On a map of a state, the scale shows that 1 cm is approximately 5 km. If the distance between two cities is 1.7 cm on the map, how many kilometers separate them?

The most strait forward way to visualise what is happening is to use ratios. The methods adopted by others will be based on this being the relationship between the values.

Adopting the fractional format of ratios.

As we require the actual distance as the answer we put that as the top value (numerator).

`("actual distance")/("measurement on the map")-> (5" km")/(1" cm")`

Let the unknown distance be `x` giving:

`" "color(green)(5/1=x/1.7)`

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`color(blue)("Method 1")`
This will be used by most people.

Multiply both sides by `color(red)(1.7)`
`color(white)()`

`" "color(green)((5color(red)(xx1.7))/1=x xx (color(red)(1.7))/1.7)`

Note that `1.7/1.7=1` giving:

`" "color(green)((5xx1.7)/1=x)`

`" "color(green)(x=8.5)`
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`color(blue)("Method 2")`

This is the same as method 1 but looks different

`" " color(green) (5/1" "=" "x/1.7)`

`" "color(green)( [color(white)(2/2)5/1color(red)(xx1)" "]" "=" "x/1.7`

`" "color(green)( [color(white)(2/2)5/1color(red)(xx1.7/1.7)" "]" "=" "x/1.7`

`color(green)(" "8.5/1.7" "=" "x/1.7)`

`x=8.5`