Two quantities can move together without being proportional. A direct-proportion model makes a stronger claim: one quantity is a constant multiple of the other. An inverse-proportion model makes a different strong claim: their product remains constant. Secondary 3 Mathematics becomes more reliable when students test those structures instead of deciding from appearance alone.
This guide develops direct proportion, inverse proportion, constants of proportionality, tables, equations, graphs, model selection and contextual limits. It extends the broader Ratio, Proportion, Percentage, Rate and Speed in Real Contexts guide by making the proportional model itself the object of study.
Official scope: the current 2027 SEC G3 Mathematics syllabus listing identifies Mathematics as K310. Within K310, Ratio and Proportion includes map scales together with direct and inverse proportion. The examples below are original teaching examples; school sequencing can vary.
Decision routine: identify the changing quantities → test the ratio or product → write the constant → form the equation → solve → return to units and context → check whether the model assumptions still make sense.
Direct Proportion Means a Constant Ratio
If y is directly proportional to x, then y/x is constant for nonzero x. We write y∝x and therefore y=kx, where k is the constant of proportionality.
The equation y=kx says more than “y increases when x increases”. It says doubling x doubles y, tripling x triples y, and scaling x by any allowed factor scales y by the same factor.
Worked Example 1: Find the Direct-Proportion Constant
y is directly proportional to x. When x=6, y=42. Find y when x=11.
Write y=kx. Using 42=6k gives k=7. Therefore y=7x. When x=11, y=77.
The constant k has meaning. If y is cost in dollars and x is mass in kilograms, k could be dollars per kilogram. A proportionality constant is often a rate hidden inside an equation.
The Origin Test on a Graph
A graph of y=kx is a straight line through the origin. Its gradient is k. A straight line with a nonzero intercept, such as y=5x+12, is linear but not direct proportion because y is not zero when x is zero.
This distinction matters in real contexts. A taxi model with a fixed booking fee plus a charge per kilometre can be linear without being directly proportional to distance.
Worked Example 2: Linear but Not Proportional
A service costs C=18+4n dollars for n sessions. Increasing n increases C, but C/n is not constant because the fixed $18 is shared differently as n changes.
At n=2, C=26 and C/n=13. At n=5, C=38 and C/n=7.6. Therefore the model is not direct proportion even though its graph is a straight line.
Use Tables to Test Direct Proportion
When a table is given, calculate y/x for several nonzero rows. If the ratio is constant, a direct-proportion model is supported by the table.
| x | y | y/x |
|---|---|---|
| 2 | 9 | 4.5 |
| 4 | 18 | 4.5 |
| 7 | 31.5 | 4.5 |
The table supports y=4.5x. Do not judge only from the fact that both columns increase.
Inverse Proportion Means a Constant Product
If y is inversely proportional to x, then xy is constant. We write y∝1/x and therefore y=k/x, or equivalently xy=k.
Doubling x halves y. Multiplying x by five divides y by five. The variables move in opposite directions, but that opposite movement alone is not enough: the product must remain constant.
Worked Example 3: Find the Inverse-Proportion Constant
y is inversely proportional to x. When x=5, y=12. Find y when x=8.
Write y=k/x. Using 12=k/5 gives k=60. Therefore y=60/x. When x=8, y=7.5.
Check the product: 5×12=60 and 8×7.5=60.
Worked Example 4: Test an Inverse-Proportion Table
| x | y | xy |
|---|---|---|
| 3 | 20 | 60 |
| 4 | 15 | 60 |
| 10 | 6 | 60 |
The constant product is 60, so y=60/x. If one row produced product 58, the exact inverse model would not fit all three rows as stated.
Direct and Inverse Models Look Different on Graphs
A direct-proportion graph y=kx is a straight line through the origin. An inverse-proportion graph y=k/x is curved and approaches the axes without reaching them when k≠0.
The graph shape is a consequence of the equation, not a picture to memorise separately. For inverse proportion, equal increases in x do not produce equal decreases in y.
Worked Example 5: Fixed Distance and Travel Time
A vehicle covers a fixed 240 km distance at constant average speed v km/h. Express travel time t in terms of v.
Since distance=speed×time, 240=vt. Hence t=240/v. Under this fixed-distance model, time is inversely proportional to speed.
At 80 km/h, t=3 h. At 96 km/h, t=2.5 h. The product vt remains 240.
The model assumes a constant average speed over the full trip and the same fixed distance. Real traffic, stops and route changes can break that simplified relationship.
Worked Example 6: Workers and Completion Time
Six equally productive workers complete an idealised task in 15 days. How long would 10 equally productive workers take, assuming total work is fixed and productivity is unchanged?
Worker-days are constant: 6×15=90. For 10 workers, time=90/10=9 days.
The inverse model depends on assumptions: workers contribute equally, can work independently without congestion, and adding workers does not create coordination losses. Those assumptions are part of the mathematics.
Worked Example 7: A Direct-Proportion Cost Model
A material costs $7.80 per kilogram with no fixed fee. If mass is m kilograms, cost C=7.8m. This is direct proportion.
For 3.5 kg, C=7.8×3.5=$27.30. For a budget of $62.40, m=62.40/7.8=8 kg.
If a delivery fee of $5 were added, the model would become C=7.8m+5 and would no longer be direct proportion.
Worked Example 8: Recipe Scaling
A recipe uses 420 g of flour for 6 portions. Assuming exact direct scaling, how much flour is needed for 15 portions?
Flour per portion=420/6=70 g. For 15 portions, flour=70×15=1050 g.
The mathematical model treats ingredient quantity as directly proportional to portion count. In real cooking, some ingredients or equipment constraints may not scale perfectly, but the school model states the relationship clearly.
Worked Example 9: Recognise a False Inverse Claim
Suppose x takes values 2,4,8 while y takes 18,10,6. y decreases as x increases, but products are 36,40,48. The product is not constant.
Therefore the data are not exactly inversely proportional. “One goes up while the other goes down” is only a visual trend, not the definition.
Worked Example 10: Find an Unknown From a Proportional Equation
P is directly proportional to q. When q=14, P=35. Find q when P=52.5.
P=kq and k=35/14=2.5. Therefore 52.5=2.5q, giving q=21.
Notice the reverse direction: direct proportion does not always mean “find y from x”. Once the constant is known, either variable can be recovered.
Worked Example 11: Inverse Proportion With Units
For a fixed rectangular area of 72 m², width w and length l satisfy lw=72. Find l when w=4.5 m.
l=72/w=72/4.5=16 m. The product has units m², matching the fixed area.
This model does not say every rectangle in the world has inverse side lengths. It says the inverse relationship appears because this particular family of rectangles has fixed area.
A Useful Model-Choice Test
- If doubling x doubles y, test direct proportion.
- If doubling x halves y, test inverse proportion.
- If y/x stays constant, direct proportion is supported.
- If xy stays constant, inverse proportion is supported.
- If neither is constant, do not force one of the two models.
- Check whether a fixed fee, threshold, changing rate or other condition breaks the simple model.
Common Errors
Calling every increasing relationship direct proportion: test y/x.
Calling every decreasing relationship inverse proportion: test xy.
Forgetting the origin: a direct-proportion graph must pass through (0,0).
Dropping units: the proportionality constant often carries a useful rate unit.
Using inverse proportion when the fixed condition changes: state what quantity is being held constant.
Independent Practice
1. y∝x and y=54 when x=9. Find y when x=14.
2. y∝1/x and y=18 when x=4. Find y when x=9.
3. Determine whether the table (x,y)=(2,7),(4,14),(6,21) shows direct proportion.
4. Determine whether (x,y)=(2,24),(3,16),(8,6) shows inverse proportion.
5. A product costs $4.60 per unit with no fixed fee. Write a formula for C in terms of n and find C for 35 units.
6. A 180 km journey is modelled at constant average speed. Find time at 72 km/h.
7. Eight equal-output machines complete a fixed job in 15 hours. Find the idealised time for 12 machines.
8. A rectangle has fixed area 96 cm². Find length when width is 7.5 cm.
9. Explain why y=3x+5 is not direct proportion.
10. A table has x=2,4,8 and y=30,15,7.5. State the model and constant.
11. A recipe uses 360 mL for 8 portions. Find the amount for 14 portions under direct scaling.
12. State one realistic reason a worker-time inverse model might fail when many more workers are added.
Explained Answers
1. k=54/9=6, so y=84.
2. k=72, so y=72/9=8.
3. Yes. y/x=3.5 throughout, so y=3.5x.
4. Yes. xy=48 throughout, so y=48/x.
5. C=4.6n; C=$161 when n=35.
6. t=180/72=2.5 h.
7. Machine-hours=120; time=120/12=10 h.
8. l=96/7.5=12.8 cm.
9. Its nonzero intercept means y/x is not constant and the graph does not pass through the origin.
10. Inverse proportion with constant 60.
11. 360/8×14=630 mL.
12. Coordination losses, limited workspace, unequal productivity or task dependencies can break the idealised constant-product assumption.
Continue the Secondary 3 Learning Route
Continue with Pythagoras, 3D Trigonometry, Elevation and Depression, Composite Solids, Surface Area, Volume and Unit Conversion, and Coordinate Geometry, Distance, Gradient, Line Equations and Intersections.
A proportional model is secure when the invariant—ratio or product—is identified, the constant has meaning, and the context still supports the model. Return to the Secondary Mathematics Hub.
