Decimals

What are Decimals?

A decimal is a way of writing numbers that are not whole, using a decimal point to separate the whole number part from the fractional part. The word comes from the Latin "decem" meaning ten — the decimal system is based entirely on powers of ten.

For example, 3.742 means:

• 3 whole units
• Plus a fractional part: 7 tenths + 4 hundredths + 2 thousandths

Decimals are used constantly in real life — in money (£3.75), measurements (1.8m), and scientific data.

Decimal Place Value

Each position in a decimal number has a specific place value. Moving left from the decimal point, values get ten times larger. Moving right, they get ten times smaller.

Thousands	Hundreds	Tens	Ones	.	Tenths	Hundredths	Thousandths
1,000	100	10	1	.	1/10	1/100	1/1000

Example: In 3.742:

• 3 is in the ones place
• 7 is in the tenths place (value: 7/10)
• 4 is in the hundredths place (value: 4/100)
• 2 is in the thousandths place (value: 2/1000)

Trailing zeros after the last non-zero decimal digit do not change the value: 0.50 = 0.5.

Comparing Decimals

To compare decimals, compare digit by digit from left to right, starting with the largest place value.

Example: Compare 0.3 and 0.29

• Tenths: 3 vs 2 → 0.3 is larger

Example: Compare 0.309 and 0.35

• Tenths: 3 = 3 (same)
• Hundredths: 0 vs 5 → 0.35 is larger

A helpful trick: pad with trailing zeros so all numbers have the same number of decimal places, then compare as whole numbers:

• 0.3 → 0.300
• 0.09 → 0.090
• 0.35 → 0.350
• 0.309 → 0.309

Order: 0.090, 0.300, 0.309, 0.350 → 0.09, 0.3, 0.309, 0.35

Adding and Subtracting Decimals

The key rule: align the decimal points before adding or subtracting. Add trailing zeros as placeholders so columns line up cleanly.

Example: 2.7 + 1.45

• Write 2.7 as 2.70
• Add column by column: 0 + 5 = 5; 7 + 4 = 11 (write 1, carry 1); 2 + 1 + 1 = 4
• Answer: 4.15

Example: 5.6 - 2.38

• Write 5.6 as 5.60
• Subtract column by column (borrowing as needed)
• Answer: 3.22

Multiplying Decimals

To multiply decimals:

1. Ignore the decimal points and multiply as whole numbers.
2. Count the total number of decimal places in both numbers combined.
3. Place the decimal point that many places from the right in the answer.

Example: 0.6 × 0.4

• 6 × 4 = 24
• Total decimal places: 1 + 1 = 2
• Place decimal point 2 places from the right: 0.24

Example: 3.5 × 1.2

• 35 × 12 = 420
• Total decimal places: 1 + 1 = 2
• Answer: 4.20 = 4.2

Key rule for multiplying by powers of 10:

• × 10: move decimal point 1 place right
• × 100: move 2 places right
• × 0.1: move 1 place left (same as ÷ 10)

Dividing Decimals

To divide by a decimal, convert the divisor to a whole number by multiplying both numbers by an appropriate power of 10.

Example: 3.6 ÷ 0.4

• Multiply both by 10: 36 ÷ 4 = 9

Example: 2.4 ÷ 0.06

• Multiply both by 100: 240 ÷ 6 = 40

Key rule for dividing by powers of 10:

• ÷ 10: move decimal point 1 place left
• ÷ 100: move 2 places left

Rounding Decimals

To round a decimal to a given number of decimal places:

1. Identify the digit at the required decimal place.
2. Look at the next digit (one place to the right).
3. If it is 5 or more, round up. If it is 4 or less, round down (keep the digit as it is).

Example: Round 4.736 to 2 decimal places.

• The second decimal place is 3.
• The next digit is 6 (≥ 5), so round up.
• Answer: 4.74

Converting Between Decimals and Fractions

Decimal to fraction: Write the decimal digits over the appropriate power of 10, then simplify.

• 0.375 = 375/1000 = 3/8 (dividing numerator and denominator by 125)

Fraction to decimal: Divide the numerator by the denominator.

• 3/8: 3 ÷ 8 = 0.375

Common Mistakes

• Not aligning decimal points when adding or subtracting — this is the most common source of errors.
• Misplacing the decimal point when multiplying — count decimal places in both numbers and place accurately.
• Treating 0.3 as smaller than 0.29 — compare digit by digit; tenths digit 3 > 2, so 0.3 > 0.29.
• Forgetting trailing zeros — 0.50 = 0.5; removing trailing zeros does not change the value.
• Dividing by a decimal without converting — always make the divisor a whole number first.

Tips and Tricks

• When comparing a list of decimals, write them all with the same number of decimal places (pad with zeros) — they then compare like whole numbers.
• A quick check for multiplication: the answer should be smaller than both original numbers when both are between 0 and 1.
• For money calculations, think in pence to avoid decimal errors: £3.75 = 375p, then convert back.
• Memorise common fraction-decimal equivalents: 1/2 = 0.5, 1/4 = 0.25, 3/4 = 0.75, 1/8 = 0.125, 1/5 = 0.2.