MISSION · Number sense
Proportions and Cross Multiplication
Two ratios set equal, and the one move that solves them
LEVEL 1 · EXPLORE
Make the idea move
Change the controls and watch what happens. Follow your curiosity and see what you notice.
Loading the activity…
LEVEL 2 · LEARN
Unlock the explanation
Open the chapters in any order. Try the examples as you go.
The idea in one sentence
A proportion says two ratios are the same size, and because they are the same size you can cross multiply to find whichever number is missing.
What makes it a proportion
A ratio compares two quantities. A proportion is a statement that two ratios are equal:
\[\frac{3}{4} = \frac{x}{20}\]Read it out loud as “3 is to 4 as \(x\) is to 20”. Nothing has been asked yet — the equation just claims the two comparisons match. Solving it means finding the \(x\) that makes the claim true.
Scaling, before any algebra
Often you can see the answer without a method. To turn 4 into 20 you multiply by 5, and a ratio only stays the same if you do the same thing to both parts:
\[\frac{3}{4} \; \xrightarrow{\times 5} \; \frac{15}{20}\]So \(x = 15\). This is the honest picture of what a proportion is, and for friendly numbers it is faster than anything else. Reach for it first.
Cross multiplication
When the scale factor isn’t whole, use the general move. Multiply each numerator by the other denominator:
\[\frac{a}{b} = \frac{c}{d} \quad \Longrightarrow \quad ad = bc\]For \(\frac{3}{4} = \frac{x}{20}\) that gives \(3 \times 20 = 4x\), so \(60 = 4x\) and \(x = 15\) — the same answer.
It is not a magic rule. Multiplying both sides by \(b\) and then by \(d\) clears both denominators at once, and that is all cross multiplication is: two ordinary steps done in one.
Setting one up from words
This is where the marks are lost, and there is one rule that prevents it: the same kind of thing goes on top of both fractions.
Four notebooks cost $10. What do eight cost?
\[\frac{\text{notebooks}}{\text{dollars}}: \quad \frac{4}{10} = \frac{8}{x}\]Cross multiplying, \(4x = 80\), so \(x = 20\). You could equally have written \(\frac{10}{4} = \frac{x}{8}\) — dollars on top in both — and got the same 20. What you cannot do is mix them: \(\frac{4}{10} = \frac{x}{8}\) puts notebooks over dollars on one side and dollars over notebooks on the other, and it gives the wrong answer.
Label the fractions before you solve. Two words written above the line have saved more marks than any amount of care with the arithmetic.
Where proportions come from
- Unit pricing. 3 for $4.50 against 5 for $7.00 — scale both to one item and compare.
- Maps and scale drawings. 1 cm on the map is 50,000 cm on the ground, in every direction.
- Similar triangles. Matching sides are in proportion, which is what lets you measure a tree from its shadow.
- Recipes and mixtures. Doubling a recipe means doubling everything, which is the whole reason ratios matter to anyone.
- Percents. \(\frac{\text{part}}{\text{whole}} = \frac{p}{100}\) is just a proportion with a 100 in it.
Common mix-ups
- Flipping one side and not the other. The units on top must match.
- Cross multiplying an addition. \(\frac{a}{b} + \frac{c}{d}\) is not a proportion — there is no equals sign to multiply across. Cross multiplication works on \(\frac{a}{b} = \frac{c}{d}\) and nothing else.
- Adding instead of multiplying. If 4 notebooks cost $10, 8 notebooks do not cost $14. Ratios scale by multiplying.
- Assuming everything is proportional. One painter takes 6 hours, so two painters take 3, not 12. Some relationships are inverse, and a few — like a taxi fare with a fixed charge — are neither.
Check yourself
Solve \(\frac{3}{4} = \frac{x}{20}\), find the cost of eight notebooks when four cost $10, and work out the flour for 12 cookies when 2 cups make 3.
\(3 \times 20 = 4x\) gives \(x = 15\). Eight notebooks double four, so $20. And \(\frac{2}{3} = \frac{f}{12}\) gives \(3f = 24\), so 8 cups.
LEVEL 3 · REFLECT
Add this quest to your trail
When you feel ready, mark this idea as explored. You can always come back and try it again.
EXPLORER STRATEGIES