What you'll learn
A decimal is a way of writing numbers smaller than one using the same place value columns as whole numbers, continued past a decimal point: in 4.75 the 7 is worth 7 tenths and the 5 is worth 5 hundredths. This topic covers reading and writing decimals, comparing and ordering them, rounding, and adding, subtracting, multiplying and dividing them. Decimals matter more than their share of the paper suggests, because every money question on the SEA Mathematics paper is a decimal question in disguise — $4.50 is a decimal with a dollar sign in front of it.
Key terms and definitions
Decimal point — the dot that separates the whole number part from the part smaller than one.
Tenths — the first column after the decimal point. In 0.7 there are 7 tenths.
Hundredths — the second column after the decimal point. In 0.45 there are 4 tenths and 5 hundredths, or 45 hundredths altogether.
Thousandths — the third column after the decimal point.
Place value — the worth of a digit because of the column it sits in. The columns keep dividing by ten as you move right.
Expanded notation — the number written as the sum of its parts: 3.09 = 3 + 0.09.
Rounding — replacing a decimal with a simpler nearby number, to the nearest whole number or the nearest tenth.
Equivalent decimal — the same value written with extra zeros on the end: 0.5 and 0.50 are equal.
Core concepts
The columns continue past the point
Whole number columns get ten times bigger as you move left. Decimal columns get ten times smaller as you move right. The decimal point simply marks where the whole numbers stop.
| Ones | . | Tenths | Hundredths | Thousandths |
|---|---|---|---|---|
| 4 | . | 7 | 5 | 0 |
So 4.75 is 4 ones, 7 tenths and 5 hundredths. Read aloud it is "four point seven five", but its meaning is four and seventy-five hundredths.
There is no "oneths" column. The column immediately right of the point is tenths, and forgetting this is the root of most decimal errors.
Decimals and fractions are the same idea
One decimal place means tenths, so 0.6 = 6/10. Two decimal places means hundredths, so 0.45 = 45/100, which simplifies to 9/20.
Going the other way, 3/4 becomes 0.75 because three quarters is seventy-five hundredths. The conversions worth learning by heart are 1/2 = 0.5, 1/4 = 0.25, 3/4 = 0.75, 1/10 = 0.1 and 1/5 = 0.2.
Comparing and ordering decimals
Compare column by column, starting at the tenths, exactly as you would with whole numbers starting from the left.
Is 0.7 or 0.65 larger? Look at the tenths: 7 against 6. So 0.7 is larger, even though 65 looks like a bigger number than 7.
A safe method is to give every decimal the same number of places by adding zeros, which does not change any value: 0.7 becomes 0.70, and 70 hundredths clearly beats 65 hundredths.
The instinct that "more digits means a bigger number" is true for whole numbers and false for decimals. This is the single most common misunderstanding in the topic.
Adding and subtracting
Line up the decimal points, one directly under the other, and fill any gaps with zeros. Then add or subtract as usual, keeping the point in the same place in the answer.
To work out 5 − 2.35, write the 5 as 5.00 first. Without that step the columns do not match and the answer comes out wrong.
Lining up the last digits instead of the points is a frequent error: adding 12.5 and 7.05 by matching the 5 and the 5 gives nonsense.
Multiplying
Ignore the decimal point, multiply as whole numbers, then count the decimal places in the question and put that many back into the answer.
3.6 × 4: work out 36 × 4 = 144. The question has one decimal place, so the answer has one: 14.4.
0.7 × 0.3: work out 7 × 3 = 21. The question has two decimal places altogether, so the answer has two: 0.21.
Multiplying by a number smaller than 1 makes the answer smaller, which surprises children who have learned that multiplication makes things bigger.
Dividing
Dividing by a whole number keeps the decimal point in line. For 9.6 ÷ 8, note that 96 ÷ 8 = 12, so 9.6 ÷ 8 = 1.2 — the same digits with the point restored.
Dividing by a decimal asks how many of the small pieces fit inside. 4.5 ÷ 0.25 asks how many quarter-litres fit into four and a half litres; each whole litre holds 4, so 4 litres hold 16 and the extra half litre holds 2, giving 18.
Rounding decimals
To round to the nearest whole number, look at the tenths digit. To round to the nearest tenth, look at the hundredths digit. In both cases, 5 or more rounds up and 4 or less stays the same.
12.36 to the nearest tenth: the hundredths digit is 6, so the tenths digit rises from 3 to 4, giving 12.4. 8.47 to the nearest whole number: the tenths digit is 4, so the 8 stays, giving 8.
Only one digit is ever consulted — the one immediately to the right of the place being rounded.
Worked examples
Example 1: Money as a decimal
Riya buys a bottle of water for $4.75 and a bake for $6.50. How much does she spend?
Line up the decimal points and add. The cents give 75 + 50 = 125 cents, which is $1.25, so one whole dollar carries into the dollars column.
$4.75 + $6.50 = $11.25.
The wrong answer $10.25 comes from adding the dollars and cents separately and forgetting to carry the extra dollar.
Example 2: Multiplying with a decimal
A maxi taxi fare is $3.50 per person. What do 6 passengers pay altogether?
Split it up: six lots of $3 is $18, and six lots of 50 cents is $3. Adding gives $21.00.
Checking by the general method: 350 × 6 = 2 100, and the question has two decimal places, so the answer is 21.00. The two methods agree, which is a useful habit to build.
Example 3: A two-step decimal problem
A plank is 3.5 m long. Devon cuts off 0.9 m, then cuts the rest into 2 equal pieces. How long is each piece?
First the subtraction: 3.5 − 0.9. Counting up from 0.9 to 1.0 is 0.1, and from 1.0 to 3.5 is 2.5, so the answer is 2.6 m.
Then halve it: 2.6 ÷ 2 = 1.3 m.
Halving the plank before removing the 0.9 m would give 1.75 m, which is the wrong order and a wrong answer. Follow the order the question gives.
Common mistakes and how to avoid them
Thinking more digits means a bigger decimal. Choosing 0.65 over 0.7. Fix: add zeros so both have the same number of places, then compare.
Lining up the last digits when adding. Fix: the decimal points go one under the other, and gaps are filled with zeros.
Forgetting to write 5 as 5.00 before subtracting. Fix: give the whole number as many decimal places as the number being taken away.
Losing the decimal point after multiplying. Writing 3.6 × 4 = 144. Fix: count the decimal places in the question and put the same number back.
Reading the first column after the point as ones. Fix: it is tenths. There is no oneths column.
Rounding in stages. Rounding 8.47 to the nearest whole number by first making it 8.5. Fix: look only at the tenths digit, which is 4, so the answer is 8.
How parents can help at home
Decimals are money, and money is everywhere, so this is one of the easiest topics to support without any special knowledge.
Read prices out properly. Ask your child to say $9.05 aloud — "nine dollars and five cents" — and then write it. The zero in the tenths column is what stops it being nine dollars and fifty cents, and hearing the difference fixes it faster than any rule.
Ask the comparing question at the shelf. "Which is cheaper, $0.70 or $0.65?" The wrong instinct is strong, and being asked repeatedly in a real setting is what wears it down.
Use a tape measure. Lengths in metres give decimals with a purpose: 1.38 m is easier to feel than 1.38 on a page. Ask your child to measure something and then round it to the nearest tenth of a metre.
Check the estimate, not the working. If your child says three roti at $18.50 cost $555, you do not need to check the arithmetic to know it is wrong. Ask "roughly how much should that be?" and let them catch it. Estimating first is the habit that catches misplaced decimal points.
Exam technique for decimals
Write money with two decimal places always. An answer of "$6.5" invites a mark to be lost where "$6.50" does not.
Estimate before calculating. Round each decimal to the nearest whole number, do the sum roughly in your head, and check the real answer sits near it. This catches answers that are ten or a hundred times too big, which is exactly what a misplaced point produces.
When comparing or ordering decimals, write them all with the same number of places before you decide. It takes seconds and removes the guesswork.
In multiple-choice items the distractors are usually the same digits with the point in a different place — 0.12, 1.2, 12 and 120. Getting the digits right is only half the question, so check the size of your answer as carefully as its digits.
Quick revision summary
- The columns after the decimal point are tenths, hundredths, thousandths — each ten times smaller than the last.
- One decimal place means tenths, two means hundredths: 0.6 = 6/10 and 0.45 = 45/100.
- Learn by heart: 1/2 = 0.5, 1/4 = 0.25, 3/4 = 0.75, 1/5 = 0.2.
- Compare from the tenths column; adding zeros on the end changes nothing, so 0.7 = 0.70.
- More digits does not mean a bigger decimal.
- Line up the decimal points when adding or subtracting, and write 5 as 5.00 when needed.
- To multiply, ignore the point, multiply, then put back as many places as the question had.
- Multiplying by a number less than 1 makes the answer smaller.
- To round, look at exactly one digit — the one immediately to the right of the place named.
- Money is a decimal: always two places after the point.