# Working with Decimals

> Add, subtract, multiply and divide decimals, including tiny numbers like 0.001, and find missing numbers.

From Selective School Maths on WhizzNotes, free study notes for NSW students: https://whizznotes.com/selective-school-maths/decimals/. Updated 3 November 2026.
Year 11 and Year 12 boxes go deeper than the Year 10 text around them.

### Story time: The stopwatch race
At the school swimming carnival, Oliver swims 50 m in 41.25 seconds. His friend Tom swims it in 39.8 seconds.

"I only lost by about 1 second," says Oliver. Tom's mum, the timekeeper, writes it out properly. She lines up the decimal points:

41.25 minus 39.80 is 1.45 seconds.

Swimming races are often won by a few hundredths of a second, so timekeepers have to be very careful with decimals.

**In short:** To add or subtract decimals, line up the decimal points. To multiply, ignore the points, multiply, then give the answer as many decimal places as the question has in total, so $0.3 \times 0.2 = 0.06$.

## Before you start

- **Place value after the point:** in $3.465$ the 4 is 4 tenths, the 6 is 6 hundredths and the 5 is 5 thousandths.
- $0.1 = \frac{1}{10}$, $0.01 = \frac{1}{100}$, $0.001 = \frac{1}{1000}$.
- Multiplying and dividing by 10, 100 and 1000 (see the previous chapter).

## Adding and subtracting: line up the points

### Key idea: Points in a line
Write the numbers in a column with the **decimal points lined up**. Fill empty places with zeros so both numbers have the same number of decimal places. Then add or subtract as usual.

### Worked example: Fill the gaps with zeros
$5 - 1.125$: write 5 as $5.000$.

$5.000 - 1.125 = 3.875$

$4.225 - 0.009 = 4.216$

$11.09 - 4.569$: write $11.090 - 4.569 = 6.521$

### Watch out: Line up the points, not the right-hand ends
For $2.009 - 0.01$, do not line up the 9 and the 1. Write $2.009 - 0.010 = 1.999$.

## Missing numbers

Use the opposite operation.

### Worked example: Find the box
$1 - \square = 0.499$. The box is $1 - 0.499 = 0.501$.

$\square - 0.075 = 2.234$. The box is $2.234 + 0.075 = 2.309$.

$6 - \square = 3.259$. The box is $6 - 3.259 = 2.741$.

## Multiplying decimals

### Key idea: Ignore the points, then put them back
1. Multiply the numbers as if there were no decimal points.
2. Count the **total** number of decimal places in the question.
3. Give the answer that many decimal places.

### Worked example: Counting places
$0.11 \times 0.9$: $11 \times 9 = 99$. There are $2 + 1 = 3$ decimal places. Answer: $0.099$.

$1.2 \times 0.98$: $12 \times 98 = 1176$. There are 3 decimal places. Answer: $1.176$.

$0.1 \times 0.01 \times 0.001$: $1 \times 1 \times 1 = 1$, with $1 + 2 + 3 = 6$ places. Answer: $0.000001$.

### Exam tip: Multiplying by 0.1, 0.01, 0.001
$\times 0.1$ is the same as $\div 10$. $\times 0.01$ is $\div 100$. $\times 0.001$ is $\div 1000$.

$32.65 \times 0.01 = 0.3265$ and $250 \times 0.001 = 0.25$.

Also, $\times 0.5$ is the same as halving.

## Dividing decimals

Dividing by 10, 100 or 1000 moves the digits right, so the number gets smaller.

$5 \div 10 = 0.5$ and $2.25 \div 20 = 2.25 \div 2 \div 10 = 1.125 \div 10 = 0.1125$.

## Thousandths in real life

$0.001$ is one thousandth. Lots of metric units use it:

| Fact | Because |
|---|---|
| $1 \text{ mm} = 0.001 \text{ m}$ | 1000 mm make 1 m |
| $0.001 \text{ km} = 1 \text{ m}$ | 1000 m make 1 km |
| $0.001 \text{ L} = 1 \text{ mL}$ | 1000 mL make 1 L |

### Challenge: Selective level: halfway and missing factors
"Find the number halfway between 12.37 and 8.31." Add them and halve: $\frac{12.37 + 8.31}{2} = \frac{20.68}{2} = 10.34$.

$\square \times 0.5 = 3.5$: multiplying by 0.5 halves a number, so the box is $7$.

$0.001 \times \square = 0.1$: to go from a thousandth to a tenth you multiply by 100.

**Mind map**

```text
Decimals
  Add and subtract
    Line up the points
    Fill gaps with zeros
  Multiply
    Ignore the points
    Count all decimal places
    $0.11 \times 0.9 = 0.099$
  Shortcuts
    x 0.1 means ÷ 10
    x 0.5 means halve
  Missing numbers
    Use the opposite operation
  Thousandths
    1 mm = 0.001 m
```

## Quick summary

- To add or subtract, line up the decimal points and fill gaps with zeros.
- To multiply, ignore the points, multiply, then give the answer as many decimal places as the question has in total.
- $\times 0.1$, $\times 0.01$ and $\times 0.001$ are the same as $\div 10$, $\div 100$ and $\div 1000$.
- Find a missing number with the opposite operation.
- Halfway between two numbers: add them and halve.

## Quiz

Try these questions from the chapter quiz.

1. $2.05 + 1.75 = $
   - A. 3.70
   - B. 3.8
   - C. 3.08
   - D. 2.80
   - Answer: B. 3.8. Line up the points: $2.05 + 1.75 = 3.80$.

2. $0.001 \times 1000 = $
   - A. 0.1
   - B. 1
   - C. 1000
   - D. 0.000001
   - Answer: B. 1. A thousandth times a thousand is one whole.

3. $2.05 - 1.75 = $
   - A. 1.3
   - B. 0.3
   - C. 0.03
   - D. 3.8
   - Answer: B. 0.3. $2.05 - 1.75 = 0.30$.

The whole chapter quiz is on the page, for signed-in students: https://whizznotes.com/selective-school-maths/decimals/#quiz