Fraction Study Guide

📖 Core Concepts Fraction – a ratio of two integers $a/b$; numerator $a$ counts parts taken, denominator $b$ (≠0) counts parts that make a whole. Sign rule – a single negative sign anywhere makes the whole fraction negative; two negatives give a positive. Proper vs. improper – |fraction| < 1 → proper; ≥ 1 → improper. Reciprocal – flip numerator and denominator; $ \frac{a}{b} \times \frac{b}{a}=1$. Invisible denominator – any integer $n$ is $n/1$. Mixed number – whole + proper fraction, e.g., $2\frac34 = \frac{11}{4}$. 📌 Must Remember Equivalent fractions: multiply or divide numerator & denominator by the same non‑zero integer. Simplify: divide numerator & denominator by their greatest common divisor (GCD). Cross‑multiply to compare: $\frac{a}{b} > \frac{c}{d}$ iff $ad > bc$. Add/Subtract: find a common denominator (LCD = LCM of denominators), then add/subtract numerators. Multiply: $\frac{a}{b}\times\frac{c}{d}= \frac{ac}{bd}$. Cancel before multiplying: remove any common factor between any numerator and any denominator. Divide by a fraction: multiply by its reciprocal, $\frac{a}{b}\div\frac{c}{d}= \frac{a}{b}\times\frac{d}{c}$. Decimal → fraction (terminating): $0.75 = \frac{75}{100} = \frac{3}{4}$ after simplifying. Repeating decimal → fraction: $0.\overline{37}= \frac{37}{99}$; for $0.12\overline{3}$ use $(123-12)/990 = \frac{111}{990}$. 🔄 Key Processes Simplifying a fraction Find GCD$(a,b)$. Divide both by GCD → lowest terms. Adding unlike fractions Compute LCD = LCM$(b,d)$. Rewrite: $\frac{a}{b}=\frac{a\cdot(LCD/b)}{LCD}$, $\frac{c}{d}=\frac{c\cdot(LCD/d)}{LCD}$. Add numerators, keep LCD. Cross‑multiply comparison Multiply across: $a d$ vs. $b c$. Larger product ↔ larger fraction. Dividing by a fraction Flip divisor → reciprocal. Multiply (apply cancellation first). Converting a repeating decimal to a fraction (algebraic method) Set $x = $ repeating decimal. Multiply by $10^{k}$ where $k$ = length of repeat (or $10^{m+k}$ if non‑repeat part of length $m$). Subtract original equation, solve for $x$. Rationalizing a binomial radical denominator Multiply numerator & denominator by the conjugate $(\sqrt{a}\pm\sqrt{b})$ to eliminate radicals in the denominator. 🔍 Key Comparisons Proper vs. improper – proper: $|a|<|b|$; improper: $|a|\ge|b|$. Negative sign location – $-\frac12$, $\frac{-1}{2}$, $\frac{1}{-2}$ all equal $-0.5$; two negatives → positive. Mixed number vs. improper fraction – $2\frac34 = \frac{11}{4}$ (mixed easier to read; improper easier for arithmetic). Decimal fraction vs. percentage – $0.75 = \frac{75}{100} = 75\%$. ⚠️ Common Misunderstandings “Canceling across the addition sign.” You may only cancel common factors between numerators and denominators before multiplication, not after adding fractions. Assuming $ \frac{a}{b}=a\cdot b$ – the slash means division, not multiplication. Treating $0.\overline{3}= \frac{3}{10}$ – the correct fraction is $\frac{1}{3}$ (use $9$ in denominator). Mixing up LCD and LCM. LCD is the least common denominator; using a larger common denominator works but makes the arithmetic longer. 🧠 Mental Models / Intuition “Fraction = slice of a pizza.” Numerator = number of slices you have; denominator = total slices that make a whole pizza. Cross‑multiply = “stretch both fractions to a common width.” If the stretched widths (products) differ, the taller one is larger. Reciprocal = “flip the picture.” Multiplying by a reciprocal is the same as “undoing” a division. 🚩 Exceptions & Edge Cases Zero denominator – undefined; never allowed. Negative denominator – move the sign to the numerator for a standard form. Repeating block of zeros (e.g., $0.25000\overline{0}$) is just a terminating decimal; treat as $0.25$. Complex fractions – always simplify by multiplying numerator and denominator by the LCD of the inner fractions. 📍 When to Use Which Add/Subtract → find LCD; use cross‑multiply only for comparison. Multiply → cancel first, then multiply numerators/denominators. Divide → flip divisor (reciprocal) → multiply. Convert decimal → fraction → if terminating, use powers of 10; if repeating, use algebraic method. Rationalize denominator → when a radical appears in a denominator and the answer must be “rationalized” (common in exact‑answer problems). 👀 Patterns to Recognize Same denominator → larger numerator = larger fraction. Identical numerators → smaller denominator = larger fraction. Repeating block of length k → denominator will be $99…9$ (k nines). When a fraction appears inside another fraction, the whole expression can be cleared by multiplying numerator & denominator by the LCD of all inner denominators. 🗂️ Exam Traps “Canceling across a plus sign.” Example: $\frac{1}{2} + \frac{1}{3}$ → you cannot cancel the 2 and 3. Mistaking $ \frac{a}{b} $ for $a \times b$. Look for the slash; it means division. Choosing the wrong sign for negative fractions. Remember: a single negative anywhere = negative overall; two negatives = positive. Using the wrong denominator when converting a repeating decimal. Always use 9s for the repeat length, plus 0s for any non‑repeating part (e.g., $0.12\overline{3}$ → denominator $990$). Leaving a mixed number un‑converted before multiplication/division. Convert to an improper fraction first; otherwise you’ll multiply whole numbers with fractions incorrectly. --- This guide pulls directly from the provided outline; no external information was added.