# Difficult proportional examples

**Example 1**

6 workers can build 4 bikes in 2 days. How many days would it take 8 workers to build 6 bikes?

Using the rules of inversely proportional it is easy to work out how long it would take 8 workers to build 4 bikes.

Because

`6wo\rkers\times2\ days=8wo\rkers\timesx\ days`

Simplifying we get

`x\ days=cancel(12)/cancel8=cancel6/cancel4=3/2`

`x\ days=1\1/2days`

So it takes `1\1/2` days for 8 workers to build 4 bikes.

To build 6 bikes would therefore take `1\1/2` this time or `(6/4)`

Therefore `1\1/2times1\1/2=3/2times3/2=9/4`

`=2\1/4\ days`

**Example 2**

Tracey completes a task in 2 hours. Brenda completes the same task in 3 hours how long would it take Tracy and Brenda to complete the task together?

To answer this try and find the work rate per hour for each person. This would then provide you with the work rate per hour together.

Tracy is `1/2` task per hour

Brenda is `1/3` task per hour

Together Tracy `+` Brenda per hour `=1/2` task `+1/3` task

`1/2+1/3=(1times3)/(2times3)+(1times2)/(3times2)=3/6+2/6`

`3/6+2/6=5/6`

So in one hour together they can complete `5/6`ths of the task.

Using our maths section on percentages we can find.

`1`hour`=5/6` task

`x=1` task done

Put the divide sign in

`1/x=(5/6)/1`

re-arranging to put `x` on its own

`x=(1times1)/(5/6)=1/(5/6)`

`x=6/5`

`x=1`hours and `1/5`of an hour

or `x=1`hour `12` minutes

**Example 3**

`x` is directly proportional to `y^2` when `x=4` and `y=6`

What is the value of `y` when `x=8`?

Using our knowledge on ratios

`x:y^2`

then `4:6^2`

and then using our knowledge on direct proportions

`4:6^2=8:y^2`

or `4/6^2=8/y^2`

rearranging the formula to get `y` on its own.

`y^2=(cancel8times6^2)/cancel4`

`y^2=2times6^2`

`y^2=2times36`

`y^2=72`

`y=sqrt72`