Distributive Property - What is it?
The distributive property in multiplication states that when a number is multiplied by the sum or difference of two numbers, then the first number can be distributed to both of those numbers with the relevant sign. It is then multiplied by each of them separately, and then we add or subtract the two products together to obtain the same result as multiplying the first number by the sum or difference of two numbers.
For example, 2 × (3 + 4) is the same as (2 × 3) + (2 × 4)
Or in the case of subtracting numbers, the distributive property for multiplication can be shown as, 2 × (5 – 3) is the same as, (2 × 5) – (2 × 3)
To make it clear let us discuss the distributive property of multiplication formula in detail.
Formulas for the Distributive Property
There are two formulas that we will focus on to understand the formula for the distributive property of multiplication.
Here are the two formulas:
- a × (b + c) = (a × b) + (a × c)
- a × (b – c) = (a × b) – (a × c)
The distributive property for numbers, especially with reference to multiplication, is better understood using the area model as shown. Let us consider the multiplication as shown below:
11 × (13 + 20) = 11 × 13 + 11 × 20
This can also be represented as a rectangle, which has a length of (13 + 20) units and width of 11 units. Now the bigger rectangle can be broken into smaller rectangles as shown. To find out the product 11 × (13 + 20) we need to find the area of the rectangle. The area of the smaller rectangle would be 11 × 13, and area of the bigger rectangle would be 11 × 20. So, the total area is 11 × 13+11 × 20 . This can be further calculated as follows:
The distributive property for multiplication formula can also be applied to algebraic expressions and equations. We can also prove algebraic identities using this formula.
Let us try and see how we can prove the distributive property for multiplication formula involving algebraic expressions:
Let us try and find the product of (a + b) and (a + b)
Here distributive property can be used and the operation can be distributed as:
(a + b) × (a + b) = (a + b) × a + (a + b) × b Use the distributive property of Multiplication formula.
= \(a^2+ab+ab+b^2\) Simplify
= \(a^2+2ab+b^2\)
Algebraic equations, or linear equations can also be solved using the distributive property.
Solved Examples
Example 1: Solve 10 × (27 + 34) using distributive property and showing this using the area model.
Solution:
10 × (27 + 34) = 10 × 27 + 10 × 34 Use the distributive property of Multiplication formula.
= 270 + 340 Add
= 610
Now let us represent using an area model,
Make a rectangle which has a length of (27 + 34) units and width of 10 units.
Example 2: The cost of a pack of 20 crayons is $4, while the cost of a pack of 15 colored pencils is 5$. Represent this by writing a numerical expression that shows the cost of buying 5 packs of crayons and 5 packs of colored pencils. Then, also find the total cost.
Solution:
We know that the cost of buying 5 packs of crayons and colored pencils is:
5 (4 + 5)
= 5 × 4+5 × 5 Use the distributive property of Multiplication formula.
= 20 + 25 Add
= $45
The total cost to buy 5 packs of crayons and colored pencils is $45.
Example 3: Find the following products using distributive property.
- 16 × 108
- 32 × 92
- 12 × 999
Solution:
1. 16 × 108 = 16 × (100 + 8) As, 100 = 100 + 8
= 16 × 100 + 16 × 8 Use the distributive property of Multiplication formula.
= 1600 + 128 Add
= 1728
2. 32 × 92 = 32 × (100 – 8) As, 92 = 100 – 8
= 32 × 100 – 32 × 8 Use the distributive property of Multiplication formula.
= 3200 – 256 Subtract
= 2944
3. 12 × 999 = 12 × (1000 – 1) As, 999 = 1000 – 1
= 12 × 1000 – 12 × 1 Use the distributive property of Multiplication formula.
= 12000 – 12 Subtract
= 11988
Example 4: Solve the following linear equations
- 6 = 16 × (- 3x + 4)
- 3y(8 + 9x) = 27xy
Solution:
1) 6 = 16 × (- 3x + 4)
6 = 16× (– 3x) + 16 × 4 Use the distributive property of Multiplication formula.
6 – 64 = – 48x + 64 – 64 Subtract 64 from both sides
– 58 = – 48x
\(\frac{-58}{-48}=\frac{-48x}{-48}\) Divide both the sides by -48
\(\frac{29}{24}=x\)
2) 3y (8 + 9x) = 27xy
3y × 8 + 3y × 9x = 27xy Use the distributive property of Multiplication formula.
24y + 27yx = 27xy
24y + 27xy – 27xy = 27xy – 27xy Subtract 27xy from both sides
24y = 0
\(\frac{24y}{24}=\frac{0}{24}\) Divide both the sides by 24
y = 0 Substituting the value of y = 0 in the original equation we get x = 0.
Example 5: Simplify the following expression.
\(~~~~~~~~~~~~~~~~~~~ 3y + 6x (6y – 3x) \)
Solution :
\( ~~~~~~~~~~~~~~~~~~~3y + 6x (6y – 3x) \)
\( ~~~~~~~~~~~~~~~~~~~= 3y (6y – 3x) + 6x (6y – 3x) \) Use distributive property of Multiplication formula
\(~~~~~~~~~~~~~~~~~~~ = 3y \times 6y – 3y \times 3x + 6x \times 6y – 6x \times 3x \) Use distributive property of Multiplication formula
\(~~~~~~~~~~~~~~~~~~~= 18y^2 – 9yx + 36xy – 18x^2\) – 9yx = – 9xy
\(~~~~~~~~~~~~~~~~~~~= 18y^2+ 27xy – 18x^2\) Add – 9yx + 36xy to get 27xy
\(~~~~~~~~~~~~~~~~~~~= 18y^2 – 18x^2+ 27xy\) Rearrange
\(~~~~~~~~~~~~~~~~~~~= 18(y^2 – x^2)+ 27xy\) Use distributive property of multiplication to club the terms together
\(~~~~~~~~~~~~~~~~~~~= 18 (y + x)(y – x) + 27xy \) Use Identity \(a^2-b^2= (a + b)(a – b) \)
The distributive property of multiplication formula also holds true for calculations using fractions as can be seen in the following example.
\(3 = \frac{6}{7}\times(4z-7)\)
\(3 = \frac{6}{7}\times4z-\frac{6}{7}\times7 \) Use the distributive property of Multiplication Formula.
\(3 = \frac{24z}{7}-6\)
\(3 + 6 =\frac{24z}{7}-6+6\) Add 6 on both the sides
\(9 =\frac{24z}{7}\)
\(9 =\frac{24z}{7}\) Multiply 7 to both sides.
\(9 \times 7 = \frac{24z}{7}\times 7 \)
\(63 = 24z \) Divide both sides by 24.
\(\frac{63}{24} =\frac{24z}{7}\)
\(\frac{63}{24}= z\) Simplify.
\(\frac{21}{8}= z\)
The distributive property of the division formula does not hold true. This can be seen below:
If distributive property of division formula were to hold true then:
\(a (b + c) = (a\div b)+(a \div c)\)
Let us try to verify this by using different values of a, b and c.
Let a = 16, b = 8 and c = 2
So by distributive property of division formula:
\(16\div (8 + 2)\) should be equal to \((16\div 8) + (16 \div 2)\)
But clearly,
\(\frac{16}{10}\neq 10\)
So, we have verified that the property of the division formula does not hold true.
The errors mostly occur due to negligence towards the sign of the number.
This can also be seen below:
