drawn, with the halfway mark 3.45. Step 2: a point labelled 3.46 is marked one tick after 3.45. Step 3: a note says it rounds up to 3.5.3.403.453.503.46rounds up to 3.5

    3 3.46 is past halfway, so it is nearer to 3.5 than to 3.4. Round up: 3.46 ≈ 3.5.

Try it

Round a length you measure

You need: a ruler marked in millimetres, a pencil or a stick.

  1. Measure the length of the pencil in centimetres. Read the millimetre marks as tenths of a centimetre, such as 14.6 cm.
  2. Look at the tenths digit. Is it 5 or more?
  3. If it is, round up to the next whole centimetre. If not, keep the whole number of centimetres.
  4. Check on the ruler. Is the end of the pencil nearer to the mark above or the mark below?

What you will see: A length of 14.6 cm is nearer to 15 cm than to 14 cm. So it rounds to 15 cm. Any length from 14.5 cm up to just under 15 cm rounds to 15 cm.

If you cannot try it, this is what happens.
A number line from 14.0 to 15.0 with a tick for each tenth. The caption says lengths in cm. A point labelled 14.6 cm is marked between 14.5 and 15.0.14.014.515.014.6 cmlengths in cm

A wrong idea many people have

To multiply by 10, add a zero. So 2.5 × 10 = 2.50.

Why it feels right: With whole numbers it works. 25 × 10 = 250 has a 0 added.

The test that breaks it: 2.50 and 2.5 are the same number. But 2.5 × 10 must be 10 times bigger. Move each digit one place left and you get 25.

What is true: Multiplying by 10 moves every digit one place to the left. 2.5 × 10 = 25. A zero is added to a whole number only because it fills the empty ones place.

Key words

decimal places
the digits after the decimal point. 3.47 has 2 decimal places.
round
to change a number to a nearby one that is shorter, such as 3.46 to 3.5.

Back to your guess

The answer is C: 3.25. Line up the points, so tenths are added to tenths.

The answer is 3.25. Tenths must be added to tenths, and hundredths to hundredths. Lining up the decimal points puts every digit under a digit of the same worth. 2.50 + 0.75 = 3.25.

Check yourself

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

  1. What is 3.4 + 0.27?

    Tap your answer.

    1. Yes, that is right. Write 3.4 as 3.40. Hundredths: 0 + 7 = 7. Tenths: 4 + 2 = 6. Ones: 3. The sum is 3.67.

    2. Not this one. You added 34 and 27 as whole numbers and ignored the points. The 4 is 4 tenths and the 27 is hundredths, so they are in different columns. Go to M3.3.5. Try another.

    3. Not this one. You added 4 tenths and 27 hundredths as if they were the same: 4 + 27 = 31. Write 3.4 as 3.40, then add each column. Go to M3.3.5. Try another.

    See the answer

    3.67

    Write 3.4 as 3.40. Hundredths: 0 + 7 = 7. Tenths: 4 + 2 = 6. Ones: 3. The sum is 3.67.

    • If you chose B: You added 34 and 27 as whole numbers and ignored the points. The 4 is 4 tenths and the 27 is hundredths, so they are in different columns. Go to M3.3.5.
    • If you chose C: You added 4 tenths and 27 hundredths as if they were the same: 4 + 27 = 31. Write 3.4 as 3.40, then add each column. Go to M3.3.5.

  2. What is 5.2 − 1.75?

    Tap your answer.

    1. Not this one. You took the smaller digit from the bigger one in each column. 2 tenths cannot lose 7 tenths, so you must borrow. Go to M3.3.5. Try another.

    2. Yes, that is right. Write 5.2 as 5.20. Hundredths: 10 − 5 = 5. Tenths: 11 − 7 = 4. Ones: 4 − 1 = 3. The answer is 3.45.

    3. Not this one. After you borrow 1 tenth for the hundredths, only 1 tenth is left, not 2. So the tenths are 11 − 7 = 4. Go to M3.3.5. Try another.

    See the answer

    3.45

    Write 5.2 as 5.20. Hundredths: 10 − 5 = 5. Tenths: 11 − 7 = 4. Ones: 4 − 1 = 3. The answer is 3.45.

    • If you chose A: You took the smaller digit from the bigger one in each column. 2 tenths cannot lose 7 tenths, so you must borrow. Go to M3.3.5.
    • If you chose C: After you borrow 1 tenth for the hundredths, only 1 tenth is left, not 2. So the tenths are 11 − 7 = 4. Go to M3.3.5.

  3. What is 6.4 × 100?

    Tap your answer.

    1. Not this one. Adding zeros after the last decimal digit does not change a number. 6.400 is still 6.4. Move the digits left. Go to M3.3.5. Try another.

    2. Not this one. That is 6.4 × 10. 100 has two zeros, so move the digits 2 places left. Go to M3.3.5. Try another.

    3. Yes, that is right. 100 has 2 zeros, so move each digit 2 places left. The 6 moves to the hundreds, the 4 moves to the tens, and a 0 fills the ones.

    See the answer

    640

    100 has 2 zeros, so move each digit 2 places left. The 6 moves to the hundreds, the 4 moves to the tens, and a 0 fills the ones.

    • If you chose A: Adding zeros after the last decimal digit does not change a number. 6.400 is still 6.4. Move the digits left. Go to M3.3.5.
    • If you chose B: That is 6.4 × 10. 100 has two zeros, so move the digits 2 places left. Go to M3.3.5.

  4. What is 35 ÷ 1 000?

    Tap your answer.

    1. Yes, that is right. Move each digit 3 places to the right. 35 = 35.0, so the 5 moves to the thousandths and the 3 to the hundredths: 0.035.

    2. Not this one. That is 35 ÷ 100. 1 000 has three zeros, so move the digits 3 places right. Go to M3.3.5. Try another.

    3. Not this one. That moved the digits 4 places. Three zeros mean three places. Go to M3.3.5. Try another.

    See the answer

    0.035

    Move each digit 3 places to the right. 35 = 35.0, so the 5 moves to the thousandths and the 3 to the hundredths: 0.035.

    • If you chose B: That is 35 ÷ 100. 1 000 has three zeros, so move the digits 3 places right. Go to M3.3.5.
    • If you chose C: That moved the digits 4 places. Three zeros mean three places. Go to M3.3.5.

  5. What is 8.62 rounded to the nearest whole number?

    A number line from 8.0 to 9.0 with a tick for each tenth. A point labelled 8.62 is marked between 8.5 and 9.0.8.08.59.08.62
    Tap your answer.

    1. Not this one. You cut off the decimals. Look at the tenths digit: 6 is 5 or more, so round up. Go to M3.3.5. Try another.

    2. Yes, that is right. The tenths digit is 6, which is 5 or more, so round up. On the line, 8.62 is nearer to 9 than to 8.

    3. Not this one. 8.6 is 8.62 rounded to one decimal place. The question asks for a whole number. Go to M3.3.5. Try another.

    See the answer

    9

    The tenths digit is 6, which is 5 or more, so round up. On the line, 8.62 is nearer to 9 than to 8.

    • If you chose A: You cut off the decimals. Look at the tenths digit: 6 is 5 or more, so round up. Go to M3.3.5.
    • If you chose C: 8.6 is 8.62 rounded to one decimal place. The question asks for a whole number. Go to M3.3.5.

  6. What is 7.449 rounded to one decimal place?

    Tap your answer.

    1. Not this one. 7.45 has two decimal places. The question asks for one. Go to M3.3.5. Try another.

    2. Not this one. You looked at the 9 and rounded up. Only the digit right after the place you keep counts. That digit is 4, so keep the 4. Go to M3.3.5. Try another.

    3. Yes, that is right. Keep one decimal place: 7.4. The next digit is 4, which is 4 or less, so the 4 stays.

    See the answer

    7.4

    Keep one decimal place: 7.4. The next digit is 4, which is 4 or less, so the 4 stays.

    • If you chose A: 7.45 has two decimal places. The question asks for one. Go to M3.3.5.
    • If you chose B: You looked at the 9 and rounded up. Only the digit right after the place you keep counts. That digit is 4, so keep the 4. Go to M3.3.5.

  7. Add. 6.8 + 0.45

    See the answer

    7.25

    Write 6.8 as 6.80, line up the points and add each column. Hundredths: 0 + 5 = 5. Tenths: 8 + 4 = 12, so write 2 and carry 1. Ones: 6 + 0 + 1 = 7.

  8. Work out 4.37 × 1 000.

    See the answer

    4 370

    1 000 has 3 zeros, so move every digit 3 places left. The 4 moves to the thousands, and a 0 fills the ones place.

  9. From an earlier chapter

    In 5.206, what is the 2 worth? Write it as a decimal.

    See the answer

    0.2

    The 2 is in the tenths place. 2 tenths is 2 ÷ 10 = 0.2.

Keep it

Copy this into your exercise book.

Line up the decimal points. Fill empty places with 0. × 10 moves digits left, ÷ 10 moves them right. To round, look at the next digit: 5 or more rounds up.

Draw: A chart with ones, tenths and hundredths. Write 2.50 above 0.75 with the points in a line, and add.

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.