Fractions: Complete Lesson (addition, multiplication, comparison, simplification)

What is a fraction?

A fraction $\dfrac{a}{b}$ expresses the ratio between two numbers. Also called a quotient.

• $a$: numerator (top)
• $b$: denominator (bottom, must be $\neq 0$)

Simplify a fraction

Irreducible means $\gcd(a, b) = 1$.

To simplify: divide numerator and denominator by a common factor.

Example: $\dfrac{12}{18}$. $\gcd(12, 18) = 6$. Divide top and bottom by 6: $\dfrac{2}{3}$.

Trick: divide by 2 as long as possible, then 3, then 5, etc.

Adding (or subtracting) fractions

Step 1: find a common denominator (often the product, or the LCM). Step 2: rewrite each fraction with that denominator. Step 3: add numerators. Step 4: simplify.

$\dfrac{a}{b} + \dfrac{c}{d} = \dfrac{ad + bc}{bd}$

Example: $\dfrac{2}{3} + \dfrac{1}{4} = \dfrac{8}{12} + \dfrac{3}{12} = \dfrac{11}{12}$.

Multiplying fractions

$\dfrac{a}{b} \times \dfrac{c}{d} = \dfrac{ac}{bd}$

Top × top, bottom × bottom. No common denominator needed.

Example: $\dfrac{2}{3} \times \dfrac{5}{7} = \dfrac{10}{21}$.

Trick: simplify BEFORE multiplying if possible.

Dividing fractions

Divide by a fraction = multiply by its reciprocal.

$\dfrac{a}{b} \div \dfrac{c}{d} = \dfrac{a}{b} \times \dfrac{d}{c} = \dfrac{ad}{bc}$

Example: $\dfrac{3}{5} \div \dfrac{2}{7} = \dfrac{3}{5} \times \dfrac{7}{2} = \dfrac{21}{10}$.

Comparing fractions

Method 1 (cross product): $\dfrac{a}{b} = \dfrac{c}{d} \iff ad = bc$

Method 2 (common denominator): rewrite both with same denominator, compare numerators.

Example: $\dfrac{3}{4}$ vs $\dfrac{5}{7}$. $3 \times 7 = 21 > 4 \times 5 = 20$. So $\dfrac{3}{4} > \dfrac{5}{7}$.

Fraction to decimal

Every fraction is a finite or repeating decimal.

• $\dfrac{1}{2} = 0.5$
• $\dfrac{1}{4} = 0.25$
• $\dfrac{1}{3} = 0.333\ldots$ (repeating)

Percent to fraction

$x\% = \dfrac{x}{100}$

E.g., $25\% = \dfrac{1}{4}$.

Common mistakes

1. “Cross-adding”: $\dfrac{1}{2} + \dfrac{1}{3} \neq \dfrac{2}{5}$. Need common denominator.
2. Divide by 0: undefined.
3. Simplifying a non-factored sum: $\dfrac{2+3}{2+4}$ ≠ $\dfrac{3}{4}$! Only simplify products.
4. Forgetting to simplify at the end.

Takeaways

• Addition: common denominator mandatory.
• Multiplication: top × top, bottom × bottom.
• Division: multiply by reciprocal.
• Comparison: cross product or common denominator.
• Always simplify.

Related

• Tips: Adding two fractions fast, Equal fractions cross product.