Ratios and proportional reasoning in context
Keep units consistent, share quantities fairly and recognise what stays constant in a proportional relationship.
Core and Extended.
Before you begin
- Find common factors.
- Calculate a fraction of a quantity.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
Simplify the ratio 2 m : 50 cm.
What you will learn
- Simplify and compare ratios with compatible units.
- Divide an amount in a given ratio.
- Use proportional reasoning for recipes, scales and best value.
A ratio compares quantities in order
If red to blue beads is 3:5, each group of three red beads corresponds to five blue. The order matters: blue to red is 5:3. To simplify 18:30:42, divide all three values by the same common factor six, giving 3:5:7. Adding the same number to every part does not preserve a ratio.
For a ratio of quantities with the same dimension, convert to the same unit first. The ratio 1.5 m to 60 cm is 150:60 = 5:2. The unsimplified labels 1.5:60 compare different unit sizes and give the wrong relationship. Ratios of different dimensions, such as kilometres to hours, describe rates and need their units stated.
Find the size of one part
To share $84 in the ratio 2:5, there are seven equal parts. One part is $84/7 = $12, so the shares are $24 and $60. Check both the total and the ratio. If only one share is given, its number of parts determines the part size; do not divide that share by the combined number of parts.
The fraction belonging to the first group in a 2:5 ratio is 2/7 of the total, not 2/5. The denominator for a share of the whole counts all parts. For three groups in the ratio 2:3:4, the whole has nine parts. State whether a question compares groups or asks for a fraction of the whole.
Scale every relevant quantity together
A recipe using 240 g of flour for six portions uses 40 g per portion under the proportional model. Ten portions require 400 g. The factor 10/6 must also be applied to every other ingredient. This model assumes portion size stays constant; changing serving size would require a different factor.
A map scale 1:25 000 means one map length unit corresponds to 25 000 of the same units in reality. A 4 cm line represents 100 000 cm = 1 km. Convert after applying the scale, and distinguish length scaling from area scaling. Scaling both sides of a rectangle by two multiplies its area by four.
Compare a common amount
For best value, compare prices for the same quantity. A 750 g pack for $3.60 costs $4.80 per kilogram; a 1.2 kg pack for $5.40 costs $4.50 per kilogram. The second has the smaller unit price even though its total price is higher. Budget, waste and quality may still affect a real purchase.
A directly proportional relationship has a constant quotient, such as cost/amount when there is no fixed fee. If a delivery service charges a fixed fee plus a per-kilogram price, total cost is not proportional to mass. In an inverse relationship with fixed total work, more equally productive workers reduce time, but only if the total-work and equal-productivity assumptions hold.
Pause and explain
In the ratio boys:girls = 2:3, what fraction of the group are girls?
Worked example
Share 96 counters among three groups in the ratio 3:4:5.
Show the worked solution
- There are 3 + 4 + 5 = 12 equal parts.
- Each part contains 96/12 = 8 counters.
- The groups receive 24, 32 and 40. Check their sum is 96 and their ratio simplifies to 3:4:5.
Answer 24, 32 and 40 counters
From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Share $75 in the ratio 2:3.
Give me a hint
Find the size of one part before multiplying for each share.
Compare my reasoning
- Total parts: 2 + 3 = 5.
- One part is $75/5 = $15.
- The shares are $30 and $45, totalling $75.
$30 and $45
Look for these in your work
- Divided by the total number of parts.
- Checked both shares add to the stated whole.
A recipe uses 360 g of rice for eight portions. Find the amount for fourteen identical portions, and the real distance represented by 6 cm at scale 1:50 000.
Give me a hint
Use a per-portion amount for the recipe and common length units for the map.
Compare my reasoning
- Rice per portion is 360/8 = 45 g; fourteen need 630 g.
- The map distance represents 6 × 50 000 = 300 000 cm.
- There are 100 000 cm in a kilometre, so this is 3 km.
630 g; 3 km
Look for these in your work
- Kept the portion size fixed.
- Converted centimetres to kilometres after scaling.
Paint A has red:white = 2:3. Paint B has red:white = 3:5. Which mixture has the larger red fraction, and how much white must be added to 10 L of A to match B?
Give me a hint
Compare fractions of each whole; the red amount stays fixed when white is added.
Compare my reasoning
- A is 2/5 red and B is 3/8 red; 2/5 is greater.
- Ten litres of A contains 4 L red and 6 L white. A 3:5 ratio with 4 L red needs (5/3) × 4 = 20/3 L white.
- Add 20/3 − 6 = 2/3 L white; the final ratio is 4:(20/3) = 3:5.
A has more red proportionally; add 2/3 L of white.
Look for these in your work
- Compared each share against its own whole.
- Kept the red quantity fixed while adjusting white.
Common mistakes
- Comparing quantities before matching their units.
- Using one ratio part as the denominator for a fraction of the whole.
- Assuming a relationship with a fixed fee is directly proportional.
What can you explain now?
Simplify 18:24:30.
Compare with the explanation
3:4:5
Divide every part by the greatest common factor six.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Explain why a 3:7 ratio gives 3/10 of the whole. Create a best-value comparison where the more expensive pack has the smaller unit price.
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.
Original StudyForA teaching content · AI-assisted checks · Human teacher review pending.
Number contains 19 sequenced lessons; Algebra and graphs contains 21; Coordinate geometry contains 10 teaching lessons and a mixed checkpoint. Nine earlier overviews support selected topics across the syllabus. The other six syllabus areas, full cumulative assessment and human teacher review are not yet complete. Written lesson practice is self-checked, not automatically graded. The checkpoint samples skills and does not save an exam grade or certify mastery.
Check the official syllabus 2025–2027 Open related practice and resources