Proportion Equations

Proportion Equations

GCSE(F), GCSE(H),

Where two terms are in direct proportion to one another, it is written as

`y prop x`

where `y` and `x` are general terms; for example, `x` could be `x^3`. This direct proportion can be re-written as:

`y = kx`

where `k` is a number called the constant of proportionality. Given two values for `x, y text( and ) k`, the third value can be derived.

Similarly, for terms that are in inverse proportion:

`y prop frac(1)(x)`

can be re-written as

`y = k xx frac(1)(x), text( or ) y = frac(k)(x)`

Examples

1. `y` is proportional to `x`. When `y = 30, x = 10`. What is the constant of proportionality?

Answer: 3

`y prop x`

Re-write as `y = kx`

Substituting for `y text( and ) x`

`30 = k xx 10` and solve for `k = 3`

2. `y` is inversely proportional to `x^2`. When `y = 3, x = 5`. What is the value of `x` when `y = 5`? give your answer to 1 decimal place.

Answer: 3.9

`y prop frac(1)(x^2)`

Therefore `y = k frac(1)(x^2)`

Substituting: `3 = frac(k)(5^2)` and solve for `k = 75`

When `y = 5`, and using `k = 75`, `5 = frac(75)(x^2)`

Rewrite as `x^2 = frac(75)(5)`

`x^2 = 15` and solve for `x = 3.873`