Decimals
Decimals are revised in this NSW Year 9 Core maths subtopic, from Stage 4 work under NESA outcome MA4‑FRC‑C‑01, then taken a step further with significant figures, a Stage 5 way of rounding (NESA outcome MA5‑MAG‑C‑01) that scientists use to show how accurate a measurement is.
You will add, subtract, multiply and divide decimals without a calculator, including \(0.65 \times 0.32\) and \(1.855 \div 0.07\), multiply by powers of ten, round to a set number of decimal places, round to significant figures (\(48\,213\) becomes \(48\,000\) to two significant figures), write a fraction as a decimal and estimate before calculating. You will also see why rounding twice can give the wrong answer. Everyday uses include pay, temperatures and sharing a quantity equally. It is part of Year 9 Core Working with Numbers.
Theory
A decimal uses place value after the decimal point: tenths, hundredths, thousandths. This Year 9 maths page revises adding, subtracting, multiplying and dividing decimals, rounding to decimal places and to significant figures, and estimating to check an answer.
Every digit after the decimal point has a place value: tenths, then hundredths, then thousandths. Each place is ten times smaller than the one to its left.
To add or subtract, line up the decimal points and fill any gaps with zeros. To multiply two decimals, multiply the digits and then count the decimal places in the question. To divide by a decimal, multiply both numbers by \(10\), \(100\) or \(1000\) until you are dividing by a whole number.
Rounding to decimal places keeps a set number of digits after the point. Rounding to significant figures counts from the first non-zero digit, wherever it is. Zeros in front of it are only placeholders.
Write \(\approx\) for a rounded answer. Always round once, from the original number.
An estimate rounds each number to something easy. It tells you roughly what size the real answer should be.
Multiplying or dividing by a power of ten moves the point:
Dividing by a decimal: multiply both numbers by the same power of ten first:
How to work with decimals
- Estimate first: round each number to something easy and work it out in your head.
- Add or subtract with the points lined up. Multiply the digits and count the places. Divide by making the divisor whole first.
- Round once, from the full answer, to the places or significant figures asked for. Use \(\approx\).
- Check the answer against the estimate. It should be about the same size.
(i) \(12.4 + 3.75\)
(ii) \(8 - 2.36\)
(i) Line up the decimal points. Write \(12.4\) as \(12.40\) so both numbers have two places:
| \(12.40 + 3.75\) | \(=\) | \(\mathbf{16.15}\) |
(ii) Write \(8\) as \(8.00\), then subtract:
| \(8.00 - 2.36\) | \(=\) | \(\mathbf{5.64}\) |
Check (ii) by adding back: \(5.64 + 2.36 = 8.00\).
(i) \(0.7 \times 0.04\)
(ii) \(5.6 \div 0.8\)
(iii) \(0.0581 \times 100\)
(i) Multiply the digits, then count the decimal places in the question:
| \(7 \times 4\) | \(=\) | \(28\) |
| \(\text{places}\) | \(=\) | \(1 + 2 = 3\) |
| \(0.7 \times 0.04\) | \(=\) | \(\mathbf{0.028}\) |
(ii) Multiply both numbers by \(10\) so the divisor is a whole number. The answer does not change:
| \(5.6 \div 0.8\) | \(=\) | \(56 \div 8\) |
| \(\) | \(=\) | \(\mathbf{7}\) |
(iii) Multiplying by \(100\) moves the point two places to the right:
| \(0.0581 \times 100\) | \(=\) | \(\mathbf{5.81}\) |
(i) Round \(3.0472\) to two decimal places.
(ii) Round \(0.0725\) to two significant figures.
(i) Cut after the second decimal place and look at the next digit:
| \(3.04\) | \(|\) | \(72\) |
The next digit is \(7\), which is \(5\) or more, so round up:
| \(3.0472\) | \(\approx\) | \(\mathbf{3.05}\) |
(ii) The zeros in front are placeholders. The first significant figure is the \(7\). Count two figures from there and cut:
| \(0.072\) | \(|\) | \(5\) |
The next digit is \(5\), so round up:
| \(0.0725\) | \(\approx\) | \(\mathbf{0.073}\) |
(i) Four concert tickets cost \(\$18.75\) each. Estimate the total, then find it exactly.
(ii) The four friends share a \(\$49.80\) taxi fare equally. How much does each pay?
(i) Estimate by rounding to the nearest dollar: \(19 \times 4 = 76\). Now the exact total. Multiply the digits, then put back the two decimal places:
| \(1875 \times 4\) | \(=\) | \(7500\) |
| \(18.75 \times 4\) | \(=\) | \(\mathbf{75.00}\) |
The total is \(\$75.00\), close to the estimate of \(\$76\).
(ii) Divide by \(4\), keeping the decimal point in line:
| \(49.80 \div 4\) | \(=\) | \(\mathbf{12.45}\) |
Each pays \(\$12.45\). Check: \(12.45 \times 4 = 49.80\).
Common pitfalls
Frequently asked questions
How do you multiply decimals?
Ignore the decimal points and multiply the digits. Then count the decimal places in both numbers and put that many in the answer. For example, \(0.7 \times 0.04\): \(7 \times 4 = 28\), with \(3\) places, so the answer is \(0.028\).
How do you divide by a decimal?
Multiply both numbers by \(10\), \(100\) or \(1000\) until the number you are dividing by is a whole number, then divide as usual. For example, \(5.6 \div 0.8 = 56 \div 8 = 7\).
What are significant figures?
They are the digits that carry meaning, counted from the first non-zero digit. In \(0.0725\), the significant figures are \(7\), \(2\) and \(5\). The zeros in front only show where the decimal point is.
How do you round to two decimal places?
Keep two digits after the point and look at the third. If it is \(5\) or more, round the second digit up; otherwise leave it. For example, \(3.0472 \approx 3.05\).
Why should you estimate before calculating?
An estimate shows roughly what the answer should be. If your exact answer is very different, such as \(7.5\) instead of \(75\), you know a decimal point has slipped.