The Distributive Property: Practice

Multiply the $4$ by each term inside, then add.

$4(10 + 2) = 4 \times 10 + 4 \times 2 = 40 + 8 = 48$

Check directly: $4 \times 12 = 48$.

Multiply each term by $2$, then add.

$2(30 + 5) = 2 \times 30 + 2 \times 5 = 60 + 10 = 70$

Check: $2 \times 35 = 70$.

Distribute the $7$, then add.

$7(10 + 2) = 7 \times 10 + 7 \times 2 = 70 + 14 = 84$

Check: $7 \times 12 = 84$.

In the first expression the $5$ reaches both terms; in the second it reaches only the $20$.

$5(20 + 3) = 100 + 15 = 115, \qquad 5 \times 20 + 3 = 100 + 3 = 103$

The gap is $115 - 103 = 12$, which is the $3$ counted once instead of five times: $5 \times 3 - 3 = 12$.

The outside $6$ reaches both numbers inside, so the second term is $6 \times 4$, not a bare $4$.

$6(20 + 4) = 6 \times 20 + 6 \times 4 = 120 + 24 = 144$

Check: $6 \times 24 = 144$. Copying the $4$ down untouched would give $124$.

Distribute the $9$ to both terms, then add.

$9(10 + 1) = 9 \times 10 + 9 \times 1 = 90 + 9 = 99$

Check: $9 \times 11 = 99$.

Distributing means the outside factor multiplies every term inside, so both products are formed before anything is added.

$3(10 + 6) = 3 \times 10 + 3 \times 6 = 30 + 18 = 48$

Copying the $6$ down instead would give $36$, and $3 \times 16 = 48$ shows that $48$ is right.

Read the distributive property backward. The shared $6$ comes out in front, and the other two numbers stay inside as a sum.

$30 + 12 = 6 \times 5 + 6 \times 2 = 6(5 + 2) = 6 \times 7 = 42$

Checking straight through, $30 + 12 = 42$ as well.

The $2$ multiplies both terms inside the parentheses.

$2(40 + 6) = 2 \times 40 + 2 \times 6 = 80 + 12 = 92$

The form $2 \times 40 + 6$ multiplies only the first term and gives $86$, and $2 \times 4 + 2 \times 6$ uses $4$ in place of $40$.

With a difference inside, the $5$ still reaches both numbers, and the minus sign stays between the products.

$5(10 - 2) = 5 \times 10 - 5 \times 2 = 50 - 10 = 40$

Check the inside first: $10 - 2 = 8$, and $5 \times 8 = 40$. Subtracting only $2$ would leave $48$.

A cut from top to bottom changes only the widths, so both pieces are still $6$ tall.

$6 \times 20 = 120, \qquad 6 \times 2 = 12$

Together they cover $120 + 12 = 132$, which is the whole rectangle's area, $6 \times 22 = 132$.

Think of a rectangle $2$ tall and $54$ wide, with its width cut into $50$ and $4$. The cut adds no area and removes none, so the two pieces must total the whole rectangle.

$2(50 + 4) = 2 \times 50 + 2 \times 4 = 100 + 8 = 108$

Check: $2 \times 54 = 108$.