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.
Let us consider the equation:
\[ \dfrac{a}{b} = \dfrac{c}{d} \]
Equations with fractions such as the above are sometimes challenging to solve because of the denominators. Let us find an equivalent equation without denominators.
The above method of transforming an equation with fractions into an equation without fractions is called cross multiplication.
Example 1: 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):
\[ \dfrac{6 \times x}{6} = \dfrac{30}{6} \]
Step 3: Simplify to find \( x \):
\[ x = 5 \]
Example 2: Are the fractions \( \dfrac{4}{3} \) and \( \dfrac{12}{9} \) 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 are all about using cross multiplication to solve for variables and check fraction equivalency.
Solve the equation: \( \dfrac{x}{6} = \dfrac{3}{2} \)
Use cross multiplication to rewrite the equation:
\[ 2 \times x = 3 \times 6 \]
Simplify:
\[ 2x = 18 \]
Divide both sides by 2:
\[ \frac{2x}{2} = \frac{18}{2} \implies \mathbf{x = 9} \]
Solve the equation: \( \dfrac{1}{3x} = \dfrac{2}{24} \)
Cross multiply denominators and numerators:
\[ 1 \times 24 = 3x \times 2 \]
Simplify:
\[ 24 = 6x \]
Divide both sides by 6:
\[ x = \frac{24}{6} \implies \mathbf{x = 4} \]
Solve the equation: \( \dfrac{3}{2} = \dfrac{12}{4x} \)
Use cross multiplication:
\[ 3 \times 4x = 2 \times 12 \]
Simplify:
\[ 12x = 24 \]
Divide both sides by 12:
\[ x = \frac{24}{12} \implies \mathbf{x = 2} \]
Solve the equation: \( \dfrac{4}{6} = \dfrac{x}{9} \)
Cross multiply:
\[ 4 \times 9 = 6 \times x \]
Simplify and solve for \(x\):
\[ 36 = 6x \implies x = \frac{36}{6} \implies \mathbf{x = 6} \]
Solve the equation: \( 2 = \dfrac{x}{14} \)
Express 2 as a fraction (\(\frac{2}{1}\)) and cross multiply:
\[ \frac{2}{1} = \frac{x}{14} \]
Cross multiply:
\[ 2 \times 14 = 1 \times x \]
Simplify:
\[ \mathbf{28 = x} \]
Solve the equation: \( \dfrac{2}{x+2} = \dfrac{1}{7} \)
Cross multiply:
\[ 2 \times 7 = (x + 2) \times 1 \]
Simplify and solve for \(x\):
\[ 14 = x + 2 \]
\[ x = 14 - 2 \implies \mathbf{x = 12} \]
Which of the following pairs of fractions are equivalent (equal)?
Definition: For two fractions to be equivalent, their cross multiplication quantities must be equal.
a) \( \dfrac{5}{6} \) and \( \dfrac{15}{18} \)
\( A = 5 \times 18 = 90 \)
\( B = 6 \times 15 = 90 \)
Since \(A = B\), the fractions are equivalent (\( \frac{5}{6} = \frac{15}{18} \)).
b) \( \dfrac{5}{3} \) and \( \dfrac{20}{13} \)
\( A = 5 \times 13 = 65 \)
\( B = 3 \times 20 = 60 \)
Since \(A \neq B\), the fractions are not equivalent.
c) \( \dfrac{25}{35} \) and \( \dfrac{5}{7} \)
\( A = 25 \times 7 = 175 \)
\( B = 35 \times 5 = 175 \)
Since \(A = B\), the fractions are equivalent (\( \frac{25}{35} = \frac{5}{7} \)).
d) \( \dfrac{23}{7} \) and \( \dfrac{46}{17} \)
\( A = 23 \times 17 = 391 \)
\( B = 7 \times 46 = 322 \)
Since \(A \neq B\), the fractions are not equivalent.