y=kx · Direct & Inverse Proportion
Multiples pass through in direct proportion; the product stays fixed in inverse.
A tap pours 2L per minute. Run it 3 times as long — how many times the water?
Distinguish increasing together from direct proportionality.
The pact that turns 1 cm on a map into 500 m of ground.
Open the immersive version →A reading path containing this lesson: Quantities that change together · 3/3
Why does this question matter?
'Buy double, pay double'; 'go twice as fast, take half the time' — most everyday arithmetic stands on these two sentences. We needed an eye for which relationships obey them and which don't.
The criteria are clean: multiples pass through → direct (y=kx); the product stays fixed → inverse (y=k/x). Master just these two and estimation and unit conversion become instant.
Graphs expose their identity — direct is a straight line through the origin; inverse is a curve hugging the axes (a hyperbola). Reading the shape of a relationship is the last step before functions.
Pause and try it yourself
Make a prediction first. Change one input at a time and compare what happens with what you expected.
Predict first — a tap pours 2L per minute. Run it 3 times as long — the water?
Change the time
2L per minute. Watch the water as the time doubles and triples.
2L
water collected
| time | 1 |
| water collected | 2 |
| water ÷ time | 2 |
Read the idea from another angle
The heart of direct proportion is passing through the origin.
Zero minutes, zero liters — do nothing, get nothing. A taxi fare grows with distance but isn't proportional (at 0km you still owe the base fare). Origin-crossing is direct proportion's ID card.
Inverse proportion is the mathematics of sharing.
Wherever a fixed whole (60L, one cake, one job) gets divided, inverse proportion appears. Twice the sharers, half the share — because the product (the whole) is constant.
Now read the definitions with their conditions
The two relationships from the bathtub experiment, formalized.
Direct proportion
Double or triple x and y follows exactly. k is the 'per one' rate (2L per minute) — the graph is a line through the origin.
Inverse proportion
Double x and y halves — the product stays k. For the 60L tub, flow × time = 60. The graph is a curve approaching the axes.
The test
y÷x always equal? Direct. x×y always equal? Inverse. Both testable straight from a table.
Make one more distinction
If both grow together, it's direct proportion.
Age and height grow together, but double the age isn't double the height. The test isn't 'grow together' but 'multiples pass through' — double one and the other doubles exactly. A far stricter condition.
Inverse proportion just means 'one grows, the other shrinks'.
Shrinking isn't enough. The test is a constant product (xy=k) — double one and the other must exactly halve. For a 60L tub, (flow) × (time) is always 60.
Where else can this idea help?
Shopping and unit price
3 for $3.60, so 5 for…? Price is proportional to count ($1.20 each) — the unit-price lab's arithmetic is direct proportion throughout.
Speed and time
Same distance, twice the speed, half the time — with distance fixed, speed and time are inversely proportional (speed × time = distance).
Recipes
A 2-serving recipe scaled to 5 — every ingredient × 2.5. Amounts are proportional to servings, so one multiplier finishes the job.
Gears
Meshed gears spin inversely to their tooth counts — half the teeth, twice the turns. The mathematics of bicycle gearing.
Explain it in your own words
Try another question
Q14 tangerines cost $2.00. How much for 10?Show answer ▾
$0.50 each (the constant k), so 10 cost $5.00. In direct proportion, find 'per one' first and any amount computes instantly.
Q2A job takes 6 people 4 hours. How long for 8 people (same pace each)?Show answer ▾
people × hours = the job = 24, constant (inverse proportion). Eight people: 24÷8 = 3 hours.
Q3Taxi fares grow with distance. Direct proportion?Show answer ▾
No — the base fare means you owe money even at 0km, so the graph misses the origin (y=0.8x+4.8). 'Grows together' isn't 'proportional': double the distance isn't double the fare.
Try answering yourself before revealing it — getting it wrong is where learning starts.
What question comes next?
A reading path containing this lesson: Quantities that change together · 3/3
Previous concept
Patterns & Correspondence
Among correspondences like □×4=△, the ones that pass multiples through are proportions.
← Patterns & Correspondence labLeads to next
Ratios
The constant k's true identity is a rate — 'per one'. Master the language of ratios and proportion completes.
Go to the Ratios lab →Related lab
Related lab
Linear Functions
Add an intercept to y=kx and you get y=ax+b — direct proportion is the special case.
Go to the Linear Functions lab →