Learn how to cross multiply to solve algebraic equations with fractions and to verify whether two fractions are equivalent. This page provides clear examples, step-by-step explanations, and practice questions with their detailed solutions to help students, parents, and teachers master this important math skill.
Step 1: Multiply both sides of the equation by the product of the denominators \( b \times d \): \[ b \times d \times \dfrac{a}{b} = b \times d \times \dfrac{c}{d} \] Step 2: Simplify both sides: \[ \cancel{b} \times d \times \dfrac{a}{\cancel{b}} \;=\; b \times \cancel{d} \times \dfrac{c}{\cancel{d}} \] Step 3: Rewrite without denominators: \[ a \times d = b \times c \] The above method of transforming an equation with fractions into an equation without fractions is called cross multiplication.
Example 2: Solve the equation \[ \dfrac{x}{3} = \dfrac{10}{6} \] Step 1: Cross multiply to eliminate denominators: \[ 6 \times x = 3 \times 10 \] Step 2: Divide both sides by the coefficient of \( x \) (which is 6): \[ \dfrac{6 \times x}{6} = \dfrac{3 \times 10}{6} \] Step 3: Simplify to find \( x \): \[ x = 5 \]
Cross Multiplication can be used to verify if two fractions are equivalent.
Example 3: Are the fractions \[ \dfrac{4}{3} \quad \text{and} \quad \dfrac{12}{9} \] equivalent?
Step 1: Cross multiply:
\[
4 \times 9 = 36 \qquad \text{and} \qquad 3 \times 12 = 36
\]
Step 2: Compare the results.
Since both products are equal, the fractions are equivalent.
Thus, we can write:
\[
\dfrac{4}{3} = \dfrac{12}{9}
\]
The exercises below with solutions and explanations are all about using cross multiplication.