We explain what the distributive property is, with examples and solved exercises

**What is the distributive property?**

The **distributive property** of multiplication with respect to addition or subtraction consists of multiplying a factor by the indicated addition or subtraction of two or more quantities.

Let there be three quantities a, b and c, which can be real numbers, algebraic or vector quantities, among others, and suppose that the following operation is solved with them:

a×(b+c)

In this expression “a” is the factor and (b + c) is the indicated sum. There are two ways to find the answer of the operation, the first one is to obtain the sum (b+c) and what is obtained is multiplied by “a”.

And the other way is to multiply «a» by each of the terms b and c, and then add the results. It is not uncommon for the same operation to be done in several ways. The following example shows that the two procedures are equivalent:

5 × (7 + 3) = 5 × 10 = 50

O well:

5 × (7 + 3) = (5 × 7) + (5 × 3) = 35 + 15 = 50

In this last procedure, 5 is multiplied by 7 and then by 3, the respective results are added to obtain the final value.

The distributive property can also be applied to subtraction, for example:

8 × (12 − 5) = (8 × 12) − (8 × 5) = 96 − 40 = 56

And in both cases, the number of terms inside the parentheses does not matter, since the multiplier factor is distributed to all, as in this other operation:

5 × (3 − 7 + 10) = (5 × 3) − (5 × 7) + (5 × 10) = 15 − 35 + 50 = 30

**The common factor: the inverse of the distributive property**

Consider the following operation:

(7 × 2) + (7 × 6)

In each parenthesis there is a 7 that multiplies another number. Well then, since the 7 is repeated in both parentheses and is multiplying, it is called *common factor*so the operation can be written as:

(7 × 2) + (7 × 6) = 7 × (2 + 6)

This operation is precisely the inverse of the distributive property and can be applied to any number of terms that have a common factor, for example:

(6 × 8) + (6 × 11) + (6 × 4) − (6 × 9)

The common factor is 6, since it is repeated in each of the terms. Therefore:

(6 × 8) + (6 × 11) + (6 × 4) − (6 × 9)= 6 × (8 + 11+ 4− 9)

**Observations**

Whenever you think about applying the distributive property it is necessary to observe the notation, in this sense it is important to highlight that:

The symbols for a cross “×” and a dot at half height “∙” are used interchangeably to denote a multiplication. Even if none of these symbols is present between the factor and the parenthesis containing the addends, it will be understood that it is a multiplication. For example, in the operation 5(4 − 9), the 5 multiplies both the 4 and the 9, in the same way as in the previous examples:

5(4− 9) = 5∙4−5∙9 = 20 − 45 = −25

In this example, the dot at half height was also used instead of the cross.

Another important fact to take into account is the presentation of the operations, 7(5+1) is not the same as 7 + (5×1). In the first case, the distributive property is applied in the same way that it has been done:

7(5+1) = 7∙5 + 7∙1 = 35+7=42

On the other hand, for the operation 7 + (5×1) we proceed according to the hierarchy of operations, which indicates that the parentheses must be eliminated first, in this way:

7 + (5×1) = 7 + 5 = 12

Multiplication is commutative, so it holds that:

a × (b + c) = (b + c) × a

The factor that multiplies the sum can be to the left or to the right of it and in any case the result is the same.

**Application Examples **

**Example 1**

The multiplication of a large number by another can be carried out, through the distributive property, if the large number is decomposed into hundreds, tens and units. For example, you are asked to do:

5×852

The number 852 breaks down into addends like:

852 = 800 + 50 + 2

And the requested operation is written as:

5×852= 5×(800 + 50 + 2)

Now we only have to apply the distributive property and obtain the resulting sum:

5×(800 + 50 + 2) = 4000 + 250 + 10 = 4260

**Example 2**

The distributive property makes it easy to calculate products of sums, products of differences, and products of sums by differences:

(a + b) × (c + d) = a∙c + a∙d + b∙c + b∙d

(a + b) × (c − d) = a∙c − a∙d + b∙c − b∙d

(a − b) × (c − d) = a∙c − a∙d − b∙c + b∙d

For example, the following operations are solved as shown:

(5 + 4) × (2 + 13) = 5∙2 + 5∙13 + 4∙2 + 4∙13 =10 + 65 + 8 +52 = 135

[(8 + (−17)] × (6 − 21) = 8∙6 − 8∙21 + (− 17)∙6 − (− 17)∙21 = 48−168−102+357 =135

(11 − 7) × (9 − 16) = 11∙9 − 11∙16 − 7∙9 + 7∙16 = 99 − 176 − 63 +112 = −28

**Example 3**

A florist’s counter has four vases of flowers, each containing 9 roses and 2 carnations. You can use the distributive property to find the total number of flowers in all four vases, simply by multiplying the sum (9 + 2) by 4:

Total flowers = 4× (9 + 2) = 36 + 8 = 44 flowers

**The distributive property in algebra**

Both the distributive property and the common factor are widely used in algebra and calculus, as they allow you to easily manipulate algebraic expressions for convenience.

Sometimes it is better to write an expression with the distributive property, while other times it may be more efficient to have the expression factored.

For example, suppose you need to develop the expression:

2(x+1)

Unlike the operation 5 × (7 + 3) = 5 × 10 = 50, the terms inside the parentheses are not like, so their sum does not reduce to a single term (instead 7 + 3 immediately reduces to 10). In such a case, the distributive property is applied to obtain:

2(x+1) = 2∙x + 2∙1=2x + 2

**Using the distributive property to solve equations**

Some algebraic equations are solved by applying the distributive property, for example:

8(x–2) = 14

Applying the distributive property to develop the left side of the equality we have:

8x – 16 = 14

8x = 14 + 16 = 30

x = 30/8 = 15/4

**Remarkable products**

The distributive property is used to prove notable products, which are widely used in algebra. For example, it can be shown that the product of the sum of two quantities multiplied by the difference of those same quantities is equal to the difference of their respective squares.

Denoting the quantities as «a» and «b» and applying the property we have:

(a + b) × (a – b) = a⋅a – a⋅b + a⋅b – b⋅b = a2 – b2

**solved exercises**

**Exercise 1**

A group of 8 friends go for a walk one afternoon to visit a museum and have a snack. Transport costs €5, admission €2 and refreshments €3 per person. Calculate the cost of the ride for the whole group.

Each participant has to spend (5 + 2 + 3) € per person, and since there are 8, the total is calculated by the following operation:_

8×(5 + 2 + 3) € = (8×5 + 8×2 + 8×3) € = (40 + 16 + 24) € = €80

**Exercise 2**

The cabin of a funicular can carry 30 seated passengers and 12 standing passengers. Calculate how many passengers are transported after 9 trips if each one carries the maximum number of people allowed.

The total number of people who go on a single trip is (30 + 12), since there are 9 trips left:

9×(30 + 12) = 9×30 + 9×12 = 270 + 108 = 378 people.

**References**

Baldor, A. 1985. Theoretical-Practical Arithmetic. Editions and Distributions Codex, Madrid. Math lessons. Solved exercises of the distributive property and taking a common factor. Recovered from: leccionesdemates.com. Math Mammoth. Distributive property or how to multiply in parts. Recovered from: mamutmatematicas.com. Smartick. Examples of distributive property. Recovered from: smartick.es. Vicen Vives. Mathematics 4, Topic: multiplication. Retrieved from: auladecarmela.com