GCSE · Maths · AQA · Spec 8300 · Higher
Construct equations of direct/inverse proportion (Higher)
Double one quantity and the other might double, halve or quadruple. Every one of those relationships hides a number that never changes — find it, and you have the equation.
Find the number that won't move
Drag the point along the curve. x changes and y changes, but x × y stays at 24. Try x = 3, then x = 6: double x and y halves. (y is shown to 2 decimal places.)
Reading equations
Direct, inverse, or neither?
For each equation, decide what it says about how y depends on x. Commit first, then read why.
Still to sort
y is directly proportional to x (or a power or root of x) (0)
y = k × (x, x², x³, √x …), with nothing added on or taken away.
Where the line is: Dividing by a plain number is still direct: y = x/3 is y = ⅓x. It is dividing by the x-part itself (x, x², √x …) that makes it inverse.
y is inversely proportional to x (or a power or root of x) (0)
y = k ÷ (x, x², √x …), which is the same as y = k × 1/(x, x², √x …).
Where the line is: y must be a fixed number divided by the x-part. "y goes down as x goes up" is not enough on its own.
Not a proportion (0)
Something is added or subtracted, so y is not simply k times the x-part, or k divided by it.
Where the line is: A straight line only shows direct proportion if it passes through (0, 0).
An equation tells you exactly how y depends on x, if you know what to look for.
Maths · Algebra
Build the equation, line by line
The same four moves every time: write the form with k, put in the pair you know, solve for k, write the finished equation. Then use it.
"Inversely proportional to the square of x", so x² goes on the bottom.
Step 1 of 9
"Inversely proportional to the square of x", so x² goes on the bottom.
WHAT YOU'VE LEARNED
A quick recap of today's lesson.
Direct or inverse, power or root: every proportion equation is a fixed number times, or divided by, something in x.
What you need to know
- How to turn words like "inversely proportional to the square of x" into an equation with a constant k
- How one known pair of values pins down k, and how to use the finished equation
- Why "inversely proportional to x" means the same as "proportional to 1/x"
- How to read an equation and say whether it shows direct proportion, inverse proportion or neither
The big picture
Every proportion equation has the same skeleton: y equals a constant k times something in x (direct), or k divided by something in x (inverse). Being inversely proportional to x is the same as being proportional to 1/x, so both kinds are built the same way: write the form, use one known pair of values to find k, and the equation is done.
Key points
Worked example
Problem
The time, t hours, that a team takes to paint a fence is inversely proportional to the number of painters, n. 4 painters take 9 hours. (a) Find an equation connecting t and n. (b) How long would 6 painters take?
⚠ Watch out
Building the equation from only half the sentence. "y is inversely proportional to the cube of x" means y = k/x³: "inversely" puts the x-part on the bottom, and "cube" makes it x³. Miss either word and every line after it is built on the wrong equation.
Memory hook
Form, fill, find, finish: write the form with k, fill in the pair you know, find k, then finish by writing the equation with k's value in.
Check yourself
y is inversely proportional to √x, and y = 3 when x = 16. Find the equation, then find y when x = 9. (Answer: y = 12/√x, so y = 4.)
Flashcards
(9)Write 'y is directly proportional to x²' as an equation.
Write 'y is inversely proportional to x' as an equation.
'y is inversely proportional to x' means the same as…?
You know the form of a proportion equation. How do you find k?
In y = k/x, what stays the same as x and y change?
Is y = 3x + 2 an example of direct proportion?
y gets smaller as x gets bigger. Does that prove y is inversely proportional to x?
Is y = x/5 direct or inverse proportion?
y = kx². What happens to y when x is doubled?
Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.
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