⚡ Grade 6 · Lesson 1 of 14
The Potion Mixer: Understanding Ratios and Proportions
A magical recipe only works if its ingredients stay in the same ratio — scale it up or down and the potion still brews true.
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The big idea: A ratio compares two quantities by division, showing how much of one thing there is for a given amount of another. As long as you multiply or divide both parts of a ratio by the same number, the relationship — and the potion's magic — stays exactly the same, whether you're brewing a single cup or a giant cauldron.
A recipe that must stay balanced
Every potion recipe is really a ratio: a fixed comparison between two ingredients. A shrinking potion might call for 2 parts moonwater to 3 parts starflower petals — written 2:3. Brew a tiny vial or a bathtub-sized batch, and as long as moonwater and starflower petals stay in that same 2:3 relationship, the potion works.
A ratio is just a way of comparing two quantities by division. It shows up everywhere outside the potion shop too: miles per gallon, students per teacher, red paint to white paint.
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A ratio compares two quantities
A ratio written a:b compares one quantity (a) to another (b); it can also be written as the fraction a/b. Order matters — 2:3 (moonwater to petals) is not the same relationship as 3:2 (petals to moonwater).
\[ 2:3 \;=\; 4:6 \;=\; 6:9 \]
Multiply both parts of a ratio by the same number and you get an equivalent ratio — same recipe, bigger or smaller batch.
🎮 Potion Ratio Mixer LIVE
Mix two ingredients and keep the ratio constant as you scale the recipe up and down.
Finding an unknown amount
Often you know a ratio and one actual amount, and need to find the matching amount of the other ingredient. Line the ratio up with the real quantities and ask: what did I multiply the known part by to get my actual amount? Multiply the other part of the ratio by that same number.
\[ \dfrac{a}{b} = \dfrac{c}{d} \quad\Longrightarrow\quad a \times d = b \times c \]
Cross-multiplication: if two ratios are equal, the product of the outer terms equals the product of the inner terms.
📝 Worked example: A shrinking potion needs moonwater and starflower petals in a ratio of 2:3. If a brewer uses 8 cups of moonwater, how many cups of starflower petals are needed?
- The ratio 2:3 means for every 2 cups of moonwater, you need 3 cups of petals.
- 8 cups of moonwater is 2 × 4, so the recipe has been scaled by a factor of 4.
- Scale the petals by the same factor: 3 × 4 = 12.
✓ <strong>12 cups</strong> of starflower petals.
📝 Worked example: A love potion uses fizzroot and morning dew in a ratio of 5:8. A batch uses 24 cups of dew. How many cups of fizzroot are needed?
- Set up equal ratios: 5/8 = x/24, where x is the cups of fizzroot.
- Cross-multiply: 8 × x = 5 × 24, so 8x = 120.
- Divide both sides by 8: x = 120 ÷ 8.
✓ <strong>15 cups</strong> of fizzroot.
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Scale both parts by the same factor
The most common mistake is changing only one part of the ratio. If you multiply moonwater by 4, you must multiply petals by 4 too — not add 4, and not leave it unchanged. A ratio only stays true if both parts scale together.
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Simplify a ratio the way you simplify a fraction
To simplify a ratio like 6:10, divide both parts by their greatest common factor (2), giving 3:5. A simplified ratio describes the exact same relationship with the smallest possible whole numbers.
Check your understanding
1. Write the ratio 6:10 in simplest form.
Divide both parts by their greatest common factor, 2: 6÷2 : 10÷2 = 3:5.
2. Which ratio is equivalent to 3:4?
Multiplying both parts of 3:4 by 2 gives 6:8 — the same relationship, a bigger batch.
3. A recipe uses flour and sugar in a ratio of 2:1. If you use 10 cups of flour, how many cups of sugar are needed?
10 cups of flour is 2 × 5, so the recipe is scaled by 5. Sugar = 1 × 5 = 5 cups.
4. In the ratio a:b, if a is multiplied by 6, what must happen to b to keep an equivalent ratio?
Both parts of a ratio must be scaled by the same factor to stay equivalent.
5. Solve for x: 4/6 = x/18.
18 ÷ 6 = 3, so the ratio was scaled by 3. x = 4 × 3 = 12. (Check: 6×12=72=4×18.)
✅ Key takeaways
- A ratio a:b compares two quantities and can be written as the fraction a/b.
- Multiplying (or dividing) both parts of a ratio by the same number gives an equivalent ratio.
- To find an unknown amount, figure out the scale factor between a known part and its match, then apply that factor to the other part.
- Cross-multiplication (a/b = c/d means a×d = b×c) solves proportions when the scale factor isn't obvious.
- Simplify a ratio by dividing both parts by their greatest common factor, just like simplifying a fraction.