Understanding Decimals

What are Decimals?

Decimals are a way of writing numbers that are not whole numbers. They use a decimal point to separate the whole number part from the fractional part.

The digits to the right of the decimal point represent parts of a whole, based on powers of ten:

Example: In the number 12.345:

Visualizing Decimals on a Number Line

A number line can help visualize decimals. For example, between 0 and 1:

Decimal Operations

Adding Decimals

To add decimals, line up the decimal points. Then add as you would with whole numbers, bringing the decimal point straight down in your answer.

Example: Calculate 12.34 + 5.6.

12.34 + 5.60 (add a zero to align) ------- 17.94

Workings:

  1. Line up decimal points:
    12.34
    + 5.6
  2. Add a trailing zero to 5.6 to make alignment clearer: 5.60.
  3. Add column by column from the right:
    • Hundredths: 4 + 0 = 4.
    • Tenths: 3 + 6 = 9.
    • Bring down the decimal point.
    • Units: 2 + 5 = 7.
    • Tens: 1 + 0 = 1.
  4. Result: 17.94.

Subtracting Decimals

To subtract decimals, line up the decimal points. Add trailing zeros if necessary so both numbers have the same number of decimal places. Then subtract as with whole numbers, bringing the decimal point straight down.

Example: Calculate 23.5 - 7.82.

23.50 (add a zero) - 7.82 ------- 15.68

Workings:

  1. Line up decimal points:
    23.5
    - 7.82
  2. Add a trailing zero to 23.5: 23.50.
  3. Subtract column by column from the right, borrowing where necessary:
    • Hundredths: 0 - 2. Cannot do. Borrow 1 from the tenths (5 becomes 4), so 0 becomes 10. 10 - 2 = 8.
    • Tenths: Now 4 - 8. Cannot do. Borrow 1 from the units (3 becomes 2), so 4 becomes 14. 14 - 8 = 6.
    • Bring down the decimal point.
    • Units: Now 2 - 7. Cannot do. Borrow 1 from the tens (2 becomes 1), so 2 becomes 12. 12 - 7 = 5.
    • Tens: 1 - 0 = 1.
  4. Result: 15.68.

Multiplying Decimals

To multiply decimals:

  1. Multiply the numbers as if they were whole numbers (ignore the decimal points initially).
  2. Count the total number of decimal places in the original numbers.
  3. Place the decimal point in the product so it has the same number of decimal places as you counted in step 2.

Example: Calculate 3.77 × 2.8.

Workings:

  1. Multiply 377 by 28 (ignoring decimals):
    377 × 28 ----- 3016 (377 × 8) 7540 (377 × 20) ----- 10556
  2. Count decimal places: 3.77 has 2 decimal places. 2.8 has 1 decimal place. Total = 2 + 1 = 3 decimal places.
  3. Place the decimal point in the product (10556) so it has 3 decimal places: 10.556.

Answer: 10.556

Dividing Decimals

a) Dividing a decimal by a whole number:

Place the decimal point in the answer directly above the decimal point in the number being divided. Then divide as usual.

Example: Calculate 12.48 ÷ 4.

3.12 ____ 4|12.48 12 --- 04 4 -- 08 8 -- 0

Answer: 3.12

b) Dividing by a decimal:

Make the divisor (the number you are dividing by) a whole number by moving its decimal point to the right. Move the decimal point in the dividend (the number being divided) the same number of places to the right (adding zeros if necessary). Then divide as usual.

Example: Calculate 15.6 ÷ 0.3.

Workings:

  1. Make the divisor (0.3) a whole number: move the decimal 1 place to the right to get 3.
  2. Move the decimal in the dividend (15.6) 1 place to the right: 156.
  3. The problem becomes 156 ÷ 3.
  4. Perform the division:
    52 ___ 3|156 15 --- 06 6 -- 0

Answer: 52

Decimals and Fractions

Converting Decimals to Fractions

To convert a decimal to a fraction:

  1. Write the decimal as a fraction with the decimal digits as the numerator.
  2. The denominator is 1 followed by as many zeros as there are decimal places.
  3. Simplify the fraction to its lowest terms.

Example 1: Convert 0.75 to a fraction.

0.75 has two decimal places, so the denominator is 100. Numerator is 75. Fraction is 75/100.

Simplify 75/100 (HCF is 25): (75÷25)/(100÷25) = 3/4.

Answer: 3/4

Example 2: Convert 0.6 to a fraction.

0.6 has one decimal place, so denominator is 10. Numerator is 6. Fraction is 6/10.

Simplify 6/10 (HCF is 2): (6÷2)/(10÷2) = 3/5.

Answer: 3/5

Converting Fractions to Decimals

To convert a fraction to a decimal, divide the numerator by the denominator.

Example 1: Convert 3/4 to a decimal.

Divide 3 by 4: 3 ÷ 4 = 0.75.

Answer: 0.75

Example 2: Convert 2/5 to a decimal.

Divide 2 by 5: 2 ÷ 5 = 0.4.

Answer: 0.4

Exam-Style Decimal Problems

1. Shopping Bill

Sarah buys three items costing £2.45, £0.89, and £12.50. She pays with a £20 note. How much change does she receive?

Answer: £4.16

Explanation:

  1. Total cost of items:
    2.45 0.89 +12.50 ------ 15.84
    £2.45 + £0.89 + £12.50 = £15.84.
  2. Change from £20:
    20.00 - 15.84 ------- 4.16
    £20.00 - £15.84 = £4.16.

2. Fuel Consumption

A car travels 245.5 miles on 5.5 gallons of fuel. How many miles per gallon does the car achieve? (Give your answer to one decimal place).

Answer: 44.6 miles per gallon

Explanation:

  1. Miles per gallon = Total miles ÷ Total gallons = 245.5 ÷ 5.5.
  2. To make the divisor a whole number, multiply both by 10: 2455 ÷ 55.
  3. Perform the division:
    2455 ÷ 55 ≈ 44.6363...
  4. Round to one decimal place: 44.6 miles per gallon.

Test Yourself: Decimal Calculations

1. Calculate: 7.82 + 15.09 + 0.3

Answer: 23.21

Explanation: Line up the decimal points and add:

7.82 15.09 + 0.30 (add trailing zero) ------- 23.21

2. Calculate: 103.5 - 67.28

Answer: 36.22

Explanation: Line up the decimal points (add a trailing zero to 103.5) and subtract:

103.50 - 67.28 -------- 36.22

3. Calculate: 6.4 × 0.35

Answer: 2.24

Explanation:

  1. Multiply 64 by 35:
    64 × 35 ---- 320 (64 × 5) 1920 (64 × 30) ---- 2240
  2. 6.4 has 1 decimal place. 0.35 has 2 decimal places. Total = 1 + 2 = 3 decimal places.
  3. Place the decimal point in 2240 to have 3 decimal places: 2.240 or 2.24.

4. Calculate: 7.2 ÷ 0.03

Answer: 240

Explanation:

  1. Make the divisor (0.03) a whole number: move decimal 2 places right to get 3.
  2. Move decimal in dividend (7.2) 2 places right: 7.2 becomes 720 (add a zero).
  3. Problem becomes 720 ÷ 3.
  4. 7 ÷ 3 = 2 remainder 1.
  5. 12 ÷ 3 = 4.
  6. 0 ÷ 3 = 0.
  7. Result: 240.

5. Convert the fraction 5/8 to a decimal.

Answer: 0.625

Explanation: Divide the numerator by the denominator: 5 ÷ 8.

0.625 _______ 8|5.000 0 -- 5 0 (bring down 0) -4 8 ---- 20 (bring down 0) -16 --- 40 (bring down 0) -40 --- 0

Interactive Decimal Ordering Tool

Enter a list of decimal numbers (separated by commas or spaces) and choose the order.

Original: ?

Ordered: ?

Key Decimal Facts