Eleven strands, seven rungs each, 77 questions. In this strand — ratio and proportion — you were secure all the way to step 4, and the first gap opened at step 5. Step 5 is the IGCSE Core standard rung: not the hard end of the paper, the ordinary middle of it.
That is a precise and useful result. It says the idea of a ratio is fine. What broke is the step where a ratio stops being something you read and starts being something you do a calculation with — sharing an amount out, scaling a recipe, working with a rate. Four rungs of secure ground sit underneath this, so we are not rebuilding from the bottom. We are repairing one storey.
| Code | Topic | Risk |
|---|---|---|
| E1.11 | Ratio and proportion | High non-calculator risk |
| E1.12 | Rates — speed, density, best buy, currency | High non-calculator risk |
| E1.14 | Working an answer out by hand | Every question on Paper 2 |
Paper 2 is non-calculator: 2 hours, 100 marks, 50% of your grade. So every method in this guide is a by-hand method. Nowhere will you be told to reach for a machine. Where a division looks ugly, you will be shown how to make it not ugly.
One more finding to carry into section 3 and section 6: sign errors recurred across several different strands. In ratio work they bite in exactly one place — the difference between two shares — and that place is flagged where it happens.
Number sense and the four operations: solid on all seven rungs. No gap anywhere. You can add, subtract, multiply and divide, you can handle negatives inside an ordinary sum, and you can hold a calculation together without dropping it.
That is the thing most students who struggle with ratio are actually missing, and you are not missing it. Every method below is built out of dividing once and multiplying once. The arithmetic in this whole guide is arithmetic you have already proved you can do. What has to change is knowing which division to do, and in which order — and that is a much smaller thing to fix than it feels like.
Every method appears four times and hands you less each time. That fading is the point of the build.
Sections 1 and 2 sit below your break, at steps 3 and 4, on purpose. They are short. Do not skip them — section 3 is built directly on top of them, and it is section 3 where the check says you fell.
A ratio is a statement about parts. When a recipe says flour and butter in the ratio 3 : 2, it means: chop the whole thing into equal-sized parts, and flour takes 3 of them while butter takes 2. Nothing more complicated than that, and everything in section 3 comes straight out of it.
5 parts in the margin. For 4 : 3 : 5 write 12 parts. That single
number is what almost every ratio question turns on, and writing it first means you can never forget to
find it.A ratio behaves like a fraction in one respect: multiplying or dividing every part by the same number does not change what it says. 6 : 4 and 3 : 2 describe the identical mixture. Simplifying is worth doing before anything else, because it makes the total number of parts small, and a small total makes the division in section 3 easy by hand.
Some questions insist on a ratio starting with 1 — map scales in particular. Divide both parts by the first part. The second part is then allowed to be a decimal; that is not a mistake, it is the point.
This is the rung the check says you fell on, so read this section properly rather than skimming it. It is also, usefully, the most mechanical thing in the whole strand. There is one method, it never changes, and it is three lines long.
“Anya gets £48 more than Ben” is the same method with one substitution. The difference in the shares is the difference in the parts. Look back at the bar model: the two shares differed by exactly one part, because 3 − 2 = 1.
Same three lines, entered at a different point. If one share is known, divide it by its own number of parts to get one part, then carry on as usual.
Two quantities are in direct proportion when doubling one doubles the other, and halving one halves the other. Ingredients and servings. Petrol and distance. Number of pens and cost. The tool is the unitary method: get down to one, then scale up to what you want. It is the same shape as section 3 — divide once, multiply once — which is why these two rungs broke together.
1 person = 50 g — because the commonest way
this goes wrong is losing track of which quantity you are holding.Extended asks the same idea in symbols. “y is directly proportional to x” is written
y ∝ x and means y = kx for some fixed number k. Find k first, always, using
the pair of values you were given. Then the equation answers everything else.
Some pairs of quantities move in opposite directions. More workers, fewer days. Faster speed, shorter time. More people at the table, smaller slice each. This is inverse proportion, and it is a different method from section 4. Using the section 4 method here is the classic wrong turn, and it produces an answer that is obviously silly if you stop to look at it.
“y is inversely proportional to x” is written y ∝ 1⁄x
and means y = k ÷ x. Same routine as direct proportion: substitute the given pair, find k,
then use the equation. The only difference is where x sits.
Extended goes further: y can be inversely proportional to x2, written
y ∝ 1⁄x², meaning y = k ÷ x².
Everything is identical — square x before you do anything else with it.
x² = 36, the honest answer
is x = ±6, because a negative squared is positive too. In a worded context — a
distance, a number of people, a length — you take the positive value and say why. In a bare algebra
question, give both. Losing that ± is the sign error that costs marks here.A rate is how much of one thing you get per one of another. Kilometres per hour. Grams per cubic centimetre. Rupees per pound. Pence per gram. Every rate question is the unitary method from section 4 wearing a different coat, which is why they sit in the same syllabus point.
An exchange rate is just a rate: so many of one currency per one of the other. Going one way you multiply, going the other way you divide. Decide which before you write anything.
Two sizes, two prices, which is better value. There are two honest routes and you may use either, but you must use the same one for both packs and compare like with like.
Every one of these is “an amount per one unit”. Find the amount for one unit, then scale it. When the formula is given in the question, your job is to substitute and rearrange, with the same triangle as speed and density: the “per” quantity goes underneath.
| rate | per | example |
|---|---|---|
| hourly pay | dollars per hour | $12.40 an hour for 38 hours |
| flow rate | litres per minute | 18 litres per minute |
| fuel consumption | km per litre (bigger is better), or litres per 100 km (smaller is better) | 6.5 litres per 100 km |
| pressure = force ÷ area | newtons per m² (or per cm²) | 600 N on 0.25 m² |
| population density = population ÷ area | people per km² | 2.4 million people on 1500 km² |
Fifteen questions drawn from all six sections, shuffled and unlabelled. Ordinary revision does one topic at a time, which quietly does the hardest part for you: deciding which method applies. Here nobody tells you. Before each one, say to yourself which it is — sharing, direct, inverse, a rate, or a simplification — and only then start writing. Sorting the type is the part of the exam that revision usually skips.
Six lines that cover almost everything an examiner can ask in this strand:
Come back to section 7 in a week without reading anything above it. If the bar sits high on a cold run, step 5 is repaired in this strand and you can move to the next one.