Ratios & unit rates
The idea
A ratio compares two quantities by division instead of subtraction: 12 bars for $9 is the relationship '12 to 9', not 'three more bars than dollars'. You already know how to divide and how to read fractions, and that is the whole engine here — a ratio is a fraction wearing work clothes. A unit rate pushes the comparison down to 'per one': dollars per bar, kilometers per hour, points per game. Once everything is per one, totally different offers become directly comparable.
The reliable mental move is: divide to find the per-one amount, then compare per-one amounts. The classic trap is additive thinking — believing that 2 cups of juice to 3 cups of soda is the same mix as 4 to 5 because you added 2 to both. Ratios scale by multiplying, not by adding: 2 to 3 matches 4 to 6. When two ratios are in question, ask what each one looks like per single unit and let those numbers settle the argument.
Worked example
A 12-pack of granola bars costs $9.00 and a 20-pack of the same bars costs $14.00. Which pack is the better deal per bar?
- Find the unit price of the 12-pack by dividing cost by count: 9 ÷ 12 = 0.75, so each bar costs $0.75.
- Do the same for the 20-pack: 14 ÷ 20 = 0.70, so each bar costs $0.70 there.
- Compare the per-one prices: $0.70 is less than $0.75, so the 20-pack is cheaper per bar by 5 cents.
- Sanity-check it: at the 12-pack rate of $0.75 per bar, 20 bars would cost 0.75 × 20 = $15, but the 20-pack actually charges $14 — so the bigger pack really does undercut it.
Answer. The 20-pack is the better deal at $0.70 per bar, compared with $0.75 per bar for the 12-pack.
Check your understanding
- Why does dividing, rather than subtracting, tell you which of two deals is better?
- When might the pack with the worse unit price still be the smarter purchase in real life?
- How would the comparison change if you computed bars per dollar instead of dollars per bar — would the better deal flip?
- Two ratios look different, like 6 to 9 and 10 to 15 — what test decides whether they describe the same relationship?
Build the foundations first
Ratios & unit rates builds on these concepts. If any feel shaky, start there.