Ratios & Proportions

1. Introduction

When you mix paint, follow a recipe, read a map, or compare prices at the store, you are using ratios and proportions — even if you don't call them that. A ratio captures the relationship between two amounts ("$2$ parts blue to $1$ part yellow"), and a proportion lets you scale that relationship up or down while keeping it the same ("so for $6$ parts blue I need $3$ parts yellow"). This pair of ideas is one of the most practical in all of mathematics.

The beauty of ratios is that they describe relative size, not absolute totals. A pancake recipe with a $2:1$ flour-to-milk ratio works whether you're making a small batch or feeding a crowd — the relationship stays fixed even as the amounts grow. A map scale of $1$ inch to $25$ miles works for any distance on that map. A paint mix of $3:2$ blue to yellow gives the same shade whether you use tablespoons or gallons, as long as you keep the ratio.

Ratios and proportions are the gateway to percentages (which are ratios out of $100$), similar figures in geometry (where corresponding sides are proportional), unit conversions, and the proportional relationships at the heart of algebra. Mastering them now means word problems feel like puzzles with a reliable toolkit instead of guesswork.

In this article we'll define ratios precisely, learn how to simplify and compare them, distinguish part-to-part from part-to-whole, master proportions and the cross-multiplication shortcut, use unit rates to breeze through comparison problems, and split quantities in a given ratio. By the end, you should be able to set up a proportion from a word problem without mixing up your units, and solve for any missing value with confidence.

2. Core Concepts

2.1 What a Ratio Is

A ratio compares two quantities by division. The ratio of $3$ cups of flour to $2$ cups of sugar can be written three ways: as $3:2$ (with a colon), as "$3$ to $2$" (in words), or as the fraction $\dfrac{3}{2}$. All three mean the same thing — for every $3$ units of flour there are $2$ units of sugar.

A ratio tells you relative size, not the actual totals. The ratio $3:2$ describes a relationship that also fits $6:4$, $9:6$, $30:20$, and infinitely many other amounts. What stays constant is how the two quantities relate, not how big they are.

Order matters. "$3$ red to $2$ blue" is $3:2$, not $2:3$. Always match the order to the wording of the problem.

2.2 Equivalent Ratios and Simplifying

Two ratios are equivalent if they describe the same relationship. You get equivalent ratios by multiplying or dividing both parts by the same number — exactly like equivalent fractions. So $3:2 = 6:4 = 15:10$.

To simplify a ratio, divide both parts by their greatest common divisor. For $12:18$, the GCD is $6$, so $12:18 = \dfrac{12}{6} : \dfrac{18}{6} = 2:3$. A ratio is in simplest form when the only number dividing both parts is $1$. Crucially, you must multiply or divide both parts — never add or subtract, which would change the relationship.

2.3 Part-to-Part vs. Part-to-Whole

Ratios come in two useful kinds. A part-to-part ratio compares one group to another: if a class has $10$ boys and $15$ girls, the boy-to-girl ratio is $10:15 = 2:3$. A part-to-whole ratio compares one group to the total: boys to all students is $10:25 = 2:5$.

Adding the parts gives the whole: with parts $2$ and $3$, the total is $2 + 3 = 5$ "shares." This is the key to splitting a quantity in a given ratio. If the ratio is boys to girls $2:3$, there are $5$ total shares, and boys get $\frac{2}{5}$ of the whole while girls get $\frac{3}{5}$.

2.4 Unit Rates

A unit rate is a ratio whose second quantity is exactly $1$, such as "miles per hour," "dollars per pound," or "words per minute." You find a unit rate by dividing the first quantity by the second. If $5$ notebooks cost $7.50$, the unit rate is $\dfrac{7.50}{5} = 1.50$ per notebook. Unit rates make comparison effortless: to find the better deal, just compare the price per single item.

The word "per" almost always signals a unit rate. "Miles per hour" means miles divided by hours, with $1$ hour in the denominator.

2.5 What a Proportion Is

A proportion is an equation stating that two ratios are equal:

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

Proportions are powerful because they let us scale a known relationship to find an unknown amount. If we know three of the four numbers, we can solve for the fourth. The main tool is cross multiplication: if $\dfrac{a}{b} = \dfrac{c}{d}$, then $a \cdot d = b \cdot c$. This works because multiplying both sides of the equation by $b$ and by $d$ clears the denominators, leaving the two "cross products" equal.

2.6 Scale Drawings and Maps

A map scale is a ratio. If $1$ inch represents $20$ miles, the ratio of map distance to real distance is $1:20$ (in consistent units). Setting up a proportion with matching units on top and bottom of each fraction lets you convert in either direction.

2.7 Ratios with Three or More Parts

A ratio can have three parts, like $2:3:4$ for splitting profit among three partners. Add all parts ($2 + 3 + 4 = 9$) to get total shares, then assign each partner their fraction of the whole.

2.8 Connecting Ratios to Fractions and Percents

The ratio $3:5$ as part-to-whole means the first part is $\frac{3}{8}$ of the total. Converting to percent: $\frac{3}{8} = 0.375 = 37.5\%$. Ratios, fractions, and percents are deeply connected — choose whichever form makes the problem clearest.

2.9 Ratios with Different Units

A ratio compares quantities measured in the same units whenever possible. Speed as $60$ miles per $2$ hours can be written $60:2$ but is cleaner as the unit rate $30$ miles per hour. When converting, write the conversion as a ratio: $1$ hour $= 60$ minutes, so $\dfrac{1 \text{ hour}}{60 \text{ min}} = 1$.

2.10 When Cross Multiplication Is Valid

Cross multiplication works because proportions are equations. If $\dfrac{a}{b} = \dfrac{c}{d}$, multiply both sides by $b$ to get $a = \dfrac{bc}{d}$, then multiply by $d$ to get $ad = bc$. This is not a magic trick — it is algebra that clears fractions. Both $b$ and $d$ must be nonzero.

2.11 Double Checking with Equivalent Fractions

After solving a proportion, plug your answer back in and simplify both sides. If $\dfrac{x}{12} = \dfrac{5}{8}$ gives $x = 7.5$, check: $\dfrac{7.5}{12} = \dfrac{75}{120} = \dfrac{5}{8}$. A quick simplification check catches most setup errors.