Ratios, Rates & Proportions

Two pitchers of lemonade

You mix one pitcher with $2$ cups of lemon juice and $3$ of water. A friend mixes theirs with $3$ cups of juice and $5$ of water. Which one tastes more lemony? The gaps don’t help — one pitcher has “one more water than juice,” the other “two more” — because taste doesn’t care about gaps. It cares about proportion: how much juice there is for each unit of water. Questions shaped like that — mixing, pricing, speed, scaling a recipe — are what this unit is for. In F5 you compared everything against a standard base of $100$; a ratio drops the standard base and compares any two quantities directly against each other.

Ratio, rate, unit rate

Three cups of flour to two cups of sugar is the ratio 3 : 2 — which is also just the fraction $\frac{3}{2}$ from F3, “one and a half cups of flour for each cup of sugar.” When the two quantities carry different units it’s called a rate — $120$ miles per $2$ hours — and a rate shrunk down to “per 1” is a unit rate: $60$ miles per hour. One family of ideas, and one tool at the end — the proportion — that solves nearly every word problem built from them.

Equivalent ratios — scale both parts together

Double the recipe: $6$ flour, $4$ sugar. Nothing about the taste changed, because both parts grew by the same factor — 6 : 4 is the same ratio as 3 : 2, exactly the way $\frac{6}{4}$ and $\frac{3}{2}$ are the same fraction. Scaling both parts by any number keeps a ratio; dividing both by their GCF simplifies it to lowest terms.

Here is where the oldest instinct in arithmetic quietly sabotages people: adding the same amount to both parts feels just as safe as multiplying — after all, it keeps the gap identical. But watch the taste: go from $2:3$ to $3:4$ (one more cup of each) and the juice climbs from $\frac{2}{5}$ of the pitcher to $\frac{3}{7}$ — from $40\%$ to about $43\%$, noticeably stronger. Equal gaps are not equal proportions; ratios live in multiplication, not addition. That’s also the answer to the two pitchers: rename both to fifteenths of a batch — $2:3 = 10:15$ against $3:5 = 9:15$ — and the first pitcher is the lemony one.

The widget opens on 6 : 4 — predict its simplest form, then scan the equivalent rows. Then enter 2 : 3 and look for 10 : 15 in its family; 3 : 5 will never produce it.

The hidden whole: part-to-part vs part-to-whole

A ratio of 3 : 2 secretly involves a total of $5$ parts. So $3$ out of every $5$ are the first kind ($\frac{3}{5}$) and $2$ out of $5$ the second ($\frac{2}{5}$). Miss that hidden whole and the fraction bar habit lays a trap: a class with boys : girls $= 3:2$ sounds like ”$\frac{3}{2}$ of the class is boys,” but $\frac{3}{2}$ of a class is more students than the class has. The $3$ and the $2$ are both parts; the whole they share is $5$. Spotting it is what turns a ratio problem into a fraction or percent problem: $\frac{3}{5} = 0.6 = 60\%$ of the students are boys — and every F5 tool applies.

Unit rate: how much “per one”

Now the shopping version of the lemonade question: $4$ pens for $$6$, or $7$ pens for $$10$ — which is the better deal? Neither raw pair is comparable, so shrink both to “per one pen”: $6 \div 4 = 1.50$ dollars per pen against $10 \div 7 \approx 1.43$. The bigger pack wins, by seven cents a pen. That’s all a unit rate is: division used to make any two deals, speeds, or mixtures comparable on a common footing — and once you have “per one,” any amount is a single multiplication away.

The widget opens on the $$6$-for-$4$ deal. Enter the rival ($10$ and $7$) and compare the per-one values yourself. Then try $120$ and $2$ — the same division turns a road trip into miles per hour.

Proportions: the same ratio, twice

“If $4$ pens cost $$6$, how much do $10$ pens cost?” The price per pen doesn’t change, so the two situations form the same ratio — written twice, with one number missing:

$\frac{4 \text{ pens}}{6 \text{ dollars}} = \frac{10 \text{ pens}}{x \text{ dollars}}$

That’s a proportion. When the numbers are friendly, solve it by pure scaling: pens went $4 \to 10$, a factor of $2.5$, so dollars go $6 \times 2.5 = 15$. When the scaling factor is ugly, cross-multiplication is the all-terrain tool: in any true proportion $\frac{a}{b} = \frac{c}{d}$, the diagonals match — $a \times d = b \times c$ — so $\frac{3}{4} = \frac{x}{20}$ becomes $4x = 60$, and $x = 15$. (The Cheat Sheet shows the one-line algebra behind why the diagonals must agree.)

The solver opens on $\frac{3}{4} = \frac{?}{20}$ — predict $x$ by scaling ($4 \to 20$ is $\times 5$) before reading its steps. Then move the empty box to a bottom position and watch the same diagonal rule handle it.

The reliable setup for word problems

Everything above assumed the ratios were written consistently — and that first move is where proportions actually go wrong. Line the same units up: top-and-top, bottom-and-bottom. Pens over dollars on the left means pens over dollars on the right, never dollars over pens. The cross-multiplication is indifferent — it will happily grind a flipped setup into a confident wrong answer — so spend your care on the setup, not the arithmetic.

The one thing to remember

Ratios compare by multiplication, never by gaps — scaling both parts keeps a ratio, adding to both parts quietly changes it. Divide to “per one” and anything becomes comparable; and when the same ratio shows up twice with a blank, line the units up and let the diagonals find it.

What they mean

A ratio compares two quantities: 3 : 2 (also written $\frac{3}{2}$). A rate is a ratio of different units, like $120$ miles per $2$ hours. A unit rate shrinks it to “per 1”: $60$ miles per hour.

The three skills

The hidden whole

A part-to-part ratio names a whole you can use. 3 : 2 means $5$ parts in all, so $\frac{3}{5}$ and $\frac{2}{5}$ of the whole — the bridge from ratios to fractions and percents ($\frac{3}{5} = 60\%$).

The reliable setup for word problems

Line the same units up top-and-top, bottom-and-bottom. “If $4$ pens cost $$6$, how much do $10$ pens cost?” → pens / dollars: $\frac{4}{6} = \frac{10}{x}$, then cross-multiply.

A proportion, step by step