Distributive Property – Meaning & Examples

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 In mathematics, the distributive property is one of the basic properties of numbers and it helps you in simplifying the mathematical expressions by splitting them into smaller values.

As the word distribute suggests, the distributive property allows us to split the numbers (generally large numbers) into smaller values while performing multiplication, thus making our calculations easy. 

What is Distributive Property?

The distributive property states that if a term is being multiplied by an expression in parentheses, then the multiplication is performed on each of the terms.

Mathematically it can be stated as

If $A$, $B$, and $C$ are any three numbers then, 

• For addition: $A \times \left(B + C \right) = A \times B + A \times C$
• For subtraction: $A \times \left(B – C \right) = A \times B – A \times C$

Note

• The two forms of this property are
  • Distristributive property of multiplication over addition
  • Distristributive property of multiplication over subtraction
• Distristributive property does not hold for division

Distributive Property of Multiplication Over Addition

The distributive property of multiplication over addition states that $A \times \left(B + C \right) = A \times B + A \times C$

Let’s understand the distributive property of multiplication over addition by this example.

Consider three numbers $A = 5$, $B = 7$ and $C = 3$.

Left Hand Side of the statement becomes $5 \times \left(7 + 3 \right) = 5 \times 10 = 50$

And Right Hand Side of the statement becomes $5 \times 7 + 5 \times 3 = 35 + 15 = 50$

The result of both is the same ($=50$).

Consider one more example by taking $A = 18$, $B = 23$ and $C = 32$.

Left Hand Side of the statement becomes $18 \times \left(23 + 32 \right) = 18 \times 55 = 990$.

And Right Hand Side of the statement becomes $18 \times 23 + 18 \times 32 = 414 + 576 = 990$.

The result of both is the same ($=990$).

So, based on the distributive property of addition we can say that the product remains the same even if the operation is distributed by splitting/breaking the numbers by adding.

Distributive Property of Multiplication Over Subtraction

The distributive property of multiplication over subtraction states that $A \times \left(B – C \right) = A \times B – A \times C$

Let’s understand the distributive property of multiplication over subtraction by this example.

Consider three numbers $A = 5$, $B = 7$ and $C = 3$.

Left Hand Side of the statement becomes $5 \times \left(7 – 3 \right) = 5 \times 4 = 20$

And Right Hand Side of the statement becomes $5 \times 7 – 5 \times 3 = 35 – 15 = 20$

The result of both is the same ($=20$).

Consider one more example by taking $A = 18$, $B = 23$ and $C = 32$.

Left Hand Side of the statement becomes $18 \times \left(23 – 32 \right) = 18 \times \left(-9 \right) = -162$.

And Right Hand Side of the statement becomes $18 \times 23 – 18 \times 32 = 414 – 576 = -162$.

The result of both is the same ($=-162$).

So, based on the distributive property of subtraction we can say that the product remains the same even if the operation is distributed by splitting/breaking the numbers by subtracting.

Is Distributive Property of Multiplication Applicable to Numbers of All Categories?

The distributive property of multiplication in both the two forms – the distributive property of multiplication over addition and the distributive property of multiplication over subtraction works well with any real number. 

Note: A set of real numbers $R$ is a superset of sets of natural numbers, a set of whole numbers, a set of integers, a set of rational numbers, and a set of irrational numbers.

Natural Numbers and Whole Numbers

Consider any three whole numbers $10$, $15$ and $12$.

Distributive Property of Multiplication Over Addition

We want to verify that $A \times \left(B + C \right) = A \times B + A \times C$

$10 \times \left(15 + 12 \right) = 10 \times 15 + 10 \times 12$

$=>10 \times 27 = 150 + 120 => 270 = 270$

Distributive Property of Multiplication Over Subtraction

We want to verify that $A \times \left(B – C \right) = A \times B – A \times C$

$10 \times \left(15 – 12 \right) = 10 \times 15 – 10 \times 12$

$=>10 \times 3 = 150 – 120 => 30 = 30$

Integers (Signed Numbers)

Consider any three integers $-11$, $B=+16$ and $C=-14$.

Distributive Property of Multiplication Over Addition

We want to verify that $A \times \left(B + C \right) = A \times B + A \times C$

$-11 \times \left(16 + \left(-14 \right) \right) = -11 \times 16 + \left(-11 \right) \times \left(-14 \right)$

$=>-11 \times \left(16 – 14 \right) = -176 + 154$

$=>-11 \times 2 = -22$ $=>-22 = -22$

Distributive Property of Multiplication Over Subtraction

We want to verify that $A \times \left(B – C \right) = A \times B – A \times C$

$-11 \times \left(16 – \left(-14 \right) \right) = -11 \times 16 – \left(-11 \right) \times \left(-14 \right)$

$=>-11 \times \left(16 + 14\right) = -176 – 154$

$=>-11 \times 30 = -330$ $=>-330= -330$

Decimal Numbers

Consider any three decimal numbers $A = 2.5$, $B = 5.6$ and $C = 0.8$.

Distributive Property of Multiplication Over Addition

We want to verify that $A \times \left(B + C \right) = A \times B + A \times C$

$2.5 \times \left(5.6 + 0.8 \right) = 2.5 \times 5.6 + 2.5 \times 0.8$

$=>2.5 \times 6.4 = 14 + 2$

$=>16 = 16$

Distributive Property of Multiplication Over Subtraction

We want to verify that $A \times \left(B – C \right) = A \times B – A \times C$

$2.5 \times \left(5.6 – 0.8 \right) = 2.5 \times 5.6 – 2.5 \times 0.8$

$=>2.5 \times 4.8 = 14 – 2$

$=>12 = 12$

Fractions

Consider any three fractions $A = \frac {1}{2}$, $B = \frac {2}{3}$ and $C = \frac {3}{4}$.

Distributive Property of Multiplication Over Addition

We want to verify that $A \times \left(B + C \right) = A \times B + A \times C$

$\frac {1}{2} \times \left( \frac {2}{3} +  \frac {3}{4} \right) = \frac {1}{2} \times  \frac {2}{3} + \frac {1}{2} \times  \frac {3}{4}$

$=>\frac {1}{2} \times \frac{17}{12} = \frac {1}{3} + \frac {3}{8}$

$=>\frac {17}{24} = {17}{24}$

Distributive Property of Multiplication Over Subtraction

We want to verify that $A \times \left(B – C \right) = A \times B – A \times C$

$\frac {1}{2} \times \left( \frac {2}{3} –  \frac {3}{4} \right) = \frac {1}{2} \times  \frac {2}{3} – \frac {1}{2} \times  \frac {3}{4}$

$=>\frac {1}{2} \times \left(-\frac{1}{12}\right) = \frac {1}{3} – \frac {3}{8}$

$=>-\frac {1}{24} = -\frac {1}{24}$

Irrational Numbers

Consider any three fractions $A = 2\sqrt{3}$, $B = 3\sqrt{2}$ and $C = 5\sqrt{2}$.

Distributive Property of Multiplication Over Addition

We want to verify that $A \times \left(B + C \right) = A \times B + A \times C$

$2\sqrt{3} \times \left(3\sqrt{2} + 5\sqrt{2} \right) = 2\sqrt{3} \times 3\sqrt{2} + 2\sqrt{3} \times 5\sqrt{2}$

$=>2\sqrt{3} \times 8\sqrt{2} = 6\sqrt{6} + 10\sqrt{6}$

$=>16\sqrt{6} = 16\sqrt{6}$

Distributive Property of Multiplication Over Subtraction

We want to verify that $A \times \left(B – C \right) = A \times B – A \times C$

$2\sqrt{3} \times \left(3\sqrt{2} – 5\sqrt{2} \right) = 2\sqrt{3} \times 3\sqrt{2} – 2\sqrt{3} \times 5\sqrt{2}$

$=>2\sqrt{3} \times \left(-2\sqrt{2}\right) = 6\sqrt{6} – 10\sqrt{6}$

$=>-4\sqrt{6} = -4\sqrt{6}$

Does Distributive Property of Division Over Addition Holds?

Let’s verify whether the distributive property of division over addition is true, i.e.,

$A \div \left(B + C \right) = A \div B + A \div C$

Again consider any three numbers $A = 5$, $B = 2$ and $C = 3$

LHS = $5 \div \left(2 + 3 \right) = 5 \div 5 = 1$

RHS = $5 \div 2 + 5 \div 3 = 2.5 + 1.67 = 4.17$

Since, LHS $\ne$ RHS, therefore, the distributive property over addition does not hold.

Does Distributive Property of Division Over Subtraction Holds?

Let’s consider the same numbers: $A = 5$, $B = 2$ and $C = 3$

LHS = $5 \div \left(2 – 3 \right) = 5 \div \left(-1 \right) = -5$

RHS = $5 \div 2 – 5 \div 3 = 2.5 – 1.67 = 0.83$

Since, LHS $\ne$ RHS, therefore, the distributive property over subtraction does not hold.

Conclusion

The distributive property of multiplication over addition and subtraction states that the result remains the same irrespective of the fact whether a number is multiplied with another number or parts of it by splitting either by adding or subtracting. But it is applicable with multiplication but not with division.

Practice Problems

Verify the distributive property of multiplication over addition and subtraction for the following set of numbers

• $A = 2$, $B = 7$, $C = 3$
• $A = -1$, $B = 4$, $C = 2$
• $A = 0.5$, $B = 2$, $C = 0.3$

Recommended Reading

• Associative Property – Meaning & Examples
• Decimal Number System – With Types & Properties
• Multiplication & Division of Decimals

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