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This page sets out the method for proportion in the order you would use it, then walks through the full lesson explanation. For three complete worked examples with the slips that cost marks, see the Proportion topic page.
When the new number of people or items is awkward (4 people to 10 people), find the amount for one first.
When the new number divides nicely by the old one (6 to 15 is ), multiply everything by the factor.
Find the amount for one, then multiply.
4 cakes need 200 g, so 1 cake needs 50 g, and 7 cakes need 350 g.
The ratio stays the same when both quantities scale.
Zero cakes need zero flour. The graph would go through the origin.
Scale every ingredient by the same multiplier.
Do not scale the flour and leave the sugar.
Half the people means half of each ingredient.
Divide by the old number of people, then multiply by the new number.
If the recipe grows from 4 to 10, the multiplier is 10/4 = 2.5.
Multiply every ingredient by 2.5.
More cakes need more flour.
If your answer went down when the recipe grew, the division was the wrong way.
Direct proportion means the second quantity is a fixed multiple of the first. If one cake needs 25 g of oats, then n cakes need 25n grams. There is no extra spoonful added on for luck. Adding the extra cakes onto the grams mixes up two different units.
The graph of a direct proportion would be a straight line through the origin. You do not need to draw it to use that fact. It tells you that doubling one side doubles the other, and that a zero batch costs nothing and needs nothing. That is the test of this topic.
Method: 1. Check that both quantities should rise together. 2. Do not add the change in the count onto the grams or the pounds. 3. Use one multiplier for the whole recipe. 4. If the count rises, the ingredient total should rise as well.
Finding one unit breaks an awkward jump into two easy operations. Eight flapjacks use 200 g, so one uses 25 g, because 200 ÷ 8 = 25. The division is exact, so the one-unit amount is trustworthy. Six flapjacks are then six copies of that amount.
The one-unit amount is a step, not always the final answer. Stop there only when the question asked for one. Otherwise multiply by the new number. A useful check is to multiply back: one unit times the original number should return the original amount.
Method: 1. Divide the known amount by the known number of units. 2. Write down the amount for one unit. 3. Multiply by the number of units you need. 4. Multiply the one-unit amount by the original count as a check.
A recipe is a bundle of ratios. If the people change from 6 to 14, the multiplier is 14/6. Chicken, stock and rice all take that same multiplier. Scaling the chicken and leaving the stock would change the dish, not just the size.
You can scale up or down. Going from 4 people to 10 is a multiplier greater than 1. Going from 6 portions to 2 is a multiplier less than 1. Write the multiplier as a fraction if that keeps the arithmetic exact, then multiply.
Method: 1. Write the new number of people over the old number. 2. Simplify that fraction if you can. 3. Multiply every ingredient by the same multiplier. 4. Check that a larger batch uses more of each ingredient.
A price list in direct proportion works like a recipe. If 3 kg cost £4.50, one kilogram costs £1.50. Eight kilograms cost 8 × 1.50 pounds. The pounds scale with the kilograms because each kilogram has the same price.
Keep the money exact. Choose the division so that the one-unit price is a whole number of pence. £4.50 ÷ 3 = £1.50 is exact, so the later multiplication does not need rounding. If your final pounds are less than the price of a smaller bag, look again at which number you divided by.
Method: 1. Divide the cost by the number of units you were given. 2. Write the cost of one unit in pounds. 3. Multiply by the number of units you want. 4. Check the one-unit cost against the original bill.
There are two reliable ways to scale a recipe. The unitary method finds the amount for one item or one person, then multiplies. The scale factor method finds how many times bigger the new batch is, then multiplies every ingredient by that number. If 8 flapjacks become 20, the scale factor is 20 ÷ 8 = 2.5. Both methods give the same answer, so use the one with friendlier numbers.
The scale factor method is quicker when the new batch is a simple multiple, such as 2.5 or 1.5, because you only work out the factor once. The unitary method is safer when the factor is awkward, such as 5/3. Avoid rounding a scale factor like 1.666dots to 1.7 because the error is carried into every ingredient. Use a fraction, or divide at the last step.
Method: 1. Identify the quantity in the recipe and the quantity you want, such as portions or biscuits. 2. Find the scale factor, or the amount for one, whichever is friendlier. 3. Multiply every ingredient by the same factor. 4. Check that the new amounts are larger if the batch is larger, and smaller if the batch is smaller.
Tip: If the factor is not a neat decimal, keep it as a fraction until the final multiplication.
Exam recipes usually give a list of several ingredients, such as butter, sugar, flour and eggs. Every amount changes by the same scale factor, so work out the factor once and apply it to each line. Set your working out as a small table with the headings "recipe" and "new amount", so each ingredient has its own row.
Be careful with units. Flour in grams and milk in millilitres are scaled separately, but they use the same factor. Some amounts become more than 1000 g or 1000 ml, and the question may ask for the answer in kilograms or litres. Whole items such as eggs need a sensible answer. If the scaled amount of eggs is 7.5, a cook would use 8, but only round if the question says so.
Method: 1. Work out the scale factor once. 2. Multiply each line of the list by that factor. 3. Convert large amounts to kg or litres only if asked. 4. Check that whole items such as eggs make sense.
A common 4-mark question lists a recipe, then lists what a person has in the cupboard, and asks for the greatest number of items they can make. For each ingredient divide the amount they have by the amount used in one batch. That gives how many batches that ingredient allows. The smallest of these numbers controls the answer, because the cook runs out of that ingredient first.
Another variant says the cook wants to make a given number and asks "how much more of each ingredient does he need?" Scale the recipe to the number wanted, then subtract what is already in the cupboard. Only the ingredients with a shortage need an answer. Take care to convert kilograms into grams before you divide or subtract, as mixing units is the usual error.
Method: 1. Change all amounts to the same unit, such as grams. 2. For each ingredient, divide what is available by what one batch needs. 3. The smallest result tells you the number of batches possible. 4. For "how much more", scale up the recipe and subtract what is already there.
Tip: Write the answers for each ingredient in a column. The smallest one stands out at once.
The same direct proportion idea appears for cost, pay, paint, fuel and travel. If 9 hours of work pays £68.40, then 1 hour pays £68.40 ÷ 9 = £7.60 and 14 hours pays 14 × 7.60 = £106.40. The structure is always the same: find the amount for one unit, then multiply. A table with two columns, one for each quantity, helps to keep the matching numbers in line.
Multi-step problems chain direct proportion with other skills, such as areas and rounding. To paint walls you may need the area, then the litres from the coverage, then the number of whole tins. Tins and packs are whole objects, so after working out the amount needed you always round up, not to the nearest whole number. Buying slightly too much is fine, but buying too little is not.
Method: 1. Identify the two matching quantities, such as hours and pay. 2. Find the amount for one unit with a division. 3. Multiply by the number of units you want. 4. Round up if the answer is a number of tins, packs or boxes.
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