Cross Product
The cross product property states that in two equal ratios (a proportion), the product of the numerator of the first ratio and the denominator of the second equals the product of the denominator of the first and the numerator of the second: if a/b = c/d, then a × d = b × c. This cross-multiplication method is the standard tool for solving proportion equations. For example, if 3/4 = n/12, then 3 × 12 = 4 × n, giving 36 = 4n, so n = 9. This 7th grade skill is covered in Saxon Math, Course 2.
Key Concepts
Property Multiply the upper term of one ratio by the lower term of the other. The cross products of equal ratios are equal.
Examples Check $\frac{4}{5} = \frac{8}{10}$: $4 \times 10 = 40$ and $5 \times 8 = 40$. Equal! Check $\frac{2}{3} = \frac{5}{8}$: $2 \times 8 = 16$ and $3 \times 5 = 15$. Not equal!
Explanation Think of the cross product as a super fast trick to check if two ratios are truly equal partners. Multiply diagonally! If the answers match, it’s a valid proportion. It's a trusty lie detector for ratios.
Common Questions
What is the cross product in math?
The cross product (cross multiplication) states that for equal ratios a/b = c/d, multiplying diagonally gives equal products: a × d = b × c. It’s the main method for solving proportion equations.
How do you use cross multiplication to solve a proportion?
Multiply the numerator of each ratio by the denominator of the other (diagonally), set the products equal, then solve. For 5/8 = n/24: 5 × 24 = 8 × n → 120 = 8n → n = 15.
Why does cross multiplication work?
Cross multiplication works because multiplying both sides of the equation a/b = c/d by bd gives ad = bc. It eliminates the fractions and produces a simple equation to solve.
What is a proportion?
A proportion is an equation stating that two ratios are equal: a/b = c/d. Solving a proportion means finding the value that makes the two ratios equivalent.
When do you use cross multiplication?
Use cross multiplication when you have a proportion with one unknown value. It’s faster than finding a common denominator and works for any proportion.
What are common mistakes with cross multiplication?
The most common mistake is multiplying straight across instead of diagonally. Remember: numerator of first × denominator of second, and denominator of first × numerator of second.
Which textbook covers cross products?
Saxon Math, Course 2 covers the cross product property for solving proportions.