Mathematics · 06 / 23
Proportionality and percentages
Middle school · Worked example and 4 exercises with solutions, at your pace.
The principle, simply
Quantities are proportional when one constant factor maps every value of the first to the second. Their graph is a line through the origin. A percentage is a fraction out of 100: taking 15% multiplies by 0.15. A 15% discount multiplies by 0.85 because 85% remains.
- 01Constant factor
- 02Same unit
- 03Multiply or divide
- 04Check the ratio
WORKED EXAMPLE
From principle to reasoning
Three notebooks cost €7.50, so each costs €2.50 and five cost €12.50. A €40 item discounted 15% costs 40 × 0.85 = €34.
Solve a problem in this subject
- Define the objects
State unknowns, domain and data. Real square roots need nonnegative arguments; denominators must be nonzero.
- Choose a transformation
Name the rule: expansion, equivalent equations, factoring, differentiation. Check its conditions. Squaring an equation requires testing candidates in the original.
- Show the reasoning
Each line should follow from the previous one. Keep exact values where possible and distinguish equality, approximation and implication.
- Test and conclude
Substitute into the original expression. Check graph points and domain boundaries. State a solution set or the required unit.
YOUR TURN
01 · Try before checking
A recipe for four uses 300 g. How much for six?
One hint
Use one person or multiply by 6/4.
Reveal and understand the solution
300/4 = 75 g per person; six need 450 g. The quantity ratio 1.5 matches 6/4.
Compare method, conditions and answer; identify the exact part to revisit.
IMPROVE THROUGH PRACTICE
Three additional challenges with solutions
Start with the first challenge. Write your method before opening a hint; use the solution to find the first step that needs work.
Four identical notebooks cost €12. What do seven cost?
A hint to get started
Find the price per notebook.
Step-by-step solution
- 12/4 = €3 per notebook.
- Seven cost 7 × 3 = €21.
- This assumes a constant unit price, without discounts.
Which step did you miss? Retry without help, then explain why the method works.
An €80 price rises by 15%. Find the new price.
A hint to get started
Use a factor of 1 + 15/100.
Step-by-step solution
- The increase is 80 × 0.15 = €12.
- New price: 80 × 1.15 = €92.
- 15 is a percentage, not an amount in euros.
Which step did you miss? Retry without help, then explain why the method works.
Does a 20% rise followed by a 20% fall restore a price?
A hint to get started
The two percentages use different bases.
Step-by-step solution
- The combined factor is 1.2 × 0.8 = 0.96.
- An original 100 becomes 96.
- This is a net 4% fall, not cancellation.
Which step did you miss? Retry without help, then explain why the method works.
The trap to avoid
Fixed fees break proportionality. Equal percentage increases and decreases do not cancel.
Without notes, can you…
- Explain the principle in your own words?
- Rebuild the diagram and its connections?
- Solve the exercise and justify the method?
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