Distributive property of multiplication over addition and subtraction

To “distribute” means to divide something or give a share or part of something.

So what does distributive property mean in math?

The distributive law of multiplication over basic arithmetic, such as addition and subtraction, is known as the distributive property. 

What Is Distributive Property?

According to this property, multiplying the sum of two or more addends by a number will give the same result as multiplying each addend individually by the number and then adding the products together. 

In other words, according to the distributive property, an expression of the form A (B $+$ C) can be solved as A (B $+$ C) $=$ AB $+$ AC.

This property applies to subtraction as well. 

A (B $–$ C) $=$ AB $–$ AC

This indicates that operand A is shared between the other two operands.

Let’s look at the formula for distributive property:

Where A, B, and C are any real numbers.

Here’s an example of how the result does not change when solved normally and when solved using the distributive property:

This property helps in making difficult problems simpler. You can use this property of multiplication to rewrite an expression by distributing or breaking down a factor as a sum or difference of two numbers.

Distributive Property of Multiplication over Addition

When we have to multiply a number by the sum of two numbers, we use this property of multiplication over addition. Let’s understand how to use the distributive property better with an example:

Example: Solve the expression: $6$ $(20 + 5)$ using the distributive property of multiplication over addition. 

Let’s use the property to calculate the expression $6$ $(20 + 5)$, the number 6 is spread across the two addends. To put it simply, we multiply each addend by 6 and then the products can be added.

$6 20 + 6 5 = 120 + 30 = 150$

Let’s take another example:

Example: Solve the expression $2$ $(2 + 4)$ using the distributive law of multiplication over addition.

Solution: $2 (2 + 4) = 2 2 + 2 4 = 4 + 8 = 12$

If we try to solve this expression using the PEMDAS rule, we’ll have to add the numbers in parentheses and then multiply the total by the number outside the parentheses. This implies:

$2 (2 + 4)$ $= 2 \times 6 =$ $12$

Thus, we get the same result irrespective of the method used.

Distributive Property of Multiplication over Subtraction

The distributive property of multiplication over subtraction is equivalent to the distributive property of multiplication over addition, except for the operations of addition and subtraction.

A(B − C) and AB − AC are equivalent expressions.

Consider these distributive property examples below.

Example: Solve the expression $6 (20 – 5)$ using the distributive property of multiplication over subtraction.

Solution: Using the distributive property of multiplication over subtraction,

$6 (20 – 5) = 6 20 – 6 5 = 120 – 30 = 90$

Let’s take another example to understand the property better.

Example: Solve the expression 2 (4 – 3) using the distributive law of multiplication over subtraction.

Solution: $2 (4 – 3) = 2 4 – 2 3 = 8 – 6 = 2$

Again, if we try to solve the expression with the order of operations or PEMDAS, we’ll have to subtract the numbers in parentheses, then multiply the difference with the number outside the parentheses, which implies:

$2 (4 – 3) = 2 1 = 2$

The distributive property of subtraction is proven since both techniques give the same result.

Fun Facts

Even though division is the inverse of multiplication, the distributive law only holds true in case of division, when the dividend is distributed or broken down into partial dividends, which are completely divisible by the divisor. 

For instance, using the distributive law for 1326 

132 can be broken down as $60 + 60 + 12$, thus making division easier. 

We cannot break 132 6 as $(50 + 50 + 32) 6$.

Also, we cannot break the divisor:  $132(4+2)$ will give you the wrong result. 

Conclusion

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Solved Examples

Example 1:  Solve $(5 + 7 + 3) 4$.

Solution: Using the distributive property of multiplication over addition,

A (B $+$ C) = AB $+$ AC

$(5 + 7 + 3) 4 = 5 4 + 7 4 + 3 4 = 20 + 28 + 12 = 60$

Or,

$(5 + 7 + 3) 4 = 15 4 = 60$

Example 2: Solve the following distributive equation $−2 (−$x$ − 7)$. 

Solution: Using the distributive property,

A (B $–$ C) $=$ AB $–$ AC

$−2 (−$x$ − 7) = (−2)(−$x$) − (−2)(7) = 2$x$ − (−14) = 2$x$ + 14$

Example 3: Which property does the equation $3 (4 − 9) = 3 4 − 3 9$ show?
Solution: The above equation shows the distributive property of multiplication over subtraction.

Practice Problems

x $+ 42$

$7$x $+ 13$

$7$x $+ 42$

$7$x $+ 6$

Correct answer is: $7$x $+ 42$
Using the distributive property of multiplication over addition,
A (B $+$ C) $=$ AB $+$ AC
$7 ($x $+ 6) = 7($x$) + 7(6) = 7$x $+ 42$

$13$x

$7$x$ – 24$

$21$x$ – 24$

$21$x$ – 8$

Correct answer is: $21$x$ – 24$
Using the distributive property of multiplication over subtraction,
A (B $–$ C) $=$ AB $–$ AC
$3 (7$x $– 8) = 3 (7$x$) – 3 (8) = 21$x$ - 24$

$3$mn $– 9$n

$3$mn $– 9$

$3$mn $– 9$m

$3$mn$+9$m

Correct answer is: $3$mn $– 9$m
Using the distributive property of multiplication over subtraction,
A (B $–$ C) $=$ AB $–$ AC
m $(3$n $– 9)$ $=$ m $(3$n$) –$ m $(9) = 3$mn $– 9$m

4260

3550

2130

426

Correct answer is: 4260
The total number of bananas harvested is given by the expression $355 x 12$.
The dozen or 12 can be distributed as 10 and 2.
The total number of bananas harvested $= 355 \times (10 + 2)$
Using the distributive property of multiplication over addition,
A (B $+$ C) $=$ AB $+$ AC
$= 355 \times 10 + 355 \times 2$
$= 3550 + 710 = 4260$
In total, 4260 bananas were harvested at the farm.

Frequently Asked Questions

Does distributive property apply to division too?

The distributive property applies to division in the same way that it applies to multiplication. However, the concept of “breaking apart” or “distributing” can be applied with division only by dividing the numerator into smaller amounts that are exactly divisible by the divisor. 

For example, to solve $\frac{125}{5}$, we can divide the numerator(125) as: (50 + 50 + 25) therefore:$\frac{125}{5}$ = $\frac{50}{5}$ + $\frac{50}{5}$ + $\frac{25}{5}$ = 10 + 10 + 5 = 25.

What is the rule for the distributive property?

According to the distributive property, multiplying the sum of two or more addends by a number produces the same result as when each addend is multiplied individually by the number and the products are added together.

How can distributive property help in solving complex questions?

The distributive property distributes complex expressions into simpler terms and thus makes problems, especially with multiple factors, easier to solve.

Can parentheses be removed after distributing?

Yes, when applying the distributive property, the outside factor is multiplied by each term inside the parentheses. This gets rid of parentheses.