Formula and Method for the Distributive Property
The distributive property of multiplication over addition (and subtraction) states that multiplying a sum by a number gives the same result as multiplying each addend individually and then adding the products: a × (b + c) = a × b + a × c. The property also holds for subtraction: a × (b − c) = a × b − a × c. This calculator distributes the outer factor a across the two inner terms b and c, shows each partial product, and combines them into the final expanded result.
How the calculation works
Enter the outer factor a, choose whether the inner terms are added or subtracted, then enter b and c. The calculator multiplies a by b and a by c separately, then combines the two partial products with the same operator that joins b and c inside the parentheses. Because multiplication distributes over addition and subtraction, this expanded sum always equals a × (b ± c) computed directly.
Common mistakes
- Dropping the sign: when the inner operation is subtraction, the second product must also be subtracted — a × (b − c) = ab − ac, not ab + ac.
- Distributing over multiplication: the property applies to addition and subtraction inside the parentheses, not to a product — a × (b × c) is not (a×b) × (a×c).
- Forgetting to distribute to every term: with more than two terms, such as a(b + c + d), the factor a must multiply each term individually: ab + ac + ad.
Real-world applications
- Mental math shortcuts: 6 × 102 = 6 × (100 + 2) = 600 + 12 = 612, avoiding long multiplication.
- Algebra: expanding expressions like 3(x + 4) = 3x + 12 before combining like terms or solving an equation.
- Factoring in reverse: recognizing ab + ac as a(b + c) to simplify an expression or spot a common factor.
- Splitting a total: distributing a per-unit price or tax rate across multiple categories to check a receipt or invoice.