Year 7 Maths Algebra and Equations

Distributive Law

Q: What is the distributive law?

a(b + c) = ab + ac: multiply every part inside the brackets, then add.

Q: How do you multiply by 99 in your head?

Multiply by 100, then take away the number once. 16 × 99 = 1600 - 16 = 1584.

Q: How do you multiply by 11 in your head?

Multiply by 10, then add the number. 23 × 11 = 230 + 23 = 253.

Q: What does expanding mean?

Removing brackets by multiplying every term inside. 2(x + 5) = 2x + 10.

20 practice questions 1 video lesson Theory + worked examples

Practice 20 questions

Practice questions

Every question with a fully worked solution.

Start practising

Watch 1 video(s)

Distributive Law - Video - The Distributive Property Watch

Theory

The distributive law says: to multiply a group in brackets, multiply every part inside the brackets, then add (or subtract) the results. $a \times (b + c) = a \times b + a \times c$. This Year 7 maths page shows the law as an area model, uses it for quick mental multiplication and expands brackets with pronumerals.

Brackets hold a group together. $3 \times (20 + 7)$ means $3$ lots of the whole group $20 + 7$.

With a sum: $a \times (b + c) = a \times b + a \times c$
With a difference: $a \times (b - c) = a \times b - a \times c$

Mental maths. Split the harder number into a friendly sum or difference, then multiply each part: $18 \times 12 = 18 \times (10 + 2) = 180 + 36 = 216$ and $17 \times 9 = 17 \times (10 - 1) = 170 - 17 = 153$.

The split number can come first too: $(10 + 5) \times 6 = 60 + 30 = 90$.

With pronumerals the law is written without $\times$ signs: $a(b + c) = ab + ac$. Removing the brackets is called expanding: $2(x + 5) = 2x + 10$.

The whole area equals the two parts added: $3 \times 27 = 3 \times 20 + 3 \times 7 = 60 + 21 = 81$.

The same picture with pronumerals: $a(b + c) = ab + ac$.

Quick rules from the distributive law:

To multiply by	Example
$9 = 10 - 1$: times $10$, take away the number	$13 \times 9 = 130 - 13 = 117$
$11 = 10 + 1$: times $10$, add the number	$13 \times 11 = 130 + 13 = 143$
$8 = 10 - 2$: times $10$, take away double	$13 \times 8 = 130 - 26 = 104$
$12 = 10 + 2$: times $10$, add double	$13 \times 12 = 130 + 26 = 156$
$101 = 100 + 1$: times $100$, add the number	$13 \times 101 = 1300 + 13 = 1313$
$99 = 100 - 1$: times $100$, take away the number	$13 \times 99 = 1300 - 13 = 1287$

Multiply every part. $6 \times (10 + 3) = 60 + 18 = 78$, not $60 + 3$.

          Two-digit numbers. Split into a multiple of ten plus or minus a small number: \(23 \times 31 = 23 \times (30 + 1) = 690 + 23 = 713\).
        

How to use the distributive law

Split one number into a friendly sum or difference ($99 = 100 - 1$).
Multiply every part by the other number.
Add (or subtract) the results.
Check the size with an estimate.

Example 1 — Three parts

The straight of a running track is $6$ m wide and $125$ m long. Split $125$ into three friendly parts to find its area.

Solution

$6 \times 125$	$=$	$6 \times (100 + 20 + 5)$
	$=$	$600 + 120 + 30$
	$=$	$750$

6 \times 125 = 6 \times (100 + 20 + 5) = 600 + 120 + 30 = 750

The area is $750$ m².

Example 2 — Split into a sum

Use the distributive law to evaluate
(i) $23 \times 11$
(ii) $32 \times 101$

Solution

$23 \times 11$	$=$	$23 \times (10 + 1)$
	$=$	$230 + 23 = 253$
$32 \times 101$	$=$	$32 \times (100 + 1)$
	$=$	$3200 + 32 = 3232$

23 \times 11 = 230 + 23 = 253, 32 \times 101 = 3200 + 32 = 3232

(i) $253$
(ii) $3232$

Example 3 — Split into a difference

Use the distributive law to evaluate
(i) $7 \times 99$
(ii) $16 \times 99$

Solution

$7 \times 99$	$=$	$7 \times (100 - 1)$
	$=$	$700 - 7 = 693$
$16 \times 99$	$=$	$16 \times (100 - 1)$
	$=$	$1600 - 16 = 1584$

7 \times 99 = 700 - 7 = 693, 16 \times 99 = 1600 - 16 = 1584

(i) $693$
(ii) $1584$

Example 4 — Times 8 and times 12

Evaluate
(i) $23 \times 8$
(ii) $14 \times 12$

Solution

$23 \times 8$	$=$	$23 \times (10 - 2)$
	$=$	$230 - 46 = 184$
$14 \times 12$	$=$	$14 \times (10 + 2)$
	$=$	$140 + 28 = 168$

23 \times 8 = 230 - 46 = 184, 14 \times 12 = 140 + 28 = 168

(i) $184$
(ii) $168$

Example 5 — Money

Tickets to a netball final cost $\$39$ each. Find the cost of $15$ tickets.

Solution

$15 \times 39$	$=$	$15 \times (40 - 1)$
	$=$	$600 - 15$
	$=$	$585$

15 \times 39 = 15 \times (40 - 1) = 600 - 15 = 585

The tickets cost $\$585$.

Example 6 — Two-digit numbers

Evaluate
(i) $26 \times 19$
(ii) $34 \times 102$

Solution

$26 \times 19$	$=$	$26 \times (20 - 1)$
	$=$	$520 - 26 = 494$
$34 \times 102$	$=$	$34 \times (100 + 2)$
	$=$	$3400 + 68 = 3468$

26 \times 19 = 520 - 26 = 494, 34 \times 102 = 3400 + 68 = 3468

(i) $494$
(ii) $3468$

Example 7 — Expanding brackets

Expand $5(2k - 3)$. Check your answer with $k = 4$.

Solution

$5(2k - 3)$	$=$	$5 \times 2k - 5 \times 3$
	$=$	$10k - 15$

5 (2 k - 3) = 10 k - 15

$10k$ and $15$ are not like terms, so stop here. Check with $k = 4$: $5 \times (8 - 3) = 25$ and $40 - 15 = 25$. So $5(2k - 3) = 10k - 15$.

Common pitfalls

Multiplying only the first part. $6 \times (10 + 3) = 60 + 18 = 78$, not $60 + 3$.

Subtracting just 1. $24 \times 99 = 2400 - 24 = 2376$, not $2400 - 1$.

Joining unlike terms. $3(a + 4) = 3a + 12$. It cannot be simplified to $15a$.

Splitting the wrong way. Times $8$ is times $10$ take away double the number, not take away $2$.

Frequently asked questions

What is the distributive law?

$a(b + c) = ab + ac$: multiply every part inside the brackets, then add.

How do you multiply by 99 in your head?

Multiply by $100$, then take away the number once. $16 \times 99 = 1600 - 16 = 1584$.

How do you multiply by 11 in your head?

Multiply by $10$, then add the number. $23 \times 11 = 230 + 23 = 253$.

What does expanding mean?

Removing brackets by multiplying every term inside. $2(x + 5) = 2x + 10$.

Why does the area model show the distributive law?

The area of the whole rectangle equals the areas of its parts added together.

To multiply by	Example
\(9 = 10 - 1\): times \(10\), take away the number	\(13 \times 9 = 130 - 13 = 117\)
\(11 = 10 + 1\): times \(10\), add the number	\(13 \times 11 = 130 + 13 = 143\)
\(8 = 10 - 2\): times \(10\), take away double	\(13 \times 8 = 130 - 26 = 104\)
\(12 = 10 + 2\): times \(10\), add double	\(13 \times 12 = 130 + 26 = 156\)
\(101 = 100 + 1\): times \(100\), add the number	\(13 \times 101 = 1300 + 13 = 1313\)
\(99 = 100 - 1\): times \(100\), take away the number	\(13 \times 99 = 1300 - 13 = 1287\)

\(6 \times 125\)	\(=\)	\(6 \times (100 + 20 + 5)\)
\(\)	\(=\)	\(600 + 120 + 30\)
\(\)	\(=\)	\(750\)

\(23 \times 11\)	\(=\)	\(23 \times (10 + 1)\)
\(\)	\(=\)	\(230 + 23 = 253\)
\(32 \times 101\)	\(=\)	\(32 \times (100 + 1)\)
\(\)	\(=\)	\(3200 + 32 = 3232\)

\(7 \times 99\)	\(=\)	\(7 \times (100 - 1)\)
\(\)	\(=\)	\(700 - 7 = 693\)
\(16 \times 99\)	\(=\)	\(16 \times (100 - 1)\)
\(\)	\(=\)	\(1600 - 16 = 1584\)

\(23 \times 8\)	\(=\)	\(23 \times (10 - 2)\)
\(\)	\(=\)	\(230 - 46 = 184\)
\(14 \times 12\)	\(=\)	\(14 \times (10 + 2)\)
\(\)	\(=\)	\(140 + 28 = 168\)

\(15 \times 39\)	\(=\)	\(15 \times (40 - 1)\)
\(\)	\(=\)	\(600 - 15\)
\(\)	\(=\)	\(585\)

\(26 \times 19\)	\(=\)	\(26 \times (20 - 1)\)
\(\)	\(=\)	\(520 - 26 = 494\)
\(34 \times 102\)	\(=\)	\(34 \times (100 + 2)\)
\(\)	\(=\)	\(3400 + 68 = 3468\)

\(5(2k - 3)\)	\(=\)	\(5 \times 2k - 5 \times 3\)
\(\)	\(=\)	\(10k - 15\)

Distributive Law

📖 Theory

How to use the distributive law

Common pitfalls

Frequently asked questions

Theory