Arithmetic
The Distributive Property
The distributive property says a × (b + c) = a × b + a × c: to multiply a number by a sum, multiply it by each part and add the results. So 3 × (4 + 5) = 12 + 15 = 27. This guide shows why it works, how it turns hard products into easy mental math, how it expands and factors algebra, and what it does not apply to.
What is the distributive property?
The distributive property says that multiplying a number by a sum gives the same result as multiplying it by each part of the sum and then adding. The multiplier is “distributed” to every term inside the parentheses.
Example: 3 × (4 + 5)
Work it out both ways and the answers match.
It works the same way with subtraction, because subtracting is adding a negative: a × (b − c) = a × b − a × c. And it works with any number of terms: 5 × (2 + 3 + 1) = 10 + 15 + 5 = 30.
The distributive property is the one rule that links multiplication and addition. Its companions, the commutative and associative properties, cover order and grouping within a single operation.
You can test it with any numbers on a basic four-function calculator: work out 6 × 15, then work out 6 × 7 and 6 × 8 separately and add them. Both routes give 90.
Why the distributive property works
Multiplication counts equal groups. 3 × (4 + 5) means three groups, each holding 4 red counters and 5 blue counters. You can count all the counters at once (3 groups of 9), or count the reds and blues separately (3 groups of 4, plus 3 groups of 5). The counters are the same either way.
The area model
Picture a rectangle 6 units tall and 34 units wide. Its area is 6 × 34. Draw a line splitting the width into 30 and 4. Now there are two smaller rectangles, and their areas add up to the whole.
This picture is why the standard written method for multiplication works. When you multiply 6 by 34 on paper, you multiply 6 by the 4, then by the 30, and add. The multiplication calculator lays out exactly those partial products.
Using the distributive property for mental math
This is where the property earns its keep. Break an awkward number into a round number plus or minus a small one, multiply each part, and combine.
Example: 7 × 23
Example: 8 × 99
99 is one less than 100, so split it as a subtraction.
Example: 6 × 48
Example: 4 items at $1.99
Pricing just under a round number makes this split especially handy at the store.
Splits worth knowing
| Problem | Split | Working | Answer |
|---|---|---|---|
| 9 × 47 | (10 − 1) × 47 | 470 − 47 | 423 |
| 12 × 13 | 12 × (10 + 3) | 120 + 36 | 156 |
| 15 × 14 | 15 × (10 + 4) | 150 + 60 | 210 |
| 25 × 36 | 25 × (40 − 4) | 1,000 − 100 | 900 |
| 35 × 102 | 35 × (100 + 2) | 3,500 + 70 | 3,570 |
| 7 × 999 | 7 × (1,000 − 1) | 7,000 − 7 | 6,993 |
| 4 × 2.25 | 4 × (2 + 0.25) | 8 + 1 | 9 |
The same trick is how most people learn the hard rows of the multiplication chart: 12 × 7 = 70 + 14 = 84 is 7 × (10 + 2) in disguise.
Multiplying two-digit numbers
When both numbers have more than one digit, split both. Every part of the first number multiplies every part of the second, and the products are added.
Example: 23 × 47
Four partial products, one for each pairing. The long multiplication method on paper produces the same four numbers, just arranged in columns.
In algebra the same pattern is often taught as FOIL (first, outer, inner, last): (x + 2)(x + 3) = x² + 3x + 2x + 6 = x² + 5x + 6. FOIL is just the distributive property applied twice.
The distributive property in algebra
In algebra you usually cannot work out the inside of the parentheses first, because it contains a variable. The distributive property is how you remove the parentheses.
Watch the signs in the last two lines. The multiplier reaches every term, and a negative multiplier flips every sign inside. −2 × −5 = +10, by the negative number rules. A minus sign in front of parentheses is a multiplier of −1.
Checking with numbers
You can test an expansion by picking a value for x. With x = 6, −2(x − 5) = −2 × 1 = −2, and −2x + 10 = −12 + 10 = −2. Same answer, so the expansion is right.
Combining like terms
Adding 3x + 5x to get 8x is also the distributive property, used backwards: 3x + 5x = (3 + 5)x = 8x.
Using it backwards: factoring
Read the formula from right to left and it becomes factoring: pull a common factor out of each term and write it in front of the parentheses. Spotting that shared factor is easier when you can list each number’s factors, which the factor calculator does.
To factor completely, take out the greatest common factor, not just any common factor. 12 + 18 = 2(6 + 9) is true but unfinished; 6(2 + 3) is fully factored. The GCF calculator finds that factor for you, and the guide on finding the GCF and LCM shows the methods.
What does and does not distribute
Multiplication distributes over addition and subtraction. Plenty of other combinations look as if they should and do not.
| Does it distribute? | Test | Result |
|---|---|---|
| Multiplication over addition | 3 × (4 + 5) vs 12 + 15 | Yes: 27 = 27 |
| Division over addition, sum on top | (12 + 18) ÷ 6 vs 2 + 3 | Yes: 5 = 5 |
| Division over addition, sum on the bottom | 6 ÷ (2 + 3) vs 6 ÷ 2 + 6 ÷ 3 | No: 1.2 vs 5 |
| Multiplication over multiplication | 2 × (3 × 4) vs (2 × 3) × (2 × 4) | No: 24 vs 48 |
| Squaring over addition | (3 + 4)² vs 3² + 4² | No: 49 vs 25 |
The division rows explain a classic fraction rule. You can split (a + b)/c into a/c + b/c, which is how fractions with a common denominator are added, but you cannot split c/(a + b). The squaring row is a common algebra error; the exponent rules list what powers do allow.
Distributive property and the order of operations
PEMDAS says to work out parentheses first, so 3 × (4 + 5) becomes 3 × 9. The distributive property gives a second, equally valid route: distribute first, then add. The two always agree, so you can pick whichever is easier.
When the inside can be simplified, PEMDAS is usually quicker. When it cannot, as in 3(x + 4), distributing is the only way forward. The order of operations calculator follows the PEMDAS route step by step if you want to check a result.
Common mistakes
- Multiplying only the first term.
3(x + 4)is3x + 12, not3x + 4. Draw an arrow from the multiplier to every term. - Dropping a sign.
−2(x − 5)is−2x + 10. A negative multiplier changes every sign inside. - Distributing over multiplication.
2(3 × 4)is 24, not6 × 8. Inside the parentheses must be a sum or a difference. - Squaring term by term.
(a + b)²is nota² + b²; it isa² + 2ab + b². - Splitting a sum in a denominator.
6/(2 + 3)is 1.2, not6/2 + 6/3. - Stopping short when factoring. Take out the greatest common factor, not just any factor.
Frequently asked questions
What is the distributive property in math?
The distributive property says that multiplying a number by a sum equals multiplying it by each part and adding: a(b + c) = ab + ac. For example, 3 × (4 + 5) = 3 × 4 + 3 × 5 = 12 + 15 = 27, the same as 3 × 9. It also works with subtraction inside the parentheses.
What is an example of the distributive property?
A mental-math example: 7 × 23 = 7 × 20 + 7 × 3 = 140 + 21 = 161. An algebra example: 3(x + 4) = 3x + 12. In both, the number outside the parentheses multiplies every term inside, and the results are added together.
How do you use the distributive property with negative numbers?
Multiply each term inside by the negative number and follow the sign rules. −2(x − 5) = −2x + 10, because −2 × −5 = +10. A minus sign in front of parentheses is a multiplier of −1, so −(x − 7) = −x + 7. Every sign inside flips.
What is 8 times 99 using the distributive property?
8 × 99 = 792. Write 99 as 100 − 1, then distribute: 8 × 100 − 8 × 1 = 800 − 8 = 792. Rounding up to a friendly number and subtracting the extra is one of the quickest mental-math uses of the distributive property.
Does division have a distributive property?
Only when the sum is being divided. (12 + 18) ÷ 6 = 12 ÷ 6 + 18 ÷ 6 = 2 + 3 = 5 works. Dividing by a sum does not: 6 ÷ (2 + 3) = 1.2, but 6 ÷ 2 + 6 ÷ 3 = 5. That is why you can split a fraction over its numerator but not over its denominator.
What is the difference between the distributive and associative property?
The distributive property mixes two operations: multiplication over addition, as in a(b + c) = ab + ac. The associative property involves one operation and only changes grouping, as in (a + b) + c = a + (b + c). If two different operations appear, it is the distributive property.
How is factoring related to the distributive property?
Factoring is the distributive property run backwards. Instead of multiplying a factor into the parentheses, you pull a common factor out: 6x + 9 = 3(2x + 3). To factor fully, take out the greatest common factor. You can check by distributing again to get the original expression.
Calculators for this
Every one of these shows the working, so you can check the method as well as the answer.
More guides
https://fourfunctioncalculator.com/distributive-property/