The Distributive Property: Practice

Split $98$ as $100 - 2$ and distribute the $6$ to each part.

$6 \times 98 = 6(100 - 2) = 6 \times 100 - 6 \times 2 = 600 - 12 = 588$

Split $103$ as $100 + 3$ and distribute the $7$.

$7 \times 103 = 7(100 + 3) = 7 \times 100 + 7 \times 3 = 700 + 21 = 721$

Any split that adds back to $99$ gives the right answer, so both work.

$8(100 - 1) = 800 - 8 = 792, \qquad 8(90 + 9) = 720 + 72 = 792$

Ana's split is the lighter one, because the correction $8 \times 1$ is easier to hold in mind than $8 \times 9$. Choosing a split is about effort, not about which answer is correct.

Write $102 = 100 + 2$, then distribute the $4$.

$4 \times 102 = 4(100 + 2) = 4 \times 100 + 4 \times 2 = 400 + 8 = 408$

Split $19$ as $20 - 1$ and distribute the $7$.

$7 \times 19 = 7(20 - 1) = 7 \times 20 - 7 \times 1 = 140 - 7 = 133$

Since $96 = 100 - 4$, the $7$ reaches the $4$ as well as the $100$, so the amount taken away is $7 \times 4$.

$7 \times 96 = 7(100 - 4) = 700 - 28 = 672$

Subtracting a bare $4$ would leave $696$, which is $24$ too large.

Replacing $39$ by $40 - 1$ is fine, but the $5$ has to reach the $1$ as well as the $40$.

$5 \times 39 = 5 \times 40 - 5 \times 1 = 200 - 5 = 195$

The student subtracted $1$ instead of $5 \times 1 = 5$, so the answer came out $4$ too large.

Split $198$ as $200 - 2$ and distribute the $7$.

$7 \times 198 = 7(200 - 2) = 7 \times 200 - 7 \times 2 = 1400 - 14 = 1386$

Both products share the factor $8$, so read the rule backward and collect it.

$8 \times 47 + 8 \times 53 = 8(47 + 53) = 8 \times 100 = 800$

The inside happens to add to a round number, which is what makes the collected form so much less work than $376 + 424$.

Split $47$ as $50 - 3$ and distribute the $6$.

$6 \times 47 = 6(50 - 3) = 6 \times 50 - 6 \times 3 = 300 - 18 = 282$

Write $99 = 100 - 1$ and distribute the $4$ to each part.

$4 \times 99 = 4(100 - 1) = 4 \times 100 - 4 \times 1 = 400 - 4 = 396$

Subtracting only $1$ instead of $4 \times 1$ leaves $399$, which is wrong.

A split is only legal if the numbers inside come back to the original factor. Here $20 + 6$, $25 + 1$, and $30 - 4$ all make $26$, but $20 + 7$ makes $27$.

$7 \times 26 = 7(20 + 6) = 140 + 42 = 182$

The illegal split would give $7 \times 27 = 189$ instead. Always check the split before distributing.