MISSION · Number sense
Ratios and Proportions
How to make more of something without changing what it is
LEVEL 1 · EXPLORE
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LEVEL 2 · LEARN
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The idea in one sentence
A ratio compares two amounts, and a proportion says two ratios are the same comparison written with different numbers.
Ratios keep their shape
Lemonade made with 2 lemons for every 5 cups of water is a ratio of \(2:5\). Double everything and you get \(4:10\) — twice as much lemonade, tasting exactly the same. Triple it, \(6:15\), still the same drink.
That’s what a ratio protects. The amounts change; the relationship doesn’t. Add one lemon without adding water and you’ve broken it, and you’ll taste the difference immediately.
The unit rate is the one to remember
Divide both sides until one of them is 1, and you get the unit rate: \(0.4\) lemons per cup of water. Now any batch is one multiplication away. Unit rates are how you compare deals that don’t line up — $3.60 for 4 apples versus $5.00 for 6 is unanswerable until you turn both into price per apple, at which point 90 cents beats 83 cents and the bigger bag wins.
Solving a proportion
When three numbers of a proportion are known, the fourth is forced:
\[\frac{2}{5} = \frac{x}{20}\]Ask what took 5 to 20 — multiplying by 4 — and do the same on top: \(x = 8\). If the scale factor isn’t a whole number, cross-multiply instead: \(5x = 2 \times 20\), so \(x = 8\). Same answer, and cross-multiplication is really just multiplying both sides by both denominators.
Part-to-part or part-to-whole
A class with 3 cats for every 2 dogs is a part-to-part ratio of \(3:2\). But cats as a fraction of all the animals is \(\frac{3}{5}\), because the whole is \(3 + 2\). Both are correct statements about the same class, and mixing them up is the single most common ratio error there is. Read carefully whether the question asks about the other part or about the total.
Common mix-ups
- Adding instead of multiplying to scale. Going from \(2:5\) to \(3:6\) by adding one to each side changes the ratio. Multiply both sides by the same factor.
- Reading \(3:2\) as three-halves of the class. It’s three fifths.
- Flipping the order. “Three cats to two dogs” is \(3:2\), not \(2:3\). The order of the words sets the order of the numbers.
- Comparing unit rates in different units. Price per apple and price per kilogram are not comparable numbers.
Check yourself
A recipe uses 2 cups of flour for every 3 eggs. You have 12 eggs. How much flour? And what is the unit rate of flour per egg?
\(\frac{2}{3} = \frac{x}{12}\), and 3 became 12 by multiplying by 4, so \(x = 8\) cups. The unit rate is \(\frac{2}{3} \approx 0.67\) cups of flour per egg.
LEVEL 3 · REFLECT
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