Proportion describes how two quantities are linked. In direct proportion the quotient y ÷ x stays constant, so y = kx. In inverse proportion the product xy stays constant, so xy = k. A map scale 1 : n is proportion in disguise: it fixes a constant multiplier between map lengths and real lengths.
Map scales, rates and speed–time problems all run on proportion. Sec 2 school papers test whether you can recognise proportion from a table or graph before you calculate anything. On map questions, most marks are lost through unit conversion, not through the proportion idea itself.
What these notes cover
- •Direct and inverse proportion defined
- •Testing a table: quotient or product
- •Reading direct and inverse proportion graphs
- •Direct proportion: y = kx and the three-step routine
- •Constant-k method and unitary method
- •Inverse proportion: xy = k
- •Workers–days problems
- •Speed–time for a fixed distance
- •Map scales 1 : n for distances
- •Area scale 1 : n²
- •Finding the scale from given lengths
- •Common mistakes and exam technique
Key definitions
- Direct proportion
- Two quantities x and y are in direct proportion if y increases as x increases in such a way that the quotient y ÷ x stays the same. Doubling x doubles y, and tripling x triples y.
- Inverse proportion
- Two quantities x and y are in inverse proportion if y decreases as x increases in such a way that the product xy stays the same. Doubling x halves y, and tripling x cuts y to a third.
- Constant of proportionality
- The fixed number k, with k ≠ 0. In direct proportion k is the common quotient y ÷ x. In inverse proportion k is the common product xy.
- Map scale 1 : n
- 1 cm on the map represents n cm on the actual ground. Both sides of the ratio are in the same unit, so the scale carries no units.
- Asymptote
- A line that a curve gets closer and closer to without ever touching. For the inverse proportion graph y = k ÷ x, both axes are asymptotes.
Key formulas
Direct proportion
k is the constant of proportionality. To test a table, check that y ÷ x is the same for every pair. The graph of y against x is a straight line through the origin with gradient k.
Inverse proportion
The product xy is the same for every pair. To solve, multiply a complete pair to find k, then divide k by the new value. The graph is a hyperbola branch that never touches either axis.
Workers–days
number of workers × number of days = k
k is the total amount of work, measured in worker-days. This assumes all workers work at the same rate.
Speed–time for a fixed distance
v × t = distance (= k)
v is the speed and t is the time. When the distance is fixed, v and t are in inverse proportion, and the constant k is the distance.
Reading a map scale
Use 1 km = 100 000 cm. To go from map length to actual length, multiply by the km-per-cm number. To go from actual length to map length, divide by it.
Area scale
linear scale 1 : n ⇒ area scale 1 : n²
Square the whole length statement. For example, if 1 cm represents 0.5 km, then 1 cm² represents 0.25 km². Unit chains: 1 km² = 10¹⁰ cm² and 1 km² = 10⁶ m².
Worked example
A map of an island is drawn to a scale of 1 : 250 000. (a) The coastal highway is 8.4 cm long on the map. Calculate its actual length in kilometres. (b) A national park on the island has an actual area of 45 km². Calculate the area, in cm², representing the park on the map.
- 1First write the usable statement: 1 cm represents 250 000 cm = (250 000 ÷ 100 000) km = 2.5 km.
- 2(a) This is map to actual, so multiply: actual length = 8.4 × 2.5 km = 21 km.
- 3(b) Square the whole length statement: 1 cm² represents (2.5 km)² = 6.25 km².
- 4This is actual to map, so divide: map area = (45 ÷ 6.25) cm² = 7.2 cm².
- 5Sanity check: 7.2 cm² is a small patch that fits on paper. Multiplying instead would give 45 × 6.25 = 281.25 cm² for one park, which is clearly wrong.
Watch out: Do not write “1 cm² represents 2.5 km²”. You must square the number as well as the unit. If the area is given in m², convert it to km² first using 1 km² = 10⁶ m².
Common mistakes
Using 1 : n instead of 1 : n² for areas
Wrong: Scale 1 : 25 000, and a reservoir covers 8 cm² on the map. Actual area = 8 × 25 000 = 200 000 cm² = 20 m².
Right: The area scale is 1 : 25 000². Actual area = 8 × 25 000² cm² = 5 × 10⁹ cm² = 0.5 km². Alternatively, 1 cm represents 0.25 km, so 1 cm² represents 0.0625 km², and 8 × 0.0625 = 0.5 km².
Treating workers–days as direct proportion
Wrong: 8 workers take 15 days, so for 12 workers: 15 ÷ 8 = d ÷ 12, giving d = 22.5 days.
Right: More workers means fewer days, so the product is constant. Total work = 8 × 15 = 120 worker-days, so d = 120 ÷ 12 = 10 days.
Calling y = kx + c direct proportion because the graph is straight
Wrong: For x = 1, 2, 3 and y = 7, 9, 11: “The points lie on a straight line, so y is directly proportional to x.”
Right: Test the quotients: 7 ÷ 1 = 7, 9 ÷ 2 = 4.5, 11 ÷ 3 = 3.67 (3 s.f.). They are not constant, so y is not directly proportional to x. The line does not pass through the origin, since y = 5 when x = 0.
Questions students ask
How do you tell if a table shows direct or inverse proportion?
For direct proportion, compute y ÷ x for every pair. The quantities are in direct proportion if all the quotients are equal. For inverse proportion, compute xy for every pair, and all the products must be equal. One mismatched pair is enough to reject proportionality, but you must check every pair to confirm it.
What does a map scale of 1 : 50 000 mean?
It means 1 cm on the map represents 50 000 cm on the ground. Dividing by 100 000 turns this into 1 cm represents 0.5 km. Multiply to go from map length to actual length, and divide to go from actual length to map length.
How do you find area using a map scale?
Square the whole length statement. If 1 cm represents 0.5 km, then 1 cm² represents 0.25 km², not 0.5 km². In general, a linear scale of 1 : n gives an area scale of 1 : n².
Is a straight-line graph always direct proportion?
No. For direct proportion, the line must also pass through the origin. A line such as y = 2x + 3 is linear but is not direct proportion, because y = 3 when x = 0, not 0.
In the full PDF
- •Labelled graphs of direct and inverse proportion, including a correct sketch of xy = 12
- •Further worked examples: the paint-mixing trap, a delivery van speed–time table, a phone repair charge and a spring extension question
- •The constant-k and unitary methods compared on the same problem
- •The “additional workers” question and a three-part speed–time question with set-off clock times
- •Finding the scale 1 : n from given lengths, and handling areas given in m²
- •The remaining common mistakes, the full exam-technique checklist and a summary table