Recipes | Mathematics

Unit 1 · Ratio & Proportion

6 Recipes

Every recipe is a direct proportion in disguise: double the people, double every single ingredient. The only real skill is finding the ONE number — the scale factor — that every ingredient has to obey, even when you have to find it backwards, or when one ingredient secretly decides the whole answer.

🎯 By the end of this you can…

✓Scale every ingredient in a recipe up or down by the same factor.
✓Work the scale factor out BACKWARDS, from how much of one ingredient was used.
✓Find the one ingredient that runs out first, and use it to find the real maximum you can make.

💡 The idea

One factor, every ingredient

A recipe is a direct proportion: ingredient amount and number of servings always grow or shrink together, by the exact same multiplier. Find that one number — the scale factor — and every ingredient follows the same rule, with no exceptions and no ingredient left behind.

📋 How to scale a recipe

1
Work out the scale factor: $new servings \div old servings$ .
2
Multiply EVERY ingredient by that same factor — including odd ones like egg counts.
3
Scaling down uses exactly the same method — the factor is just smaller than 1.

✏️ Worked example

Omar is making Shortbread for 16 people. The recipe (serves 8) needs 150g butter, 75g caster sugar, 175g plain flour, 50g cornflour.

1
Scale factor: $16 \div 8 = 2$
2
Butter: $150 \times 2 = 300 g$
3
Caster sugar: $75 \times 2 = 150 g$
4
Plain flour: $175 \times 2 = 350 g$
5
Cornflour: $50 \times 2 = 100 g$

✅ Answer: 300g butter, 150g caster sugar, 350g plain flour, 100g cornflour

⚠️ The classic trap

Scaling SOME ingredients and forgetting others — especially small amounts like baking powder or a single egg. Every line in the recipe obeys the same factor; if you've multiplied four ingredients but the recipe has five, the fifth one is still wrong even if the other four are perfect.

🍳 Try it — drag the people

8people

Original serves 8 → scale factor ×1

Butter150g

Caster Sugar75g

Plain Flour175g

Cornflour50g

🎮 Practice · scale it up or down

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Shortbread (serves 8 → 16): Caster Sugar 75g → ?

Shortbread (serves 8 → 16): Plain Flour 175g → ?

Scones (serves 8 → 2): Butter 60g → ?

Scones (serves 8 → 2): Buttermilk 180ml → ?

Stuffed Turkey (serves 4 → 10): Chicken Stock 300ml → ?

Stuffed Turkey (serves 4 → 10): Garlic Cloves 2 → ?

Vegetable Soup (serves 6 → 9): Potatoes 260g → ?

Shortbread (serves 4 → 3): Caster Sugar 60g → ?

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💡 The idea

Working the factor out backwards

Sometimes you're not told the number of servings at all — only how much of ONE ingredient someone actually used. The method doesn't change: you still need the scale factor first, you just have to find it a different way — by comparing what was used to what the recipe originally asked for. Once you have it, every other ingredient (or even the number of people) follows exactly as before.

📋 How to find the factor backwards

1
Pick the ONE ingredient (or quantity) you know both the original and the actual amount for.
2
Scale factor $= amount actually used \div original recipe amount$ .
3
Apply that same factor to everything else in the recipe — including the number of people, if asked.

✏️ Worked example

Jo's Bolognese recipe needs 500g minced beef, 750g chopped tomatoes, 40g mushrooms and 150ml chicken stock. She only has 400g of mince.

1
Find the factor from the ingredient she knows: $400 \div 500 = 0.8$
2
Chopped tomatoes: $750 \times 0.8 = 600 g$
3
Mushrooms: $40 \times 0.8 = 32 g$
4
Chicken stock: $150 \times 0.8 = 120 ml$

✅ Answer: 600g tomatoes, 32g mushrooms, 120ml stock

⚠️ The classic trap

Using the WRONG ingredient to find the factor — swapping which amount is "original" and which is "actual" flips the factor upside down (0.8 becomes 1.25) and every single ingredient that follows is wrong. Always divide what was actually used by what the recipe originally said, never the other way round.

🎮 Practice · find the factor backwards

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Bolognese: recipe needs 500g mince, Jo only has 400g. What scale factor is she using?

Using that same 0.8 factor, Chopped Tomatoes 750g → ?

Chilli Con Carne: 1kg mince serves 6. How much mince (g) for 15 people?

Same recipe: 400g tomatoes serves 6. Shane uses 1200g tomatoes — how many people is that for?

Rice Krispie cakes: 120g chocolate makes 12 cakes. How much chocolate (g) for 30 cakes?

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💡 The idea

The one ingredient that decides everything

With a limited stock of SEVERAL ingredients, you can't just scale by any one of them — you have to check every single ingredient's own scale factor, because whichever one gives the smallest factor is the one that runs out first, and it alone decides the real maximum. Having plenty of the other four doesn't help at all once the fifth one is gone.

📋 How to find the real maximum

1
For EVERY ingredient, work out its own scale factor: $stock \div recipe amount$ .
2
Find the SMALLEST of those factors — that ingredient is the one that runs out first.
3
Multiply that one smallest factor by the recipe's yield.
4
Round DOWN once, at the very end, to a whole number of whatever you're actually making.

✏️ Worked example

Nancy's cupcake recipe (makes 15) needs 120g flour, 135g caster sugar, 300g icing sugar, 3 eggs, 260g butter. She has 360g flour, 450g caster sugar, 850g icing sugar, 12 eggs, 760g butter.

1
Work out EVERY ingredient's own factor: flour $360 \div 120 = 3$ , sugar $450 \div 135 \approx 3.33$ , icing $850 \div 300 \approx 2.83$ , eggs $12 \div 3 = 4$ , butter $760 \div 260 \approx 2.92$ .
2
The SMALLEST is icing sugar's 2.83 — even though butter's 2.92 is close, icing sugar runs out first.
3
Multiply by the yield: $2.8 \overline{3} \times 15 = 42.5$
4
Round down ONCE, at the end, to a whole cupcake: $42$

✅ Answer: 42 cupcakes

⚠️ The classic trap

Rounding each ingredient's factor down SEPARATELY first, then taking the smallest — for this exact recipe that mistake gives floor(2.83)=2 and floor(2.92)=2, so "2 whole batches of 15" = 30 cupcakes, which is simply wrong. Nothing forces you to scale in whole-batch jumps: ingredients scale continuously, so you only round once, at the very last step, on the OUTPUT — never on the factor itself.

🏁 Which ingredient runs out first?

Flour (360g ÷ 120g)×3.00

Caster Sugar (450g ÷ 135g)×3.33

Icing Sugar (850g ÷ 300g)×2.83

Eggs (12 ÷ 3)×4.00

Butter (760g ÷ 260g)×2.92

🎮 Practice · the limiting ingredient

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Concrete (cement : sand : gravel = 1 : 2 : 3). 43kg cement allows ×43, 83kg sand allows ×41.5, 151kg gravel allows ×50.33. Which ingredient runs out first?

Using that limiting factor, what is the maximum concrete Adil can make (kg)? (1 : 2 : 3 adds up to 6 parts.)

Doughnuts: recipe needs 45g sugar per 12 doughnuts. Harry has 290g sugar (plenty of everything else). What scale factor does the sugar allow? (2 d.p.)

Using the real (unrounded) factor, what is the maximum number of doughnuts Harry can make?

He sells every doughnut for £1.80. How much does Harry raise?

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🤔 A recipe has FIVE ingredients. You have plenty of four of them, but only enough of the fifth for ×3 the recipe. What's the real maximum you can make?

💡 The idea

Combining the skills: scale, then compare

The hardest real questions don't test one skill at a time — they ask you to scale a recipe, then do something ELSE with the result: find how much MORE you need to buy given what you already have, or decode a hidden unit (like the mass of a single bar) from a rate before you can use it at all.

✏️ Worked example

Ella's chocolate rice krispy recipe (makes 8) needs 150g chocolate and 100g butter. She wants 36 cakes, and already has 400g chocolate and 175g butter.

1
Scale factor: $36 \div 8 = 4.5$
2
Chocolate NEEDED in total: $150 \times 4.5 = 675 g$
3
Chocolate she must still BUY: $675 - 400 = 275 g$
4
Same two steps for butter: needed $100 \times 4.5 = 450 g$ , buy $450 - 175 = 275 g$

✅ Answer: Buy 275g more chocolate and 275g more butter

⚠️ The classic trap

Stopping after finding the total NEEDED, and writing that down as the answer to "how much more do you need to buy?" — the question asked for the shortfall, not the total, so the amount already owned always has to be subtracted at the very end.

🎮 Practice · combine the skills

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Chocolate Rice Krispy cakes: recipe makes 8, Ella wants 36. What scale factor does she need?

At ×4.5, the recipe's 150g chocolate becomes how much (g)?

Ella already has 400g chocolate. How much MORE does she need to buy?

Same recipe, butter: needs 450g in total, Ella has 175g. How much more?

Yoghurt bars: protein is 5.75g per 100g, and 2.3g per bar. What is the mass of ONE bar (g)?

A pack weighs 520g. How many 40g bars fit in a pack?

Rice Krispie cakes aside: write 120g out of 200g as a percentage.

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🏅

You finished this lesson!

One idea carries every one of these: find the scale factor, then apply it everywhere, all at once — forwards, backwards, or guarded by whichever ingredient runs out first.

Keep reading — this lesson marks itself done automatically.