Cross Multiplication - Algebra-1

1. Basic Concept
Proportions state that two ratios (or fractions) are equal, expressed as: $\frac{a}{b} = \frac{c}{d}$

Cross-multiplication is a method to solve proportions by multiplying the numerator of one fraction by the denominator of the other, setting the products equal:
$a \times d = b \times c$

This transforms the proportion into a simple equation, making it easier to solve for an unknown variable.

2. Key Points
Purpose: Used to solve for an unknown in a proportion (e.g., $\frac{x}{3} = \frac{4}{6}$ ).
When to Apply: Only valid when the proportion is set up as two equal fractions.
Verification: After solving, always check if both sides of the original proportion simplify to the same value.

Common Pitfalls:
Misaligning terms (e.g., multiplying $a\times c$ instead of $a \times d$ ).
Forgetting to simplify fractions before cross-multiplying.

3. Classic Example
Problem: Solve for x in $\frac{x}{5} = \frac{3}{15}$ .

Solution:
1. Cross-multiply:
$x \times 15 = 5 \times 3 \implies 15x = 15$
2. Solve for x:
$x = \frac{15}{15} = 1$
3. Verification: Substitute x = 1 back into the original proportion:
$\frac{1}{5} = \frac{3}{15} \implies \frac{1}{5} = \frac{1}{5} \quad \text{✓}$

4. Problem-Solving Tips
Simplify First: Reduce fractions if possible (e.g., $\frac{3}{15} = \frac{1}{5}$ ) to make calculations easier.

Isolate the Variable: After cross-multiplying, use basic algebra to solve for the unknown.

Check Units: Ensure ratios compare the same units (e.g., miles/hour vs. miles/hour).

Advanced Application:
Use cross-multiplication to solve real-world problems involving scales (e.g., maps) or similar triangles.

Example:
If $\frac{2\,\text{cm}}{5\,\text{km}} = \frac{x\,\text{cm}}{20\,\text{km}} $ , cross-multiply to find $x = 8 \,\text{cm} $ .

Summary
Cross-multiplication is a powerful tool for solving proportions by converting them into linear equations. Mastery of this technique ensures accuracy in algebraic and real-world applications.

$\frac{a}{b} = \frac{c}{d} \implies a \times d = b \times c$$\frac{a}{b} = \frac{c}{d} \implies a \times d = b \times c}$