Solving Proportions Using Cross Products
Grade 8 math lesson on solving proportions using the cross products property. Students learn to set up proportional equations from word problems and apply cross multiplication to find unknown values in proportional situations.
Key Concepts
Property If two ratios are equal, their cross products are equal. For a proportion $\frac{a}{b} = \frac{c}{d}$, this means $ad = bc$.
Examples For $\frac{x}{12} = \frac{4}{6}$, cross products give $6x = 12 \cdot 4$, so $6x = 48$ and $x = 8$. For $\frac{y}{2.5} = \frac{4}{10}$, cross products give $10y = 2.5 \cdot 4$, so $10y = 10$ and $y = 1$. For $\frac{5}{z} = \frac{2}{8}$, cross products give $5 \cdot 8 = 2z$, so $40 = 2z$ and $z = 20$.
Explanation Turn a tricky fraction equation into a simple one! The 'criss cross' shortcut lets you multiply diagonally across the equals sign. This gets rid of the fractions and leaves you with a basic equation, making it super easy to find the missing value. It's like a magic trick for proportions!
Common Questions
How do you solve a proportion using cross products?
In a proportion a/b = c/d, the cross products are a times d and b times c, and they are equal. To solve for an unknown, cross multiply to create a simple equation, then divide both sides to isolate the variable.
What is the cross products property of proportions?
The cross products property states that if a/b = c/d, then a x d = b x c. This means the product of the means equals the product of the extremes, and it is used to solve for unknown terms in a proportion.
When should you use cross products to solve a proportion?
Use cross products when you have a proportion with one unknown. Set up the equation with the unknown in one position, cross multiply, then solve the resulting one-step or two-step equation.
What is an example of solving a proportion with cross products?
Solve: 4/6 = x/9. Cross multiply: 4 x 9 = 6 x x, so 36 = 6x. Divide both sides by 6: x = 6. Check: 4/6 = 6/9 = 2/3, which is true.