We Distribute, Yet Things Multiply: questions with answers

By 14, which is 126 divided by 9. One more 9 means one more group of 14, so the product gains 14 -- which the distributive property says exactly: 14(9 + 1) = 14 x 9 + 14. The product does NOT gain 1: that is how addition behaves, where adding 1 to an addend adds 1 to the sum, and multiplication does not work that way. Nor does it gain 23. That is what BOTH numbers rising by 1 would give, less the corner: here only one of them has moved.

The new product is $(12 + 1)(7 - 1)$. Expanding gives $12 \times 7 + 7 - 12 - 1$. Subtracting the original product leaves $7 - 12 - 1 = -6$. So the product falls by 6. Increasing one factor and decreasing the other by the same amount preserves their sum. It does not always preserve their product or make it rise.

Two terms combine when their letters and powers match exactly. Only their coefficients are added; the letter part stays unchanged. Thus, $7mn + 2mn = 9mn$. The term $2m^2$ cannot combine with $7mn$ because one term contains n and the other does not. Writing out an expansion does not finish the work when like terms remain to be combined.

Multiply each of the three terms by 1/3 to get 2g - 5 + 7h/3. The fraction multiplies the constant and 7h as well as the first term. The term containing g and the constant are unlike terms, so they cannot be combined.

The expansion is ab + 7a + 2b + 14. Each term of the first bracket multiplies each term of the second. Two terms times two terms give four products: ab, 7a, 2b and 2 x 7. Pairing the brackets straight across gives only ab + 14. That answer is short by both middle terms. Keeping just one middle term gives ab + 7a + 14. The two middle terms 7a and 2b are different terms, so both must be written.

Each of the two terms in the first bracket multiplies both terms in the second bracket. This gives $2 \times 2 = 4$ products. Adding them gives $rw + 9r + 2w + 18$. Counting only matching terms gives 2 products and misses both middle products. Counting just one middle product gives 3 products.

$248000 - 248$. Since $999 = 1000 - 1$, multiplying by $999$ means multiplying by $1000$ and then taking away one copy of $248$. Here $248 \times 1000 = 248000$, so the calculation is $248000 - 248$. Adding $248$ instead gives $248 \times 1001$, which is too large. Taking away only $1$ removes a single unit, not one copy of $248$.

The correct step multiplies each term of a sum by the same factor. This uses the distributive property. The other steps only regroup or reorder factors in a product. All three steps are true, but a bracket by itself does not show distributivity.

16y^2 + 48y + 36. Multiplying the bracket by itself gives four products, and the two cross products are the same, so they add to twice one of them: 48y. That is where the 2 in a^2 + 2ab + b^2 comes from, and the square of side a + b shows it as TWO rectangles of sides a and b. Squaring each part and stopping gives 16y^2 + 36, which throws both rectangles away, and counting only one rectangle gives 16y^2 + 24y + 36. Substituting y = 1 separates all three at once.

The identity holds for all integers and fractions. Using the distributive property, $(a-b)^2 = (a-b)(a-b) = a^2-ab-ab+b^2 = a^2-2ab+b^2$. These steps also work with zero, negative numbers and fractions. No step requires the letters to represent counting numbers or positive numbers.

Using the distributive property, $(m - n)^2 = m^2 - mn - nm + n^2$. Since $mn = nm$, collecting like terms gives $m^2 - 2mn + n^2$. The middle term is $-2mn$. This expansion holds even when $m - n$ is negative. Reversing the difference gives $(n - m)^2 = n^2 - 2mn + m^2$, with the same middle term.

3591. The two numbers sit either side of 60, each 3 away, so the product is (60 - 3)(60 + 3) = 3600 - 9. The centre does not have to be a multiple of ten: it is simply the midpoint of the two numbers, and 43 x 45 works exactly as well as 98 x 102 with 44 in the middle. Treating both brackets as sums gives 3609 -- one factor is a subtraction, which is what makes the cross terms cancel -- and ignoring the gap altogether leaves 3600, the square of the centre.

Expanding and simplifying the left side gives $k^2 + 2k$.

$k(k + 2) = k^2 + 2k$.

These steps use the distributive property and hold for every value of $k$. Substituting $k = 4$ gives $24$ on each side, but checks only that value. Matching constant terms alone does not establish equality of the whole expressions.

The products are $17 \times 25 = 425$ and $18 \times 24 = 432$. The first product is 7 less. To explain the pattern, let the top-left entry be $a$. The other entries are $a+1$, $a+7$, and $a+8$. The first diagonal gives $a(a+8)=a^2+8a$. The other gives $(a+1)(a+7)=a^2+8a+7$. So the first product is always 7 less. Moving down a calendar row adds 7, not 1.

At Step $k$, the main square has $k^2$ circles. An extra row below and an extra column beside it each add $k$ circles. The total is $k^2 + 2k$. At Step 10, this gives $10^2 + 2 \times 10 = 120$ circles. The number 10 labels the step; it does not count the circles. The answer 110 counts the square and extra column but leaves out the extra row.