Cross multiply
Cross multiplying is the technique for solving a proportion equation by multiplying the numerator of each fraction by the denominator of the other, creating a simple multiplication equation. For the proportion 25/3 = b/60, cross multiply to get 3b = 25 x 60 = 1500, then b = 500. This Grade 7 math skill from Saxon Math, Course 2 is the primary algebraic tool for solving any proportion problem and connects directly to clearing fractions from algebraic equations — a skill used throughout algebra, geometry, and beyond.
Key Concepts
Property To solve a proportion, multiply the numerator of one fraction by the denominator of the other. For $\frac{a}{b} = \frac{c}{d}$, the result is $ad = bc$.
Examples Given $\frac{12}{40} = \frac{30}{p}$, cross multiplying gives $12p = 40 \cdot 30$. Given $\frac{25}{3} = \frac{b}{60}$, cross multiplying gives $3b = 25 \cdot 60$.
Explanation Cross multiplication is a magical trick to eliminate fractions from a proportion. Just multiply diagonally across the equals sign to turn a tricky fraction equation into a simple one that’s much faster and easier to solve for the unknown.
Common Questions
What does it mean to cross multiply?
Cross multiplying in a proportion a/b = c/d means computing a times d and b times c. Because the fractions are equal, these cross products are always equal: ad = bc.
How do I use cross multiplication to solve for an unknown?
Set the cross products equal, forming a multiplication equation. Then divide both sides to isolate the variable. For 12/40 = 30/p: 12p = 1200, so p = 100.
Why does cross multiplication work?
Multiplying both sides of the proportion a/b = c/d by the product b times d (both denominators) eliminates the fractions, producing the equivalent equation ad = bc.
When do I use cross multiplication vs other methods?
Cross multiplication is the most efficient method for proportions with one variable. For more complex equations, other algebraic methods may be needed.
When do students learn cross multiplication?
Cross multiplication is introduced as the main proportion-solving technique in Grade 7. Saxon Math, Course 2 covers it in Chapter 5 and uses it throughout the course.
What are common mistakes with cross multiplication?
Students sometimes multiply in the same direction (numerator times numerator) rather than diagonally. Always multiply across: one numerator with the opposite denominator.
How does cross multiplication connect to clearing fractions in algebra?
In algebra, clearing fractions by multiplying both sides of an equation by the LCD is the same idea as cross multiplication generalized. Both techniques eliminate denominators to create a simpler equation.