Multiplying Numbers
Multiplying finds the total of equal groups. This NSW Year 7 maths subtopic consolidates primary-school skills, so there is no Stage 4 outcome of its own; it sharpens the mental strategies and the written method you need for everything that follows.
You will split numbers using the distributive law (\(47 \times 99 = 47 \times 100 - 47\)), double and halve, multiply by 4, 5, 8 and 9 in your head, set out long multiplication with a placeholder zero, and estimate first to check the answer. Counting stadium seats or pricing excursion tickets are everyday examples. It is part of Year 7 Whole Numbers.
Every question with a fully worked solution.
- Multiplying Numbers - Video - How to multiply ANY numbers the fast way Watch
Theory
Multiplying finds the total of equal groups. Split, double or adjust the numbers to multiply in your head, and use the written method for bigger numbers. This Year 7 maths page shows the mental strategies, the distributive law, the area model and the written method with its placeholder zero.
Split into tens and ones, multiply each part, then add. This is the distributive law: \(7 \times 43 = 7 \times 40 + 7 \times 3 = 280 + 21 = 301\).
Double. \(\times 2\) is double once, \(\times 4\) is double twice, \(\times 8\) is double three times (\(8 = 2 \times 2 \times 2\)).
Halve. \(\times 5\) is \(\times 10\), then halve (or halve, then \(\times 10\)): \(5 = \dfrac{1}{2} \times 10\). Double and halve: halve one number and double the other; the answer stays the same: \(14 \times 25 = 7 \times 50 = 350\).
Adjust. \(\times 9\) is ten lots take away one lot: \(17 \times 9 = 170 - 17 = 153\). In the same way \(\times 99\) is \(\times 100\) take away one lot, \(\times 101\) is \(\times 100\) add one lot, and \(\times 102\) is \(\times 100\) add two lots.
\(\times 10\) puts one zero on the end of a whole number; \(\times 100\) puts two zeros on the end.
Written method. Estimate first. Multiply by the ones digit, carrying any tens to the next column. Then multiply by the tens: that row starts with a placeholder zero. Add the rows and compare with the estimate.
Mental strategies:
| Multiplying by | Method |
|---|---|
| \(2\) | double once |
| \(4\) | double twice |
| \(8\) | double three times |
| \(5\) | halve, then \(\times 10\) |
| \(9\) | \(\times 10\), then take away the number |
| \(10\) | put \(0\) on the end |
| \(100\) | put \(00\) on the end |
| \(99\) / \(101\) | \(\times 100\), then take away / add the number |
How to use the written method (2-digit multiplier)
- Estimate by rounding each number to its leading digit.
- Multiply by the ones digit, carrying tens.
- Write a placeholder zero, then multiply by the tens digit.
- Add the two rows and compare with the estimate.
(i) \(43 \times 2\)
(ii) \(43 \times 4\)
(iii) \(43 \times 8\)
| \(43 \times 2\) | \(=\) | \(86\) |
| \(43 \times 2 \times 2\) | \(=\) | \(86 \times 2 = 172\) |
| \(43 \times 2 \times 2 \times 2\) | \(=\) | \(172 \times 2 = 344\) |
(i) \(86\)
(ii) \(172\)
(iii) \(344\)
(i) \(34 \times 5\)
(ii) \(26 \times 9\)
(iii) \(56 \times 10\)
(iv) \(78 \times 100\)
| \(34 \times \dfrac{1}{2} \times 10\) | \(=\) | \(17 \times 10 = 170\) |
| \(26 \times 10 - 26\) | \(=\) | \(260 - 26 = 234\) |
(i) \(170\)
(ii) \(234\) (take away one lot of \(26\), not \(9\))
(iii) \(560\)
(iv) \(7800\)
| \(8 \times 50 + 8 \times 7\) | \(=\) | \(400 + 56\) |
| \(\) | \(=\) | \(456\) |
\(456\)
(i) \(18 \times 50\)
(ii) \(38 \times 99\)
(iii) \(27 \times 101\)
| \(9 \times 100\) | \(=\) | \(900\) |
| \(3800 - 38\) | \(=\) | \(3762\) |
| \(2700 + 27\) | \(=\) | \(2727\) |
(i) Halve \(18\), double \(50\): \(900\)
(ii) \(38 \times 100 - 38 \times 1 = 3762\)
(iii) \(27 \times 100 + 27 \times 1 = 2727\)
| \(58 \times 3\) | \(=\) | \(174\) |
| \(58 \times 40\) | \(=\) | \(2320\) |
| \(174 + 2320\) | \(=\) | \(2494\) |
Estimate \(60 \times 40 = 2400\). \(58 \times 43 = 2494\).
(i) What is the total cost?
(ii) The P&C pays \(\$2000\). How much do the families pay?
| \(185 \times 2\) | \(=\) | \(370\) |
| \(185 \times 40\) | \(=\) | \(7400\) |
| \(370 + 7400\) | \(=\) | \(7770\) |
| \(7770 - 2000\) | \(=\) | \(5770\) |
(i) \(\$7770\) (estimate \(200 \times 40 = 8000\))
(ii) \(\$5770\)
| \(92 \times 75\) | \(=\) | \(6900\) |
| \(95 \times 72\) | \(=\) | \(6840\) |
Put the biggest digits (\(9\) and \(7\)) in the tens places, then try both ways: \(92 \times 75 = 6900\) is the largest.
Common pitfalls
Frequently asked questions
How do you multiply by 4 in your head?
Double twice: \(26 \times 4 = 52 \times 2 = 104\).
How do you multiply by 9 quickly?
Multiply by \(10\), then take away the number: \(17 \times 9 = 170 - 17 = 153\).
What is the distributive law?
Splitting a number and multiplying each part: \(7 \times 43 = 7 \times 40 + 7 \times 3\).
Why is there a zero in the second row?
Because you are multiplying by tens. The zero holds the ones place.
How do you multiply by 5?
Multiply by \(10\) and halve: \(34 \times 5 = 170\).