Construct Inverse Proportion Equations

If the relationship is an inverse proportion, then write the proportionality equation as:

`y prop frac(1)(x)`

Then change `prop` to `=` and multiply one side of the equation by `k`:

`y prop frac(k)(x)`

Remember that the `x` or the `y` might be a value such as `x^2` or `sgrt(x)` or other power term.

Example 1

An experiment showed that the number of bacteria in a dish was inversely proportional to the square of the temperature.

One dish at 24℃ had 18,000 bacteria. How many bacteria would you expect there to be in a dish at 20℃?

Proportion is	`n`	`prop`	`frac(1)(t^2)`
Create equation with `k`	`n`	`=`	`frac(k)(t^2)`
Substitute	`18000`	`=`	`frac(k)(24^2)`
Solve	`k`	`=`	`10368000`

	`n`	`=`	`frac(10368000)(t^2)`
When temp = 20	`n`	`=`	`frac(10368000)(20^2)`
	`n`	`=`	`25920`

Answer:

Example 2

An experiment showed that a temperature, `t`, was inversely proportional to the square root of a height, `h`. When the height was 400m, the temperature was 20℃.

What was the temperature at 1000m? give your answer to 1 decimal place.

Proportion is	`t`	`prop`	`frac(1)(sqrt(h))`
Create equation with `k`	`t`	`=`	`frac(k)(sqrt(h))`
Substitute	`20`	`=`	`frac(k)(sqrt(400))`
Solve	`k`	`=`	`400`

	`t`	`=`	`frac(400)(sqrt(h))`
When height = 1000	`t`	`=`	`frac(400)(sqrt(1000))`
	`t`	`=`	`12.649`
	`t`	`=`	`12.6 (1dp)`

Answer: 12.6℃