Distributive Property of Multiplication

What is the distributive property?

The distributive property is one of the fundamental properties of multiplication that explains how multiplication interacts with addition and subtraction. It provides a rule for expanding expressions that contain parentheses.

When a number is multiplied by a group of numbers added or subtracted together, you "distribute" the outside multiplier to every single number inside the parentheses.

For a basic expression containing three numbers, the property is written mathematically as:

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

The variable $a$ is the outside multiplier, while $b$ and $c$ are the numbers being added. You can drop the multiplication sign when a number sits immediately next to parentheses, meaning $a(b+c)$ means the exact same thing as $a\times (b+c)$.

Understanding this concept also connects closely with the properties of addition, because the numbers inside the parentheses can be added first if they are known, or they can be multiplied separately and added later. Both mathematical paths lead to the same final answer.

Distributing over addition

When distributing multiplication over addition, you multiply the number outside the parentheses by each addend inside the parentheses. After finding the individual products, you add them together.

Consider the expression $4(3+2)$. There are two valid ways to evaluate it:

1. Add inside the parentheses first: $3+2=5$, then multiply $4\times 5=20$.
2. Use the distributive property: Multiply $4\times 3=12$, multiply $4\times 2=8$, and add the products $12+8=20$.

Both methods yield $20$. Distributing is especially useful when one of the terms inside the parentheses is an unknown variable, meaning you cannot complete the addition first.

Distributing over subtraction

The property works identically for subtraction. When distributing over a difference, you multiply the outside number by both numbers inside the parentheses and subtract the second product from the first.

The mathematical rule for distributing over subtraction is:

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

For example, to evaluate $5(10-2)$, multiply $5\times 10=50$, multiply $5\times 2=10$, and subtract to find $50-10=40$.

Array and area-model proof

One of the clearest ways to visualize the distributive law is by drawing an area model for multiplication. An area model uses a large rectangle that is split into smaller rectangular sections.

If a rectangle has a total height of $4$ and a total width of $(5+3)$, the total area is $4\times (5+3)$.

By splitting the rectangle into a width of $5$ and a width of $3$, you create two smaller shapes. The first has an area of $4\times 5$, and the second has an area of $4\times 3$. The sum of the two smaller areas exactly equals the original total area, physically proving that $4(5+3)=20+12$.

Use the property to multiply

You can use the distributive property as a break apart method to perform mental calculations that would normally require long multiplication. You do this by breaking a large, difficult number into simpler numbers, like tens and units.

Suppose you need to multiply $6\times 48$.

You can break $48$ into an addition problem using its place values: $40+8$.

$6\times 48=6(40+8)$

Distribute the $6$ to both terms:

$(6\times 40)+(6\times 8)=240+48=288$

Alternatively, you can break $48$ apart using subtraction by recognizing it is close to $50$:

$6\times 48=6(50-2)$

Distribute the $6$ over the difference:

$(6\times 50)-(6\times 2)=300-12=288$

Both distribution strategies simplify the calculation and produce the exact same accurate result.

Worked examples

Example 1: Using addition to break apart numbers

Question: Multiply $7\times 104$ using the distributive property.

Method:

1. Break apart the larger number into a sum of friendly numbers: $100+4$.
2. Write the new expression: $7(100+4)$.
3. Distribute the outside number to each term: $(7\times 100)+(7\times 4)$.
4. Calculate the separate products: $700+28$.
5. Add the products together.

Answer: The final product is $728$.

Check: Use standard column multiplication. $7\times 100$ is $700$, and $7\times 4$ is $28$. Their sum is $728$.

Example 2: Using subtraction to break apart numbers

Question: Multiply $9\times 39$ using the distributive property.

Method:

1. Identify that $39$ is very close to a multiple of ten. Break it apart as $40-1$.
2. Write the expression: $9(40-1)$.
3. Distribute the multiplier: $(9\times 40)-(9\times 1)$.
4. Calculate the products: $360-9$.
5. Subtract carefully.

Answer: The product is $351$.

Check: Calculate $9\times 40=360$, and count backwards by $9$ to reach $351$.

Example 3: Expanding an algebraic expression

Question: Expand the expression $4(2x+5)$.

Method:

1. Multiply the outside term by the first inside term: $4\times 2x$.
2. Multiply the outside term by the second inside term: $4\times 5$.
3. Keep the addition sign between the two products.
4. Simplify both terms.

Answer: $8x+20$.

Check: The expression cannot be simplified further because $8x$ and $20$ are not like terms. The multiplication was correctly distributed to both terms.

Common mistakes

The most common mistake when expanding expressions is forgetting to distribute the multiplication to the final term inside the parentheses.

For example, when asked to expand $3(x+4)$, a student might mistakenly write $3x+4$. The student successfully multiplied the $3$ by the $x$, but completely forgot to multiply the $3$ by the $4$.

Always verify that you have drawn a distribution arrow to every single term inside the parentheses.

Another frequent mistake involves confusing the signs when distributing over subtraction. Ensure you keep track of the minus sign. For instance, $2(y-6)$ expands correctly to $2y-12$, not $2y+12$.

Frequently asked questions

Is there a distributive property of division?

Yes, division can be distributed over addition or subtraction, but only when the sum or difference is in the numerator. For example, $\frac{10+6}{2}$ can be distributed as $\frac{10}{2}+\frac{6}{2}$, which simplifies to $5+3=8$. You cannot distribute division if the addition or subtraction is in the denominator.

Does the break apart method work for three or more numbers?

Yes. You can distribute a multiplier across as many addends as you need. For example, $a(b+c+d)=ab+ac+ad$. If you multiply $3\times 245$, you can expand it completely as $3(200+40+5)=600+120+15=735$.

What if there is a variable outside the parentheses?

The property functions the same way. If you have $x(y+3)$, you multiply the $x$ by both parts to get $xy+3x$. The mathematical rules of distribution remain consistent whether you use numbers, variables, or a combination of both.