KnowThisKey Takeaways
Key Takeaways
- 1A ratio compares two quantities directly, like 2:3 — for every 2 of one thing, there are 3 of the other.
- 2A proportion is just a statement that two ratios are equal, which is exactly what lets you scale a recipe, map, or mixture up or down.
- 3Cross-multiplication is the standard tool for solving an unknown value in a proportion — multiply diagonally, then divide.
The concept
A recipe uses flour and sugar in a ratio of 3:1. If you use 9 cups of flour, how much sugar do you need?
Worked examples
Example 1: Simplifying a ratio (baseline case)
Example 2: Solving a proportion with cross-multiplication (edge/variation case)
Example 3: Reading a map scale (applied case)
Common mistakes
Common Mistakes
Reversing the order of a ratio and treating it as equivalent.
→ 2:3 and 3:2 are different relationships. Keep the same order on both sides of a proportion — if flour comes first on one side, it must come first on the other.
Scaling only one quantity in a ratio and leaving the other unchanged.
→ Both quantities must scale by the same factor to preserve the ratio — find that factor first, then apply it to every term.
Setting up a proportion with mismatched units on each side.
→ Keep the same quantity type in the same position on both sides (e.g. flour/sugar = flour/sugar), never flour/sugar = sugar/flour.
Common misconception
“A ratio and a fraction are completely different mathematical objects.”
A ratio a:b and the fraction a/b carry the same information and follow the same simplification and scaling rules — the colon notation is really just a stylistic convention for comparing two (or more) quantities, not a fundamentally different kind of math.
What to do next
What to do next
- Practice simplifying a handful of ratios by dividing both terms by their HCF, the same way you'd simplify a fraction.
- Next time you scale a recipe up or down, write it out as a formal proportion first to avoid scaling only one ingredient by accident.
- Try reading a real map's scale and calculating an actual distance using the proportion method.