The Distributive Property: Additional Practice (Set 2)

The $7$ must multiply both terms inside, not just the first.

$7(10 + 3) = 7 \times 10 + 7 \times 3 = 70 + 21 = 91$

The student forgot to multiply the $3$ by $7$.

Write $103$ as $100 + 3$ and distribute the $14$ to each part.

$14 \times 103 = 14(100 + 3) = 14 \times 100 + 14 \times 3 = 1400 + 42 = 1442$

The second product must be $14 \times 3 = 42$. Every wrong option keeps the $1400$ and then mishandles that second product: $1403$ adds the $3$ without multiplying it, $1414$ adds $14 \times 1$, and $1430$ adds $10 \times 3$.

Multiply each of the three terms by $5$, then add.

$5(2 + 3 + 4) = 5 \times 2 + 5 \times 3 + 5 \times 4 = 10 + 15 + 20 = 45$

Check: $5 \times 9 = 45$.

The greatest common factor of $42$ and $56$ is $14$, since $42 = 14 \times 3$ and $56 = 14 \times 4$.

$42 + 56 = 14 \times 3 + 14 \times 4 = 14(3 + 4)$

Check by distributing back: $14(3 + 4) = 14 \times 7 = 98$, and $42 + 56 = 98$. Forms like $7(6 + 8)$ and $2(21 + 28)$ are true equalities but pull out only part of the common factor, so they are not fully factored.

Write $99 = 100 - 1$ and distribute the $6$ to each part.

$6 \times 99 = 6(100 - 1) = 6 \times 100 - 6 \times 1 = 600 - 6 = 594$

So the matching form is $6 \times 100 - 6$. The others evaluate to $501$, $606$, and $588$.

The parentheses hold a product, not a sum, so there is nothing to distribute. Work out the inside first.

$4(3 \times 5) = 4 \times 15 = 60$

Distributing the $4$ to both factors gives $12 \times 20 = 240$, which multiplies by $4$ twice. Treating the inside as a sum gives $32$. Replacing the outside multiplication with an addition gives $19$.

Distribute and add each option, then keep the one equal to $48$.

$4(5 + 7) = 4 \times 12 = 48$

The others give $8 \times 5 = 40$, $6 \times 7 = 42$, and $11 \times 4 = 44$, none of which is $48$.

Distribute the $6$ across the difference, keeping the minus sign.

$6(15 - 8) = 6 \times 15 - 6 \times 8 = 90 - 48 = 42$

Check the inside first: $15 - 8 = 7$, and $6 \times 7 = 42$.

All four splits are true, and all four give the same answer, so the question is only which leaves the least work.

$9 \times 52 = 9(50 + 2) = 450 + 18 = 468$

Both of those products are ones you can do at a glance. The other splits still leave a two-digit product to work out, $9 \times 26$, $9 \times 39$, or $9 \times 22$, so none of them is as light as $50 + 2$.

Write $14$ as $10 + 4$ and distribute the $11$ to both parts.

$11 \times 14 = 11(10 + 4) = 11 \times 10 + 11 \times 4 = 110 + 44 = 154$

The form $11 \times 10 + 4$ multiplies only the first part, and $11 \times 1 + 11 \times 4$ uses $1$ instead of $10$, so both are wrong.

Split $250$ as $200 + 50$ and distribute the $8$.

$8 \times 250 = 8(200 + 50) = 8 \times 200 + 8 \times 50 = 1600 + 400 = 2000$

The greatest common factor of $48$ and $80$ is $16$, since $48 = 16 \times 3$ and $80 = 16 \times 5$.

$48 + 80 = 16 \times 3 + 16 \times 5 = 16(3 + 5)$

Check: $16(3 + 5) = 16 \times 8 = 128$, and $48 + 80 = 128$. The forms $8(6 + 10)$ and $4(12 + 20)$ leave a common factor inside, so they are not fully factored.