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Decimals

Line up the decimal point to add decimals without mistakes, and divide by a decimal like 0.6 by moving the point first.

A decimal is a way to write parts of a whole with a point, like 2.5 or 0.75. You use decimals every day: a price of $3.99, a height of 1.65 meters, a race won in 9.58 seconds. They work just like whole numbers, with one extra thing to watch: the decimal point. Decimals became common in Europe after 1585, when the Flemish engineer Simon Stevin wrote a short book showing merchants how much easier they made sums. Almost every mistake with decimals is a point in the wrong place, so always know where your point is.

What the digits after the point mean

Each place to the right of the point is worth one tenth of the place before it. The first place after the point counts tenths. The second place counts hundredths.

NumberOnesTenthsHundredthsIn words
0.7070seven tenths
0.45045forty-five hundredths
3.75375three and seventy-five hundredths

Zeros at the end, after the point, change nothing. 0.7 = 0.70, because seven tenths is the same as seventy hundredths. Think of money: 70 cents is written $0.70, and it is the same as $0.7. This fact is the key to adding and subtracting.

Between 0 and 1, each long tick is one tenth and each short tick is five hundredths. 0.7 is further to the right than 0.45, so 0.7 is the bigger number, even though 45 is bigger than 7.

Adding decimals

Remember

Line up the decimal points, one under the other.

Then fill the empty places with zeros, and add as you do with whole numbers.

  1. Write the numbers in a column with the points in a straight line.
  2. Give both numbers the same number of digits after the point by adding zeros at the end.
  3. Add from the right, carrying as usual.
  4. Put the point in the answer straight under the other points.

Example 1

What is 0.7 + 0.45?

  1. Write 0.7 as 0.70, so both numbers have two digits after the point.
  2. Hundredths: 0 + 5 = 5.
  3. Tenths: 7 + 4 = 11. Write 1 and carry 1.
  4. Ones: 0 + 0 + 1 = 1.

Answer: 0.70 + 0.45 = 1.15

Example 2

What is 2.5 + 3.75?

  1. Write 2.5 as 2.50.
  2. Hundredths: 0 + 5 = 5.
  3. Tenths: 5 + 7 = 12. Write 2 and carry 1.
  4. Ones: 2 + 3 + 1 = 6.

Answer: 2.50 + 3.75 = 6.25

Subtracting decimals

Subtracting works the same way: points in a line, zeros to fill the gaps, then take away from the right, borrowing when you need to.

Example 3

What is 3.6 − 0.75?

  1. Write 3.6 as 3.60, under it 0.75, with the points in a line.
  2. Hundredths: you cannot take 5 from 0. Borrow one tenth, which is 10 hundredths: 10 − 5 = 5.
  3. Tenths: 6 became 5 after the borrow. You cannot take 7 from 5. Borrow one whole, which is 10 tenths: 15 − 7 = 8.
  4. Ones: 3 became 2: 2 − 0 = 2.

Answer: 3.60 − 0.75 = 2.85

Check a subtraction by adding back: 2.85 + 0.75 = 3.60. It matches, so the answer is right.

Multiplying decimals

Here you do not line up the points. Instead, you forget the points for a moment, multiply, and put the point back at the end.

  1. Multiply the numbers as if they had no points.
  2. Count all the digits after the point in both numbers together.
  3. Put the point in your answer so it has that many digits after the point.
  4. Only now drop any zeros at the end.

Example 4

What is 0.6 × 0.5?

  1. Without the points: 6 × 5 = 30.
  2. 0.6 has one digit after the point, and 0.5 has one. That is two in total.
  3. Two digits after the point: 0.30.
  4. Drop the end zero: 0.3.

Answer: 0.3

Example 5

What is 1.5 × 0.4?

  1. Without the points: 15 × 4 = 60.
  2. One digit after the point in each number: two in total.
  3. Two digits after the point: 0.60 = 0.6.

Answer: 0.6

Dividing by a decimal

Dividing by a decimal like 0.6 is hard. Dividing by a whole number like 6 is easy. So we change the question into an easy one that has the same answer.

Remember

Multiply both numbers by 10, or by 100, until the number you divide by is a whole number.

Moving the point the same number of places in both numbers does not change the answer.

Example 6

What is 4.8 ÷ 0.6?

  1. 0.6 has one digit after the point, so multiply both numbers by 10.
  2. 4.8 becomes 48, and 0.6 becomes 6.
  3. 48 ÷ 6 = 8.

Answer: 8

Example 7

What is 0.36 ÷ 0.03?

  1. 0.03 has two digits after the point, so multiply both numbers by 100.
  2. 0.36 becomes 36, and 0.03 becomes 3.
  3. 36 ÷ 3 = 12.

Answer: 12

Why is this fair? A division is the same as a fraction, 4.8/0.6. Multiplying the top and the bottom of a fraction by the same number keeps its value, so 4.8/0.6 = 48/6. Money shows it too. How many 5-cent coins make $2.50? In cents, that is 250 ÷ 5 = 50 coins. In dollars, it is 2.5 ÷ 0.05, and the answer is the same 50.

Try it yourself

  1. 1.What is 1.25 + 0.8?

    Show the answer

    2.05

    Write 0.8 as 0.80 and line up the points: 1.25 + 0.80 = 2.05.

  2. 2.What is 5 − 2.35?

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    2.65

    Write 5 as 5.00, then subtract with the points in a line: 5.00 − 2.35 = 2.65. Check: 2.65 + 2.35 = 5.

  3. 3.What is 0.3 × 0.4?

    Show the answer

    0.12

    3 × 4 = 12, and there are two digits after the point in total, so the answer is 0.12.

  4. 4.What is 2.4 × 0.5?

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    1.2

    24 × 5 = 120, two digits after the point give 1.20, which is 1.2. It is half of 2.4.

  5. 5.What is 6.3 ÷ 0.7?

    Show the answer

    9

    Multiply both numbers by 10: 63 ÷ 7 = 9.

  6. 6.What is 1.2 ÷ 0.04?

    Show the answer

    30

    Multiply both numbers by 100: 120 ÷ 4 = 30.

Ready to test yourself?

Play Arithmetic: 20 seconds per question, and it gets harder as you go.