Proportion
Recognise and solve problems involving direct proportion (as one quantity increases, the other increases at a constant rate) and inverse proportion (as one increases, the other decreases), including graphical and algebraic representations
What a learner can do afterwards
- Identify whether a real-world relationship is direct or inverse proportion and justify the choice
- Set up and solve a direct-proportion equation (e.g. if 4 pens cost £6, find the cost of 10)
- Sketch graphs showing direct proportion (straight line through origin) and inverse proportion (curve)
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Direct proportion means two quantities grow at exactly the same rate. Double one and the other doubles too, so their ratio never changes: one is always the same multiple of the other. You have met this on a graph already, because direct proportion is a straight line through the origin. A gym charging £8 per class is direct proportion: 2 classes cost £16, 5 classes cost £40, always £8 each. A car at a steady speed works the same way, since distance divided by time always gives the same speed.
Here is the find-one-first method. 3 notebooks cost £4.50, and you want 8 notebooks. One notebook costs £4.50 ÷ 3 = £1.50, so 8 notebooks cost £1.50 × 8 = £12. Because the price per notebook never changes, this works for any number you like, even 100.
Inverse proportion is the mirror image. One quantity goes up while the other goes down, and their product always stays the same fixed number. A job might need exactly 24 worker-days: more workers means fewer days, because workers times days must stay 24. On a graph, inverse proportion makes a curve that falls as you move right, never a straight line.
5 printers need 12 hours to finish a batch of flyers. How long would 10 printers take? First find the fixed amount of work: 5 × 12 = 60 printer-hours. Then share it over the new team: 60 ÷ 10 = 6 hours. Multiply first to find the total, then divide by the new amount.
To tell the two apart, ask what stays fixed. If dividing one quantity by the other gives the same number every time, it is direct proportion. If multiplying them together gives the same number every time, it is inverse proportion. Try it on the printers: 5 × 12 and 10 × 6 both give 60, so that pairing must be inverse.
In direct proportion two quantities grow at the same rate, their ratio stays fixed, and the graph is a straight line through the origin. In inverse proportion one goes up as the other goes down, their product stays fixed, and the graph is a curve.
Adapted from OpenStax Algebra and Trigonometry (CC-BY 4.0), openstax.org · license
More to read
4 Ways to Determine Whether Two Variables Are Directly Proportional
Two things are directly proportional when they both change by the same amount at the same time. If one number doubles, the other doubles too. If one triples, the other triples too.
You can write this as y = kx, where k is a fixed number called the constant of proportionality. It never changes, no matter what x and y are.
Here's a quick way to check: divide y by x. If you always get the same answer, no matter which pair of numbers you pick, then y and x are directly proportional.
On a graph, direct proportion always looks like a straight line that passes right through the point (0, 0). That start at zero matters. If the line does not start at the origin, the two amounts are not directly proportional, even if the graph is still a straight line.
Try it yourself: if 3 apples cost $1.50, does 6 apples cost $3.00? Divide cost by the number of apples each time. If you get $0.50 both times, you found the constant of proportionality, and yes, they are directly proportional.
How to Find the Constant of Proportionality (+ Practice Problems)
The constant of proportionality is just a number that tells you how two amounts are linked. If y is directly proportional to x, then y = kx, and k is that special number.
You can find k two ways.
From an equation: if you already have y = kx, the constant is just the number sitting in front of x. In y = 4x, the constant is 4.
From a table or graph: pick any matching pair of x and y values (not the point where both are zero) and divide y by x. You should get the same k every time, because that is what makes the relationship proportional in the first place.
On a graph, k is also the slope of the line, the steepness of that straight line through the origin.
Example: a car uses 2 liters of fuel for every 25 kilometers it drives. Divide 2 by 25 to get 0.08. That means the car uses 0.08 liters per kilometer, and that is the constant of proportionality for this trip.
7.5 Solve Applications with Rational Equations
This lesson is about solving proportions, which are just two equal fractions, like 3/4 = 6/8. To solve a proportion for a missing number, cross-multiply: multiply across the equals sign, then divide to find the unknown.
The page also compares direct variation and inverse variation. In direct variation, one amount grows as the other grows, following y = kx. Think of hours worked and total pay at a fixed hourly rate.
In inverse variation, one amount grows while the other shrinks, following y = k/x. Think of speed and travel time for a fixed distance: drive faster, and the trip takes less time.
Similar figures use proportions too. If two shapes are similar, their matching sides are always in the same ratio, so you can set up a proportion to find a missing side length.
Linear equations and functions: FAQ
A proportional relationship is when two amounts always keep the same ratio to each other. If y is directly proportional to x, doubling x doubles y. Tripling x triples y.
Every proportional relationship can be written as an equation, y = kx, where k stays the same no matter what x you plug in.
On a graph, proportional relationships always show up as a straight line running through the point (0, 0). If the line is straight but skips over the origin, the relationship is not proportional, it is just linear.
A handy check: pick two different points from a table. Divide y by x for each one. If both answers match, you have found a proportional relationship, and that matching number is your constant of proportionality.
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Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
18 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.