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Math · Fractions & Basic Operations

Ratio and proportion

Set up and solve ratio, rate, and proportion problems — scaling a recipe, reading a map, mixing a solution. The setup decides the answer; the cross-multiplication at the end is the easy part. Every question comes with a written breakdown of how the proportion was set up.

Practise ratio and proportion nowSee sample questions
3
Difficulty tiers
40s
Target pace
56%
Average first-attempt score

What's covered

  • Writing a ratio correctly — keep the order of quantities consistent with how the problem states them — 3:5 and 5:3 describe different situations.
  • Setting up a proportion — put corresponding quantities in the same position on both sides of the equation before cross-multiplying.
  • Unit rates — reduce a rate to a single unit — price per item, distance per hour — to compare two options fairly.
  • Scaling a ratio — multiply every part of a ratio by the same factor to preserve the relationship, whether scaling up or down.
  • Part-to-part vs. part-to-whole — a ratio like 2:3 splits a total into 5 parts, not 3 — converting to a fraction of the whole requires that extra step.

Where students lose marks

Mismatching the order across a proportion

If the first ratio is written as boys:girls, the second one has to stay boys:girls too — flipping the order on one side breaks the whole equation even though the numbers might still look reasonable.

Treating a part-to-part ratio as a part-to-whole fraction

In a class with a 2:3 ratio of boys to girls, boys aren't 2/3 of the class — they're 2/5, since the ratio's two parts add up to the whole.

Cross-multiplying before the units match

A rate given in different units on each side of a proportion — minutes on one side, hours on the other — needs to be converted to matching units before cross-multiplying, not after.

Three sample questions

Straight from the bank — one per difficulty tier. Reveal the answer to see the explanation you'd get in a real session.

Sample 01Foundation

A recipe uses a ratio of 2 cups of flour to 3 cups of sugar. If you use 10 cups of flour, how many cups of sugar are needed?

Sample 02Core

A car travels 180 miles in 3 hours. At the same rate, how many minutes will it take to travel 60 miles?

Sample 03Advanced

Two alloys are mixed: Alloy A is 40% copper and Alloy B is 70% copper. How many kilograms of Alloy B must be mixed with 6 kg of Alloy A to produce an alloy that is 60% copper?

How to practise this

Ratio and proportion problems are almost entirely about the setup — once the proportion is written correctly, cross-multiplying rarely goes wrong.

Label every quantity before setting up the ratio

Write what each number represents next to it — flour, sugar, hours, copper — before building the proportion. Untimed practice is where that labeling habit sticks.

Then 40 seconds a question

Move to timed sessions once setup is fast and consistent. Ratio problems run a bit longer since there's a real-world scenario to parse first.

Fold into a mock

Pair ratio and proportion with word problems in a mock — both are translation-heavy skills before any arithmetic starts.

Stop guessing the setup.
Start labeling every quantity.

  • A setup walkthrough on every item — not just the cross-multiplication
  • Timed or untimed sessions, any length you like
  • Tracked separately — see your ratio and proportion accuracy and pace over time
Start practising free

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Related Math subtopics

Decimals and percentagesOperations on fractionsOrder of operations (PEMDAS)Scientific notationSimplificationAll of Math