1. Ratios
A ratio is a comparison of two quantities by division. "Three apples for every two oranges" is the ratio of apples to oranges, written $3 : 2$, or $\tfrac{3}{2}$, or "3 to 2." The three notations mean the same thing.
An ordered comparison of two quantities of the same kind. Written $a : b$, $\tfrac{a}{b}$, or "$a$ to $b$." Order matters: the ratio of $3$ apples to $2$ oranges is $3 : 2$, not $2 : 3$.
Ratios reduce the same way fractions do: $6 : 4$, $9 : 6$, and $30 : 20$ are all equivalent to $3 : 2$. So a "ratio in simplest form" has been divided by its greatest common divisor.
Three things make ratios distinct from ordinary fractions:
- Ratios compare two parts of a bigger whole. "The boy-to-girl ratio in this class is $3 : 4$" means there are $3$ boys for every $4$ girls — out of every $7$ students. A fraction $\tfrac{3}{4}$ might mean any number of things; a ratio carries the comparison meaning explicitly.
- Ratios can involve more than two quantities. A recipe might call for flour, sugar, and butter in the ratio $4 : 2 : 1$.
- Ratios usually keep their two parts distinct. While we'd reduce $\tfrac{6}{4}$ to $\tfrac{3}{2}$, we'd also reduce the ratio $6 : 4$ to $3 : 2$ — and we typically don't collapse it to "$1.5$" the way we might a fraction.
2. Rates and unit rates
When the two quantities in a ratio have different units, the ratio is called a rate.
- $60$ miles per $1$ hour — a rate of speed.
- $\$2.50$ per $1$ pound — a rate of price.
- $120$ words per $1$ minute — a rate of typing speed.
- $3$ tablespoons per $1$ cup of milk — a recipe rate.
The word "per" is the giveaway: it's the colloquial form of "for each" or "divided by," and it converts the ratio into a unit-bearing comparison.
A rate whose denominator is $1$. To find the unit rate corresponding to a given rate, divide both quantities by the denominator. $\$10$ for $4$ pounds becomes $\$2.50$ per $1$ pound; $240$ km in $3$ hours becomes $80$ km per $1$ hour.
Unit rates are why rates are so useful — they let you compare prices, speeds, or efficiencies that came in inconveniently different "package sizes." A $12$-pack of soda for $\$4.80$ versus an $18$-pack for $\$6.30$: the unit rates are $\$0.40$ per can versus $\$0.35$ per can. The bigger pack is cheaper per unit.
Whenever you read "per" in a rate, mentally translate to "divided by." $60$ miles per hour means $\tfrac{60 \text{ miles}}{1 \text{ hour}}$. With that translation, every rate problem reduces to fraction arithmetic.
3. Proportions
A proportion is an equation stating that two ratios are equal:
$$ \frac{a}{b} = \frac{c}{d}. $$For example, $\tfrac{3}{4} = \tfrac{6}{8}$ is a proportion. So is $\tfrac{1}{2} = \tfrac{x}{10}$, where one of the four numbers is unknown and we're being asked to find it.
Proportions show up wherever scaling does. Doubling a recipe is a proportion: the ratio of flour to sugar must stay the same. Reading a map is a proportion: the ratio of map distance to real distance is fixed. Mixing concrete is a proportion. So is converting currency. The world is full of them, and being fluent in proportions translates immediately into being faster at everyday problems.
The single algebraic fact behind every proportion-solving technique is this: $\tfrac{a}{b} = \tfrac{c}{d}$ if and only if $a \cdot d = b \cdot c$. The two cross-products are equal exactly when the ratios are.
4. Solving proportions
To find an unknown in a proportion, cross-multiply, then solve the resulting equation. Let's solve $\tfrac{x}{15} = \tfrac{4}{6}$ for $x$:
Step 1. Cross-multiply: the numerator of one fraction times the denominator of the other.
$$ x \cdot 6 = 4 \cdot 15. $$Step 2. Simplify:
$$ 6x = 60. $$Step 3. Divide both sides by $6$:
$$ x = 10. $$That's it. Cross-multiplication is the workhorse — it converts a proportion (an equation with fractions) into a simpler linear equation (one without). Once the fractions are gone, it's just basic algebra.
An equivalent route: rewrite both sides with a common denominator and equate numerators. For $\tfrac{x}{15} = \tfrac{4}{6}$, reduce $\tfrac{4}{6}$ to $\tfrac{2}{3}$, then notice that $\tfrac{x}{15}$ equals $\tfrac{2}{3}$ when $x = 10$, since $\tfrac{10}{15} = \tfrac{2}{3}$. Same answer, slightly more thought required.
5. Direct vs. inverse proportion
"Proportional" actually splits into two patterns. They look similar at a distance but behave oppositely.
Direct proportion
Two quantities are in direct proportion if their ratio stays constant. Doubling one doubles the other; halving one halves the other.
$$ \frac{y}{x} = k \quad \Longleftrightarrow \quad y = kx. $$$k$ is the constant of proportionality. Examples: distance and time at constant speed ($d = vt$); cost and quantity at fixed price; circumference of a circle and its diameter ($C = \pi d$).
Inverse proportion
Two quantities are in inverse proportion if their product stays constant. Doubling one halves the other.
$$ y \cdot x = k \quad \Longleftrightarrow \quad y = \frac{k}{x}. $$Examples: time and speed for a fixed distance — drive twice as fast and the trip takes half as long; pressure and volume of a gas at constant temperature (Boyle's law); the number of workers and time to finish a fixed job (with caveats about coordination).
"If $5$ workers can build a wall in $12$ days, how many days will $10$ workers take?" The relationship between workers and time is inverse, not direct. More workers means less time. The answer is $6$ days (because $5 \cdot 12 = 10 \cdot d$ gives $d = 6$), not $24$. Always ask which way the two quantities move when the other increases.
6. Common pitfalls
A proportion is an equation, so consistency of orientation matters. If the left side is "miles per hour," the right side must also be "miles per hour" — not "hours per mile." Mixing the two flips the answer's units and gives the wrong number.
Cross-multiplication works on equations of the form $\tfrac{a}{b} = \tfrac{c}{d}$. It does not work on $\tfrac{a}{b} + \tfrac{c}{d} = e$ — adding fractions is a different operation entirely. Make sure you actually have a proportion before reaching for the move.
"Twice as many workers, twice as fast" is direct (and wrong for total time). "Twice as many workers, half the time" is inverse (and correct). When a problem describes one quantity getting bigger while the other gets smaller, it's inverse — set up $x \cdot y = k$, not $\tfrac{y}{x} = k$.
If a rate involves minutes and a problem involves hours, you must convert before computing — or the cross-multiplication will give a number with the wrong order of magnitude. Carry the units through every step; if they don't cancel sensibly, something's off.
7. Worked examples
Example 1 · Simplify the ratio $30 : 18$
$\gcd(30, 18) = 6$. Divide both:
$$ 30 : 18 \;\longrightarrow\; 5 : 3. $$Answer: $\boxed{5 : 3}$.
Example 2 · Find the unit rate: $\$10$ for $4$ pounds
Divide both quantities by $4$:
$$ \frac{\$10}{4 \text{ lb}} = \frac{\$2.50}{1 \text{ lb}}. $$Answer: $\boxed{\$2.50 \text{ per pound}}$.
Example 3 · Solve $\tfrac{x}{15} = \tfrac{4}{6}$
Cross-multiply:
$$ 6x = 4 \cdot 15 = 60. $$Divide:
$$ x = 10. $$Check: $\tfrac{10}{15} = \tfrac{2}{3} = \tfrac{4}{6}$ ✓. Answer: $\boxed{x = 10}$.
Example 4 · A car travels $180$ miles in $3$ hours. How far in $5$ hours at the same rate?
Set up the proportion (miles to hours equal on both sides):
$$ \frac{180}{3} = \frac{d}{5}. $$Cross-multiply: $3d = 180 \cdot 5 = 900$, so $d = 300$.
Sanity check: the unit rate is $60$ mph, so $5$ hours gives $300$ miles ✓. Answer: $\boxed{300 \text{ miles}}$.
Example 5 · If $5$ workers finish a job in $12$ days, how long for $10$ workers?
This is inverse proportion: doubling the workforce halves the time. The constant is workers × days:
$$ 5 \cdot 12 = 10 \cdot d \;\Longrightarrow\; d = 6. $$Answer: $\boxed{6 \text{ days}}$. The temptation to set up $\tfrac{5}{12} = \tfrac{10}{d}$ (a direct proportion) gives the wrong answer, $24$ — exactly because direct and inverse have been swapped.