Secondary Mathematics · Worked Repair Guide 24
Proportion is a statement about how quantities change together. Direct proportion means one quantity is a constant multiple of another. Inverse proportion means the product of the two quantities stays constant. Both ideas appear inside rate, scale, finance, geometry, science-style models and graphs, but they should not be reduced to cross-multiplication recipes.
This guide develops one central habit: identify what must remain constant before calculating. For direct proportion, the ratio y/x is constant. For inverse proportion, the product xy is constant. Once the invariant is known, the algebra, graph and numerical table should all agree.
All contexts below are invented teaching models. Real rates can depend on additional variables, thresholds and fixed costs. Use the simplified formulas only when the stated assumptions support them.
1. Direct proportion means a constant ratio
If y is directly proportional to x, write y=kx for some constant k.
If y=18 when x=6, then k=y/x=3. Therefore y=3x.
When x=11, y=33. When x doubles, y doubles; when x is multiplied by 5, y is multiplied by 5.
Entry check: if y=4x, then y/x=4 for every non-zero x. At x=0, y=0 too, so the graph passes through the origin.
2. A straight line is not enough for direct proportion
The equation y=3x+5 is linear but not directly proportional because y/x is not constant and the graph does not pass through the origin.
At x=1, y/x=8; at x=5, y/x=4. The ratio changes.
Direct proportion has the special linear form y=kx with zero intercept.
This connects directly to gradient: k is the gradient of the proportionality graph.
3. Find the constant before using the model
Suppose an invented printing model uses ink amount I directly proportional to the number n of identical pages. If 50 pages use 12 mL, then I=kn and k=12/50=0.24 mL per page.
The model is I=0.24n. For 140 pages, I=33.6 mL.
The units of k are useful: millilitres per page. The constant is not an abstract number detached from the context.
If real printing included a fixed cleaning loss, direct proportionality would fail because the model would need a non-zero intercept.
4. Rate is often a constant of proportionality
If distance d is directly proportional to time t under constant speed v, then d=vt. Speed is the constant ratio d/t.
At 72 km/h for 2.5 h, d=180 km.
But if speed changes during the journey, one fixed v no longer describes the entire trip.
A formula can be algebraically valid under its assumptions and still be a poor model when those assumptions fail.
5. Unit rate and proportion are two views of the same structure
If 8 identical items cost $28 in an invented linear price model with no fixed fee, the unit price is 28/8=$3.50.
Cost C=3.5n. Twelve items cost $42.
A proportion equation such as 28/8=C/12 produces the same result because both ratios equal the same constant k.
The unit-rate method often makes the invariant more visible than cross-multiplication.
6. Scale factor is a direct proportion relationship
For similar shapes, corresponding lengths are directly proportional.
If a smaller shape has side 6 cm corresponding to 15 cm in a larger shape, the length scale factor is 15/6=2.5.
A corresponding 8 cm side becomes 20 cm.
Area, however, scales by 2.5²=6.25. Do not use the length factor directly on area.
7. Direct variation can involve powers
Where variation notation is in scope, “y varies directly as x²” means y=kx², not y=kx.
If y=45 when x=3, then 45=9k, so k=5. The model is y=5x².
When x doubles from 3 to 6, y multiplies by 4 because x² has quadrupled.
Always preserve the stated power when finding k.
8. Inverse proportion means a constant product
If y is inversely proportional to x, write y=k/x for x≠0, or equivalently xy=k.
If y=12 when x=5, then k=xy=60. Therefore y=60/x.
When x=10, y=6. Doubling x halves y.
The product remains 60.
9. Inverse proportion is not a decreasing straight line
The graph y=k/x is a curve, not a straight line.
For positive k and positive x, y decreases as x increases, but it does not decrease by a constant amount per unit x.
The equation y=20−2x also decreases, but it is not inverse proportion because xy is not constant.
Do not identify inverse proportion from “one goes up while the other goes down” alone.
10. Work-rate models require clearly stated assumptions
In a simplified model, suppose completion time t is inversely proportional to the number n of identical workers because total work W is fixed and each worker maintains the same rate.
If 6 workers take 10 hours, nt=60 worker-hours under the model. Ten workers would take 6 hours.
Real tasks may not scale this way because workers can interfere, coordinate or perform different roles.
The inverse model is therefore conditional, not a universal statement about teamwork.
11. Constant area can create inverse variation
For rectangles of fixed area A, length l and width w satisfy lw=A. Therefore w=A/l.
If area is 48 cm² and length is 6 cm, width is 8 cm. If length becomes 12 cm, width becomes 4 cm.
The product remains 48.
This geometric model gives inverse proportion a concrete meaning: one dimension must shrink when the other grows if the area stays fixed.
12. Inverse variation can also involve powers
“y varies inversely as x²” means y=k/x².
If y=8 when x=3, then k=yx²=8×9=72.
Therefore y=72/x². When x=6, y=72/36=2.
Doubling x divides y by 4, not by 2.
13. Test a table by checking the invariant
Table A has x values 2,4,6 and y values 5,10,15. Ratios y/x are 2.5,2.5,2.5, so direct proportion holds.
Table B has x values 2,4,8 and y values 12,6,3. Products xy are 24,24,24, so inverse proportion holds.
Table C has x values 2,4,6 and y values 7,11,15. Neither ratio nor product is constant. It follows the linear equation y=2x+3 instead.
Testing the invariant is faster and safer than relying on the visual trend.
14. Ratios need matching quantities and units
Suppose distance is 3000 metres in 4 minutes. The rate is 750 metres per minute.
If another calculation uses kilometres per hour, convert consistently: 3 km in 4 minutes =3 km in 1/15 hour, so speed=45 km/h.
Do not compare 750 m/min numerically with 45 km/h without conversion. Different units can represent the same physical rate.
Use Units, Scale and Measurement when compound-unit conversion is the actual weakness.
15. Percentage change is multiplicative proportion
An increase of 12% multiplies by 1.12. A decrease of 12% multiplies by 0.88.
Repeated percentage change applies the multiplier repeatedly to a changing base.
This is not direct proportion between time and amount; the amount follows a multiplicative sequence under a fixed percentage rate.
Use Financial Arithmetic — Growth, Depreciation and Interest for that distinction.
16. Joint variation combines several factors
As an optional extension, if z varies directly as x and y, then z=kxy.
If z=24 when x=3 and y=4, then k=24/12=2. Thus z=2xy.
At x=5 and y=7, z=70.
If z instead varies directly as x and inversely as y, the model is z=kx/y. Translate the words into a formula before substituting numbers.
17. Linearising a relationship can reveal the constant
If y=kx², plotting y against x² gives a straight line through the origin with gradient k.
If y=k/x, plotting y against 1/x gives a straight line through the origin.
This changes the representation without changing the relationship.
It also explains the role of transformed graphs in later Mathematics: a nonlinear-looking relationship can become linear in a more suitable variable.
18. Check model plausibility before calculation
If y is directly proportional to positive x with positive k, doubling x should double y. If a proposed answer becomes smaller, the direction is wrong.
If y is inversely proportional to positive x, doubling x should halve y. A proposed result doubling both quantities contradicts the model immediately.
These directional predictions are useful before the calculator is used.
A good check asks whether the invariant ratio or product survived.
19. Capstone: distinguish fixed cost from proportional cost
An invented delivery model charges $8 fixed plus $2.50 per kilometre. The total cost is C=8+2.5d.
This is linear but not directly proportional because when d=0, C=8 rather than 0.
For 20 km, C=$58. The ratio C/d=2.9 is not the per-kilometre variable rate because it includes the fixed charge.
For 40 km, C=$108 and C/d=2.7. The changing average cost per kilometre confirms that total cost is not directly proportional to distance.
The gradient 2.5 still has a clear interpretation: each additional kilometre adds $2.50 under the model.
20. Independent practice
- y is directly proportional to x and y=21 when x=7. Find k.
- Write the direct-proportion equation for Question 1.
- Find y when x=15.
- Decide whether y=4x+2 is direct proportion.
- A proportional cost model gives 9 items for $31.50. Find the unit rate.
- At the same rate, find the cost of 14 items.
- y varies directly as x² and y=48 when x=4. Find k.
- Use Question 7 to find y when x=10.
- y is inversely proportional to x and y=9 when x=8. Find k.
- Use Question 9 to find y when x=12.
- Under an inverse-work model, 5 identical workers take 18 hours. Find the modelled time for 9 workers.
- A rectangle has fixed area 72 cm². Find width when length is 8 cm.
- For the same area, find width when length is 18 cm.
- y varies inversely as x² and y=20 when x=2. Find k.
- Use Question 14 to find y when x=5.
- Test whether x=2,4,10 and y=15,7.5,3 represent inverse proportion.
- Test whether x=3,6,9 and y=8,16,24 represent direct proportion.
- If z varies jointly as x and y, and z=30 when x=3,y=5, find z when x=4,y=7.
- Explain why y=3x+6 is linear but not directly proportional.
- An inverse-proportion model has xy=84. If x increases from 7 to 14, predict and calculate the change in y.
21. Worked answers
1. k=3. 21/7.
2. y=3x.
3. 45.
4. No. The non-zero intercept means y/x is not constant.
5. $3.50 per item.
6. $49.
7. k=3. 48=16k.
8. 300. y=3x².
9. k=72. xy=72.
10. 6. 72/12.
11. 10 hours. Worker-hours=90, so 90/9.
12. 9 cm.
13. 4 cm.
14. k=80. yx²=20×4.
15. 3.2. 80/25.
16. Yes. Products are 30,30,30.
17. Yes. Ratios are 8/3 for all three pairs.
18. 56. k=30/(15)=2, so z=2(4)(7).
19. Direct proportion requires the form y=kx and therefore passes through the origin; y=3x+6 has a fixed offset.
20. y halves from 12 to 6. Initially y=84/7=12; after doubling x to 14, y=84/14=6.
22. Diagnose proportion errors by the invariant they break
Common failures include assuming every straight line is direct proportion, identifying every decreasing relationship as inverse proportion, using the wrong power in a variation formula, or cross-multiplying without first matching quantities and units.
A useful correction note says “direct → constant ratio”, “inverse → constant product”, “fixed intercept breaks direct proportionality”, or “rate units must match”.
Then predict the direction of change before calculating. A learner who knows the invariant should be able to say what doubling one variable will do to the other.
23. Continue through the BTT learning routes
Return to the BTT Mathematics Hub or BTT Mathematical Lab. Use Ratio, Percentage and the Correct Base for foundational proportional thinking and Graphs, Tables and Relationships for graphical rate interpretation.
Within Batch 06, use Coordinate Geometry for gradient as a rate, Sequences, Patterns and Nth-Term Generalisation for repeated additive and multiplicative patterns, and Transformations, Symmetry, Coordinates and Invariants for geometric scale factors.
24. Sources and scope
The rate, work, cost, area and variation examples are original teaching models. Their conclusions apply only under the explicitly stated proportionality assumptions.
For the current Singapore Secondary curriculum doorway, see MOE: Curriculum for secondary schools. Match joint variation and transformed-graph extensions to the learner’s actual subject level and school programme.