Ratio Study Guide

📖 Core Concepts Ratio – tells how many times one quantity contains another; written $a:b$, $a:b$, or as the quotient $a/b$. Antecedent – the first term of a ratio; Consequent – the second term. Proportion – statement that two ratios are equal: $A\!:\!B = C\!:\!D$. Extremes: $A$ and $D$ Means: $B$ and $C$ Continued proportion – three or more equal ratios, e.g., $A\!:\!B = C\!:\!D = E\!:\!F$. Dimensionless ratio – when both quantities share the same unit, the units cancel (e.g., $1\text{ min}:40\text{ s}=3:2$). Rate – a ratio of different units (e.g., km/h, $ \text{g}/\text{L}$). Odds – “unfavorable : favorable”; e.g., $7\!:\!3$ means 7 failures for every 3 successes. Special irrational ratios Square‑root‑2: $d/s = \sqrt{2}$ (diagonal to side of a square) $\pi$: circumference ÷ diameter of a circle Golden ratio $\varphi$: $a:b = b:(a+b)$, $\displaystyle \varphi = \frac{1+\sqrt5}{2}\approx1.618$ --- 📌 Must Remember Ratio equality: $a/b = c/d \iff ad = bc$. Scaling rule: Multiplying all terms by the same number leaves the ratio unchanged (e.g., $3:2 = 12:8$). Simplest form: No common integer factor > 1 between terms. Odds → probability: For odds $u\!:\!v$ (unfavorable : favorable), $$P(\text{success}) = \frac{v}{u+v}.$$ Example: $7\!:\!3 \Rightarrow P = \frac{3}{10}=30\%$. Whole‑part fraction: If a ratio includes every part of a situation, each term divided by the sum gives its proportion of the whole (e.g., $2$ oranges / $5$ total fruit $=2/5$). Irrational special ratios cannot be expressed as exact fractions; remember their approximate values. --- 🔄 Key Processes Convert a ratio to a fraction of the whole Sum all terms $S = a1 + a2 + \dots + an$. Fraction for term $ai$ = $\displaystyle \frac{ai}{S}$. Scale a ratio Choose a multiplier $k$. New ratio = $(ka1) : (ka2) : \dots$. Reduce a ratio to simplest form Compute $\gcd(a,b)$. Divide both terms by the gcd. Convert odds to probability (see Must Remember). Check a proportion Cross‑multiply: verify $A\cdot D = B\cdot C$. --- 🔍 Key Comparisons Ratio vs. Rate Ratio: same units → dimensionless (e.g., $3:2$). Rate: different units (e.g., $60\text{ km}/\text{h}$). Antecedent vs. Consequent Antecedent = first term, Consequent = second term. Odds vs. Probability Odds $u\!:\!v$ → probability $v/(u+v)$. Probability $p$ → odds $p/(1-p)$. Proportion vs. Continued proportion Proportion: equality of two ratios. Continued proportion: equality of three or more ratios. --- ⚠️ Common Misunderstandings “All ratios are fractions.” Ratios with > 2 terms (e.g., $2:3:7$) cannot be reduced to a single fraction. Confusing odds with probability. Odds $7:3$ is not $70\%$; the correct probability is $3/(7+3)=30\%$. Assuming a ratio must be dimensionless. When units differ, the expression is a rate, not a pure ratio. Forgetting extremes/means in a proportion. Mixing up $A,D$ (extremes) with $B,C$ (means) leads to wrong cross‑multiplication. --- 🧠 Mental Models / Intuition Balancing scales – a proportion $A\!:\!B = C\!:\!D$ is like a seesaw: the product of opposite ends (extremes) balances the product of the middle ends (means). Slice of a pizza – a ratio $a:b$ is the same as the slice $a/(a+b)$ of the whole pizza; scaling just makes the pizza larger but the slice size stays the same. --- 🚩 Exceptions & Edge Cases Multi‑term ratios – $2:3:7$ cannot be written as a single fraction; treat each pair separately. Ratios without a “1” – writing $3:4$ as “3” can be interpreted as a multiplier rather than a formal ratio. Irrational ratios – $\sqrt{2}$, $\pi$, $\varphi$ are never exact fractions; use approximations only when the problem permits. --- 📍 When to Use Which Proportion – when you need to assert equality of two ratios (solve for an unknown term). Continued proportion – when three or more successive terms are linked (e.g., geometric sequences). Rate – when units differ; keep the units attached (e.g., $5\text{ m}/\text{s}$). Odds – when a problem states “against” or “for” in a competitive context; convert to probability if the question asks for a likelihood. Simplify – always reduce ratios before plugging into formulas to avoid arithmetic errors. --- 👀 Patterns to Recognize Same multiplier across all terms → indicates an equivalent ratio (e.g., $3:2$ and $12:8$). Cross‑product equality in a proportion → quick check for correctness. Odds format “a to b against” → look for the unfavorable term first. When the sum of terms appears in the denominator → whole‑part proportion problem. --- 🗂️ Exam Traps Distractor: “$7:3$ odds give $70\%$ probability.” – wrong because odds must be converted. Distractor: “Any ratio can be written as a single fraction.” – false for $>2$ terms. Distractor: “If units cancel, the ratio is always dimensionless.” – misses the distinction that a rate keeps its units. Distractor: “In a proportion, $A$ and $B$ are the extremes.” – extremes are $A$ and $D$, means are $B$ and $C$. Distractor: “The golden ratio equals $1.5$.” – actual value $\approx1.618$. ---