Distributive Property

The distributive law is useful in two ways:

🌟 Expanding:

Expanding means using the distributive law to remove brackets. You multiply the number outside the bracket with each term inside.

• Formula:

$$
a \times b + a \times c = a \times (b + c)
$$

• Example:

$$
\begin{align*}
15 \times (100 + 2) &= 15 \times 100 + 15 \times 2 \\[0.7em]
&= 1500 + 30 \\[0.7em]
&= 1530
\end{align*}
$$

🌟 Factorising:

Factorising means using the distributive law in reverse to add brackets. You find a common factor in both terms and write the expression as a single bracket.

• Formula:

$$
a \times b \;-\, a \times c = a \times (b \;-\, c)
$$

• Example:

$$
\begin{align*}
50 \times 24 \;-\, 50 \times 14 &= 50 \times (24 \;-\, 14) \\[0.7em]
&= 50 \times 10 \\[0.7em]
&= 500
\end{align*}
$$

The distributive law also works with three or more terms inside the brackets.

You simply multiply the number outside the bracket by every term inside the bracket.

• Formula:

$$
a \times (b + c + d) = a \times b + a \times c + a \times d
$$

• Example:

$$
\begin{align*}
5 \times (200 + 20 + 1) &= 5 \times 200 + 5 \times 20 + 5 \times 1 \\[0.7em]
&= 1000 + 100 + 5 \\[0.7em]
&= 1105
\end{align*}
$$

The distributive law generally does not apply to division.

It only works when the bracket is being divided, and a single number is the divisor.

But even then, using it often doesn’t make the calculation any easier.

For example,

$$
(19 + 2) \div 3 = 19 \div 3 + 2 \div 3 \approx 6.33 + 0.67 
$$

It’s better to simplify what’s inside the bracket first, then do the division.

$$
(19 + 2) \div 3 = 21 \div 3 = 7
$$