Multiplying and Dividing Decimals
Multiplying and dividing decimals is two jobs in NSW Year 8 maths: settle the sign with the integer rule, then work out the size (NESA outcome MA4‑FRC‑C‑01). When multiplying, the answer has as many decimal places as the two numbers together, so \(0.06 \times 0.8 = 0.048\).
You will divide by a decimal by first making the divisor a whole number, as in \(23.5 \div 0.05 = 2350 \div 5\), find powers of decimals such as \((-0.2)^3\) and use the order of operations. You will also find missing decimals and count how many whole pieces fit. Changing money for a New Zealand trip, the cost of running a heater and petrol for a drive to Canberra all multiply or divide decimals. It is part of Year 8 Working with Numbers.
Theory
Multiplying or dividing decimals is two jobs: work out the sign (same signs positive, different signs negative), then the size. To multiply, ignore the points, multiply, then count the decimal places. To divide, first make the divisor a whole number. This Year 8 maths page shows how.
Multiplying: count the places. Ignore the decimal points and multiply the digits. The answer has as many decimal places as the two numbers have together: \(5.2 \times 0.6\) comes from \(52 \times 6 = 312\) with \(1 + 1 = 2\) places, so it is \(3.12\).
The sign. Same signs give a positive answer; different signs give a negative answer, exactly as for integers. Powers: the places add up each time: \(0.5^2 = 0.25\) and \(0.5^3 = 0.125\).
Dividing: make the divisor a whole number. Move the decimal point of the divisor to the end, and move the point of the number being divided the same number of places (multiply both by \(10\), \(100\) or \(1000\)): \(41.3 \div 0.2 = 413 \div 2\). Then divide as normal, and check by multiplying back.
Short cuts. \(\times\,0.1\) is the same as \(\div\,10\), and \(\times\,0.01\) is the same as \(\div\,100\). There are \(2\) halves in every whole, so \(\div\,0.5\) is the same as \(\times\,2\), and \(\div\,0.25\) is the same as \(\times\,4\).
Split a hard multiplier: \(3.8 \times 2.5 = 3.8 \times 2 + 3.8 \times 0.5 = 7.6 + 1.9 = 9.5\). Whole pieces only: \(2.9 \div 0.6 = 4.83\ldots\), so \(4\) whole pieces fit, using \(2.4\) and leaving \(0.5\).
Rates. Speed \(=\) distance \(\div\) time; cost \(=\) quantity \(\times\) price for one; a conversion multiplies one way and divides the other. Do \(\times\) and \(\div\) before \(+\) and \(-\).
Size check. Multiplying a positive number by something between \(0\) and \(1\) makes it smaller; dividing by something between \(0\) and \(1\) makes it bigger: \(0.9 \div 0.03 = 30\).
Multiply the digits, then count the places:
| Work out | Points ignored | Places | Answer |
|---|---|---|---|
| \(1.2 \times 5\) | \(12 \times 5 = 60\) | \(1\) | \(6\) |
| \(0.05 \times 0.6\) | \(5 \times 6 = 30\) | \(3\) | \(0.03\) |
| \(1.1^3\) | \(11^3 = 1331\) | \(3\) | \(1.331\) |
| \((-2.5)^2\) | \(25^2 = 625\) | \(2\) | \(6.25\) |
How to multiply decimals
- Decide the sign.
- Multiply the digits with the points ignored.
- Count the decimal places in both numbers and put that many in the answer.
How to divide by a decimal
- Decide the sign.
- Multiply both numbers by \(10\), \(100\) or \(1000\) so the divisor is whole.
- Divide, then check by multiplying back.
| \(52 \times 6\) | \(=\) | \(312\) |
| \(5.2 \times 0.6\) | \(=\) | \(3.12\) |
There are \(2\) decimal places in total, so \(5.2 \times 0.6 = 3.12\).
| \(41.3 \div 0.2\) | \(=\) | \(413 \div 2\) |
| \(\) | \(=\) | \(206.5\) |
\(206.5\). Check: \(206.5 \times 0.2 = 41.3\).
(i) \(-0.6 \times 0.05\)
(ii) \(-1.5 \times (-0.8)\)
(iii) \((-0.7)^2\)
| \(6 \times 5\) | \(=\) | \(30\) |
| \(15 \times 8\) | \(=\) | \(120\) |
| \(7 \times 7\) | \(=\) | \(49\) |
(i) Different signs, \(3\) places: \(-0.030 = -0.03\)
(ii) Same signs, \(2\) places: \(1.20 = 1.2\)
(iii) Even index, \(2\) places: \(0.49\)
(i) \(-4.8 \div 0.08\)
(ii) \(0.96 \div (-0.004)\)
| \(4.8 \div 0.08\) | \(=\) | \(480 \div 8 = 60\) |
| \(0.96 \div 0.004\) | \(=\) | \(960 \div 4 = 240\) |
(i) \(-60\)
(ii) \(-240\). Check: \(240 \times 0.004 = 0.96\).
| \(2.4 \times 60\) | \(=\) | \(144 \text{ minutes}\) |
| \(144 \times 4.5\) | \(=\) | \(144 \times 4 + 144 \times 0.5\) |
| \(\) | \(=\) | \(576 + 72 = 648\) |
\(648\) cents \(= \$6.48\). Estimate: \(150 \times 4 = 600\) cents.
(ii) One Australian dollar buys \(1.25\) units of a foreign currency. How many units do \(80\) Australian dollars buy, and how many Australian dollars is \(45\) units worth?
| \(9.6 \div 0.8\) | \(=\) | \(96 \div 8 = 12\) |
| \(80 \times 1.25\) | \(=\) | \(100\) |
| \(45 \div 1.25\) | \(=\) | \(4500 \div 125 = 36\) |
(i) \(12\) km/h
(ii) \(100\) units; \(45\) units is worth \(\$36\).
(ii) Show that \(8.6 \div 0.5\) gives the same answer as \(8.6 \times 2\).
| \({?}\) | \(=\) | \(1.5 \times (-0.06) = -0.09\) |
| \(8.6 \div 0.5\) | \(=\) | \(86 \div 5 = 17.2\) |
| \(8.6 \times 2\) | \(=\) | \(17.2\) |
(i) \(-0.09\). Check: \(-0.09 \div (-0.06) = 9 \div 6 = 1.5\).
(ii) Both give \(17.2\): there are \(2\) halves in every whole.
Common pitfalls
Frequently asked questions
How do you multiply decimals?
Multiply the digits without the points, then give the answer as many decimal places as the two numbers have together: \(0.3 \times 0.4 = 0.12\).
How do you divide by a decimal?
Multiply both numbers by \(10\), \(100\) or \(1000\) to make the divisor whole: \(4.2 \div 0.6 = 42 \div 6 = 7\).
Why does dividing by 0.5 double a number?
There are two halves in every whole, so dividing by one half is the same as multiplying by 2.
Is a negative decimal times a negative decimal positive?
Yes. Same signs give a positive: \(-0.2 \times (-0.3) = 0.06\).
Does multiplying always make a number bigger?
No. Multiplying a positive number by a decimal between 0 and 1 makes it smaller.