Fractions & Decimals | Math Learning Guide

📋 Prerequisites

None — start here if you need to build basics

Fractions and decimals are two ways to represent parts of a whole. They are fundamental to all of mathematics and appear in every competition, from AMC 8 all the way up to AIME. Mastering fraction arithmetic is essential — you will use it in algebra, geometry, number theory, and combinatorics.

📚 Key Concepts

Definitions

Fractions are in the form of $\frac{q}{p}$, where $p$ and $q$ are integers, usually positive, with $q\neq 0$. The following rules apply to both positive and negative integers.

Cancel out

When there is an integer common divisor for both numerator and denominator, the fraction can be simplified

\[ \frac{ac}{bc} = \frac{a}{b} \]

Fraction Arithmetic: Addition & Subtraction

To add or subtract fractions, find a common denominator:

\[ \frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd} \]

Here, $a$, $b$, $c$, $d$ are both integers, $c,\, d \neq 0$

The least common denominator (LCD) is the smallest number divisible by both denominators. Using the LCD keeps numbers small and reduces simplification work later.

Multiplication

Multiply numerators together and denominators together:

\[ \frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd} \]

Tip: Always cross-cancel before multiplying — it keeps numbers small!

Division

Dividing by a fraction is the same as multiplying by its reciprocal:

\[ \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc} \]

Simplifying Fractions

Divide numerator and denominator by their GCD
A fraction is in lowest terms when GCD(numerator, denominator) = 1
Competition answers should almost always be fully simplified

Mixed Numbers & Improper Fractions

Mixed number: whole number + proper fraction, e.g. 2 1/2
Improper fraction: numerator ≥ denominator, e.g. 5/2
For calculations, convert mixed numbers to improper fractions first

Decimals ↔ Fractions Conversion

Terminating decimal: write digits over appropriate power of 10, then simplify
Fraction to decimal: divide numerator by denominator
Repeating decimals: use algebra trick (set x = decimal, multiply by 10^n, subtract)

⚠️ Common Mistake: When adding fractions, do NOT add the denominators! $ \frac{1}{2} + \frac{1}{3} \neq \frac{2}{5} $. Always find a common denominator first.

💡 Key Insight: In competitions, fractions are almost always preferred over decimals because they are exact. Only use decimals when the problem explicitly asks for them.

✏️ Example Problems

📝 Example 1 (AMC 8 Level)

Compute: $ \displaystyle \frac{3}{5} - \frac{1}{6} $

Step 1: Find the least common denominator of 5 and 6.

LCD(5, 6) = 30

Step 2: Rewrite both fractions with denominator 30:

\[ \frac{3}{5} = \frac{18}{30}, \quad \frac{1}{6} = \frac{5}{30} \]

Step 3: Subtract:

\[ \frac{18}{30} - \frac{5}{30} = \frac{13}{30} \]

Step 4: Check if reducible. GCD(13, 30) = 1, so $ \frac{13}{30} $ is the final answer.

📝 Example 2 (AMC 8 Style Word Problem)

A pizza has 12 slices. Alice eats $ \frac{1}{3} $ of the pizza, then Bob eats $ \frac{1}{4} $ of what remains. How many slices are left?

Step 1: Find how many slices Alice eats.

\[ \frac{1}{3} \times 12 = 4 \text{ slices} \]

Step 2: Find remaining slices after Alice.

$ 12 - 4 = 8 $ slices remain

Step 3: Find how many slices Bob eats (1/4 of what's left).

\[ \frac{1}{4} \times 8 = 2 \text{ slices} \]

Step 4: Calculate final remaining slices.

$ 8 - 2 = 6 $

Answer: 6

Common trap: Don't do $ 1 - \frac{1}{3} - \frac{1}{4} = \frac{5}{12} $. Bob eats $ \frac{1}{4} $ of the remainder, not $ \frac{1}{4} $ of the whole pizza.

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