ALGEBRAArithmeticMathematics Calculator
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Distributive Property

a(b+c) = ab + ac โ€” multiply the factor by each term inside the parentheses. Works with subtraction and multi-term. Essential for mental math, algebra, and factoring.

Concept Fundamentals
a(b+c) = ab+ac
Rule
a(bโˆ’c) = abโˆ’ac
Subtraction

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Mental math: 7ร—98 = 7(100โˆ’2) = 700 โˆ’ 14 = 686. Area model: rectangle width a, length (b+c) has area ab + ac. Factoring reverses it: ab + ac = a(b + c).

Key quantities
a(b+c) = ab+ac
Rule
Key relation
a(bโˆ’c) = abโˆ’ac
Subtraction
Key relation

Ready to run the numbers?

Why: The distributive property underlies multiplication of sums and is fundamental to algebra.

How: Multiply the factor a by each term inside: aร—b + aร—c = ab + ac. For subtraction: a(bโˆ’c) = ab โˆ’ ac.

Mental math: 7ร—98 = 7(100โˆ’2) = 700 โˆ’ 14 = 686.Area model: rectangle width a, length (b+c) has area ab + ac.
Sources:NCTM Standards

Run the calculator when you are ready.

Start CalculatingEnter a, b, c to expand a(b+c) or a(bโˆ’c).

Enter Values

distributive.sh
CALCULATED
$ distributive --a 3 --b 4 --c 5
Left
27
Right
27
Equal
Yes โœ“
Expression
3(4 + 5)
Distributive Property Calculator
3(4 + 5) = 3ร—4 + 3ร—5
Both = 27 โœ“
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Term Values

Proportions

๐Ÿ“ Step-by-Step Breakdown

EXPAND
Left side
3(4 + 5) = 3 ร— 9 = 27
EXPAND
Right side
3ร—4 + 3ร—5 = 27
RESULT
Equal?
Yes โ€” distributive property holds! โœ“

For educational and informational purposes only. Verify with a qualified professional.

๐Ÿงฎ Fascinating Math Facts

๐Ÿ“

The area model: a(b+c) is the area of a rectangle split into ab and ac.

โ€” Geometry

๐Ÿงฎ

Mental math: 23ร—12 = 23(10+2) = 230 + 46 = 276.

โ€” Arithmetic

๐Ÿ“‹ Key Takeaways

  • โ€ข a(b + c) = ab + ac โ€” multiply each term inside by the factor outside
  • โ€ข a(b โˆ’ c) = ab โˆ’ ac โ€” works with subtraction too
  • โ€ข Mental math: 7ร—98 = 7(100โˆ’2) = 700 โˆ’ 14 = 686
  • โ€ข Area model: rectangle width a, length (b+c) has area ab + ac

๐Ÿ’ก Did You Know?

๐Ÿ“The area model: a(b+c) is the area of a rectangle split into ab and ac.Source: Geometry
๐ŸงฎMental math: 23ร—12 = 23(10+2) = 230 + 46 = 276.Source: Arithmetic
โž–a(bโˆ’c) = ab โˆ’ ac. The negative distributes to each term.Source: Algebra
๐Ÿ“ŠMulti-term: a(b+c+d) = ab + ac + ad.Source: Algebra
๐ŸŽฏFactoring reverses it: ab + ac = a(b + c).Source: Algebra
๐Ÿ“Used in algebra to expand and factor expressions.Source: Pre-Algebra

๐Ÿ“– How It Works

To expand a(b + c), multiply the factor a by each term inside the parentheses: aร—b + aร—c = ab + ac. For a(bโˆ’c), we get ab โˆ’ ac. The distributive property is fundamental in arithmetic and algebra.

๐Ÿ“ Worked Example: 3(4+5)

Left: 3(4+5) = 3ร—9 = 27

Right: 3ร—4 + 3ร—5 = 12 + 15 = 27

Result: Both equal 27 โœ“

๐Ÿš€ Real-World Applications

๐Ÿงฎ Mental Math

7ร—98 = 7(100-2) = 700 - 14 = 686

๐Ÿ“ Area Model

Rectangle area: a(b+c) = ab + ac

๐Ÿ“Š Algebra

Expand and factor expressions

๐Ÿ’ฐ Finance

Distribute costs across items

๐Ÿ”ฌ Physics

Force distribution, work

๐Ÿ“ Measurement

Scale factors, conversions

โš ๏ธ Common Mistakes to Avoid

  • Watch signs: โˆ’2(3โˆ’5) = โˆ’6 + 10 = 4, not โˆ’6 โˆ’ 10
  • Distribute to ALL terms: a(b+c+d) = ab + ac + ad
  • Factoring is inverse: ab + ac = a(b+c), not a(bc)

๐ŸŽฏ Expert Tips

๐Ÿ’ก Watch signs

โˆ’2(3โˆ’5) = โˆ’6 + 10 = 4

๐Ÿ’ก Mental math

Break numbers: 7ร—98 = 7(100โˆ’2)

๐Ÿ’ก Verify

Substitute values to check

๐Ÿ’ก Factoring

Reverse: ab + ac = a(b + c)

๐Ÿ“Š Reference Table

FormExpansion
a(b + c)ab + ac
a(b โˆ’ c)ab โˆ’ ac
โˆ’a(b + c)โˆ’ab โˆ’ ac

โ“ FAQ

What is the distributive property?

a(b+c) = ab + ac. A number times a sum equals the sum of that number times each term.

Does it work with subtraction?

Yes. a(bโˆ’c) = ab โˆ’ ac.

How do I use it for mental math?

Break numbers: 7ร—98 = 7(100โˆ’2) = 700 โˆ’ 14 = 686.

What about three terms?

a(b+c+d) = ab + ac + ad.

What is factoring?

The reverse: ab + ac = a(b + c). Factor out the common factor a.

๐Ÿ“Œ Summary

The distributive property: a(b+c) = ab + ac. Multiply the factor by each term inside. Works with subtraction and multi-term. Essential for mental math, algebra, and factoring.

๐Ÿ”— Next Steps

Try the Associative Property Calculator or the FOIL Method Calculator for binomial expansion.

โš ๏ธ Disclaimer: Results are for educational purposes. Verify critical calculations independently.

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