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Ratio & Proportion Cheatsheet (Years 5–8)

Understanding ratios and proportions is essential for math success, especially for students in Years 5 to 8. This comprehensive cheatsheet equips learners with fast, reliable methods to solve ratio and proportion problems, making it an invaluable resource for homework, test prep, and quick revision. Let's break down the core concepts and provide practical examples for easy understanding.


Key Concepts in Ratio & Proportion


What is a Ratio?


A ratio compares two or more quantities. It's important to note that the order of parts matters. For example, if you have red and blue marbles in a ratio of 3:2, it means for every 3 red marbles, there are 2 blue marbles.


Equivalent Ratios


Equivalent ratios are found by multiplying or dividing both parts of the ratio by the same number. For instance, the ratio 3:4 can also be expressed as 6:8 by multiplying both sides by 2.


Understanding Unit Rates


A unit rate tells you how much of one item corresponds to one of another item. For instance, if a pen costs £0.80, this is the unit rate for pens. Similarly, if a car travels 60 km in one hour, the unit rate for speed would be expressed as 60 km/h.


Proportions


A proportion states that two ratios are equal. You can solve proportions using either the unitary method or cross-multiplying. For example, if you know that 1/2 = x/6, you can cross-multiply to find x.


Visualizing with Bar Models


Bar models are a helpful visual tool for understanding ratios and proportions. They allow you to see how many parts each component contributes to the whole, making it easier to solve problems.


Essential Moves in Ratio & Proportion


Students will frequently utilize the following five moves when dealing with ratio and proportion questions:


  1. Simplify the Ratio: Always start by dividing both parts by the highest common factor (HCF).


  2. Scale Up or Down: Multiply or divide both sides of the ratio by the same number to find equivalent ratios.


  3. Find One Part (Unitary Method): If you have the total, divide it by the number of parts to find the value of one part.


  4. Solve a Proportion: Set up the equation and cross-multiply to find an unknown.


  5. Use a Scale Factor: When working with maps or drawings, remember that a scale factor tells you the real-life distance represented by a measurement on the map. For example, a scale of 1:200 means 1 unit on the map equals 200 units in reality.



Quick Reference for Ratio & Proportion


  • Part-to-Total Relationship: If A:B = 3:5, the total parts equal 8. If the total equals 24, then each part equals 3 (24 ÷ 8). Thus, A = 9 (3 x 3) and B = 15 (5 x 3).

  • Fraction Link: In the ratio A:B, the fraction of A to the whole (A+B) is calculated as A/(A+B). Be cautious: the ratio itself is not a fraction of the total unless summing the parts.


  • Speed Triangle: Use the formula: Speed = Distance/Time, Time = Distance/Speed, and Distance = Speed * Time to tackle problems related to speed.


Worked Examples


Let's step through some worked examples to illustrate these concepts:


  1. Simplifying Ratios:

    • Simplify the ratio 18:24. Divide both by 6, so it becomes 3:4.


  2. Sharing Money:

    • Share £64 in the ratio 1:3:4. Total parts = 1 + 3 + 4 = 8. Each part is £8 (64 ÷ 8). So £8, £24, and £32 for each part.


  3. Scaling Recipes:

    • In a recipe, the ratio of flour to milk is 2:5. For 10 cups of milk, calculate the flour needed: 10/5 = 2, 2x2 = 4 cups.


  4. Map Measurements:

    • Using a scale of 1:50,000, if a distance on the map is 3.4 cm, the real-world distance is ( 3.4 times 50,000 = 170,000 ) cm or 1.7 km.


  5. Calculating Speed:

    • If a car travels 120 km in 3 hours, its speed is 120 / 3 = 40 km/h.


  6. Mixtures:

    • Given nuts:raisins = 4:1, if there are 2 kg of nuts, then one part equals 0.5 kg and implies raisins also weigh 0.5 kg.


Close-up view of a speed triangle for equations

Common Traps in Ratio & Proportion


While mastering ratios and proportions, students often encounter common pitfalls:


  • Mixing Part-Part and Part-Whole: Ensure you recognize whether the question provides totals or simply compares parts.

  • Getting the Direction Wrong: With a scale of 1:200, the real value is drawing × 200—not division.


  • Units Consistency: Always keep your units (km ↔ m, £ ↔ p) consistent throughout calculations.


  • Rounding Too Early: Work with exact values and delay rounding until the end of your calculations to ensure accuracy.


Practice Sets


To reinforce these skills, here is a set of practice questions tailored for Years 5-8, along with their answers:


Year 5 Questions

  1. Simplify 12:18.

  2. In a class, boys: girls = 3:5. If total = 24, how many girls?

  3. Share £64 in the ratio 1:3:4.

  4. A recipe uses 2 cups of flour to 5 cups of milk. How much flour for 10 cups of milk?

  5. On a map, 1 cm represents 5 km. What distance is 3.4 cm?

  6. A car travels 120 km in 3 hours. What is the speed?


Answers: 2:3, 15 girls, £8, £24, £32, 4 cups, 17 km, 40 km/h.


Year 6 Questions

  1. Give an equivalent ratio to 9:12 with the second term 36.

  2. Orange: red marbles = 7:3. If there are 15 red, how many orange?

  3. Syrup: water = 2:3. Make 1.25 L drink—how much of each?

  4. Scale drawing 1:200. A pillar is 6.5 cm on the plan. What is the real height?

  5. If 5 pens cost £4, what is the cost of 12 pens (same price each)?

  6. Trail mix has nuts:raisins = 4:1. If nuts = 2 kg, how much are the raisins?


Answers: 27:36, 35 orange, Syrup 0.50 L, Water 0.75 L, 13 m, £9.60, 0.5 kg.


Years 7–8 Questions

  1. A model car is 18 cm long at a scale of 1:24. What is the real length?

  2. At 60 km/h, how long to travel 150 km?

  3. Triangle angles are in the ratio 2:3:4. What are the angles?

  4. If A:B = 5:7 and A = 35, find B.

  5. y is directly proportional to x. When x = 8, y = 24. Find y when x = 14.

  6. Blue: yellow paint = 3:2. Make 2.5 L of paint. How much of each?

  7. Best buy: 6 batteries for £4.20 or 10 for £6.80?


Answers: 4.32 m, 2.5 hours, 40°, 60°, 80°, B = 49, y = 42, Blue 1.5 L, Yellow 1.0 L, 10-pack (≈ £0.68 each vs. £0.70).


Mastery Checklist


To ensure you're on the right track, using this checklist can be invaluable:


  • Ratio simplified (divide by HCF)

  • Total parts identified (if needed)

  • One-part value found (unitary method)

  • Units consistent and labeled

  • Final answer checked against the question


Continuous Learning


Want structured math practice with weekly feedback? Book a free trial lesson with Education Nation today! We'll help diagnose your gaps and build a tailored plan for the next 6-8 weeks.


High angle view of a straightforward checklist

This cheatsheet provides a strong base for mastering ratios and proportions, giving students the toolkit they need to approach their math work confidently. Happy learning!

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