Law of Arithmetic
Theory
You can change the order (commutative law) and the grouping (associative law) of the numbers in a sum or a product, and the answer stays the same. This Year 7 maths page shows both laws, how to use friendly pairs and multiples of ten to work faster, and why the laws do not work for subtraction and division.
The commutative law says the order does not matter when you add or multiply: \(3 + 5 = 5 + 3\), so \(a + b = b + a\); and \(3 \times 4 = 4 \times 3\), so \(a \times b = b \times a\) (or \(ab = ba\)).
The associative law says the grouping does not matter when you add or multiply three or more numbers. Brackets show which pair is worked out first: \((3 + 5) + 2 = 3 + (5 + 2)\) and \((3 \times 4) \times 2 = 3 \times (4 \times 2)\). So \((a + b) + c = a + (b + c)\) and \((a \times b) \times c = a \times (b \times c)\).
The pronumerals \(a\), \(b\) and \(c\) (letters that stand for numbers) can be any numbers, so the laws work for integers too: \(-8 + 15 = 15 + (-8) = 15 - 8 = 7\).
Both laws work for adding and multiplying only.
Friendly pairs make a round number:
| Friendly sums | Friendly products |
|---|---|
| \(38 + 2 = 40\) | \(2 \times 5 = 10\) |
| \(56 + 4 = 60\) | \(4 \times 25 = 100\) |
| \(62 + 8 = 70\) | \(5 \times 8 = 40\) |
| \(73 + 27 = 100\) | \(2 \times 50 = 100\) |
How to use the laws to work faster
- Look for friendly pairs that make \(10\), \(50\), \(100\), ...
- Re-order the numbers so the pairs sit together (commutative law).
- Group each pair with brackets (associative law).
- Work out the brackets, then the rest.
(i) \(43 + 8 + 7\)
(ii) \(-15 + 40\)
| \(43 + 8 + 7\) | \(=\) | \((43 + 7) + 8\) |
| \(\) | \(=\) | \(50 + 8 = 58\) |
| \(-15 + 40\) | \(=\) | \(40 + (-15)\) |
| \(\) | \(=\) | \(40 - 15 = 25\) |
(i) \(58\)
(ii) \(25\)
(i) \(8 \times 40\)
(ii) \(30 \times 70\)
| \(8 \times 40\) | \(=\) | \((8 \times 4) \times 10\) |
| \(\) | \(=\) | \(32 \times 10 = 320\) |
| \(30 \times 70\) | \(=\) | \((3 \times 7) \times (10 \times 10)\) |
| \(\) | \(=\) | \(21 \times 100 = 2100\) |
(i) \(320\)
(ii) \(2100\)
Re-order so the friendly pair sits together: \(22 + 5 + 78 + 16 = 22 + 78 + 5 + 16\). Then group the pairs.
| \((22 + 78) + (5 + 16)\) | \(=\) | \(100 + 21\) |
| \(\) | \(=\) | \(121\) |
The friendly pair is \(22 + 78 = 100\). The answer is \(121\).
| \(25 \times 17 \times 4\) | \(=\) | \(25 \times 4 \times 17\) |
| \(\) | \(=\) | \((25 \times 4) \times 17\) |
| \(\) | \(=\) | \(100 \times 17\) |
| \(\) | \(=\) | \(1700\) |
\(1700\)
(ii) Evaluate \(2 \times 13 \times 50 \times 3\).
(i) The friendly pair is \(25 \times 4 = 100\).
(ii) The friendly pair is \(2 \times 50 = 100\).
| \((25 \times 4) \times 9\) | \(=\) | \(900\) |
| \((2 \times 50) \times (13 \times 3)\) | \(=\) | \(3900\) |
(i) \(900\) muffins
(ii) \(100 \times 39 = 3900\)
| \(10 - 4\) | \(=\) | \(6\) |
| \(4 - 10\) | \(=\) | \(-6\) |
| \((20 - 8) - 2\) | \(=\) | \(10\) |
| \(20 - (8 - 2)\) | \(=\) | \(14\) |
| \(12 \div 2\) | \(=\) | \(6\) |
Changing the order or the grouping changes the answer. Also \(12 \div 2 = 6\) but \(2 \div 12 = \dfrac{1}{6}\).
Common pitfalls
Frequently asked questions
What is the commutative law?
The order does not matter when adding or multiplying: \(a + b = b + a\) and \(ab = ba\).
What is the associative law?
The grouping does not matter when adding or multiplying: \((a + b) + c = a + (b + c)\).
Does the commutative law work for subtraction?
No. \(10 - 4 = 6\) but \(4 - 10 = -6\).
What are friendly pairs?
Numbers that add or multiply to a round number, such as \(73 + 27 = 100\) or \(4 \times 25 = 100\).
How do you multiply by a multiple of ten in your head?
Split it into factors: \(9 \times 50 = 9 \times 5 \times 10 = 450\).