Solving Proportions Using Cross Products
Solve proportions using cross products in Grade 9 algebra: multiply diagonally across the equal sign (a/b = c/d → ad = bc) to find unknown values in ratio and proportion problems.
Key Concepts
Property Cross Products Property : If $\frac{a}{b} = \frac{c}{d}$ and $b \ne 0$ and $d \ne 0$, then $ad = bc$.
Examples Solve $\frac{x}{10} = \frac{4}{5}$. Cross multiply to get $5 \cdot x = 10 \cdot 4$, so $5x = 40$, which means $x=8$. Solve $\frac{y+2}{8} = \frac{5}{4}$. Cross multiply to get $4(y+2) = 8 \cdot 5$. This simplifies to $4y + 8 = 40$, so $4y=32$ and $y=8$.
Explanation Think of this as the ultimate shortcut for solving proportions! Instead of guessing the missing number, you just multiply the numbers that are diagonal from each other across the equals sign. This 'cross multiplication' trick magically turns a tricky fraction problem into a simple equation you already know how to solve. It is a powerful tool!
Common Questions
What is the cross products method for solving proportions?
If a/b = c/d, then cross-multiplying gives ad = bc. Multiply the numerator of each fraction by the denominator of the other, set them equal, then solve the resulting linear equation.
How do you solve x/5 = 12/20 using cross products?
Cross-multiply: x · 20 = 5 · 12. Simplify: 20x = 60. Divide both sides by 20: x = 3. Cross products always produce a linear equation that is straightforward to solve.
When can you use cross multiplication?
Cross multiplication applies when you have exactly one fraction on each side of an equals sign — that is, a true proportion. It does not apply when there are added fractions or expressions not in proportion form.