Question

How long would it take to bicycle 45 miles at an average speed of 15 miles per hour?

Answer 1

`3` hours

Explanation:

Bicycle will travel `15` miles in `1` hour
or
`1` mile in `1/15` hour
or
`45` miles in`(1/15)(45)` hours
or
`45/15` hours
or
`3` hours

Answer 2

So it takes 3 hours to bicycle 45 miles

Explanation:

This is a ratio type of problem.

The initial ratio is 15 miles per hour, written as:

`" "("15 miles")/("1 hour")`

But we need the time for 45 miles. So by the rules of ratio we have

`" "("15 miles")/("1 hour")-=("45 miles")/("unknown time")`

Instead of writing "unknown time" lets just write `t`

So every time you see `t` think "unknown time"

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Lets turn the whole thing upside down. It works better that way

Instead of: `" "("15 miles")/("1 hour")-=("45 miles")/t`

Write: `" "(1 hour)/("15 miles")-=t/("45 miles")`

For now, forget about the units of measurement and just write:

`" "color(brown)(1/15=t/45)`

So we need to change the 15 into 45 to get the right hand side

Note that `15xx3=45`

Multiply by 1 but where `color(blue)(1=3/3)` giving

`" "color(brown)((1/15color(blue)(xx3/3))=t/45)`

`" "3/45=t/45`

So "unknown time" `=t=3" hours"`