Question

You were able to purchase 5 gallons of gas for $19.85. How many gallons did you buy if you spent 847.64?

Answer 1

`213.51" gallons to 2 decimal places"`

Explanation:

`color(blue)("Preamble:")`

I will show you both the shortcut method and by ratio adjustment method

So much can be explained by ratios so it is a good idea to really understand them.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Using ratio but in fractional format;

`color(brown)("Initial condition")`

`("gallons")/("cost in $") -> 5/19.85`
..............................................................................................

`color(brown)("Solving for total fuel")`

Let the unknown count of gallons be `x`

Much better to put the unknown at the top of a ratio (numerator)

`=>("gallons")/("cost in $") -> 5/19.85 -= x/847.64`

Don't like decimals so lets get rid of them:

Multiply both sides by `1/100`

`=>("gallons")/("cost in $") -> 5/1985 -= x/84764`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Shortcut method")`

Multiply both sides by 84764

`" " (5xx84764)/1985 = x xx84764/84764`

but `84764/84764=1`

`" "color(red)(x=213.51" gallons to 2 decimal places")`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Maintaining the ratio method")`

`=>("gallons")/("cost in $") -> 5/19.85 -= x/847.64`

Change the denominator of 19.85 into 847.64 and apply the same process to the numerator of 5

`=(5-:19.85)/(19.85-:19.85) =(5-:19.85)/1 = (color(green)(5-:19.85xx847.64))/(1xx847.64)`

`= 213.511/847.64`

`" "color(red)(213.51" gallons to 2 decimal places")`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Foot note")`

Notice that `" "color(green)(5-:19.85xx847.64) =(5xx84764)/1985`

Which is the same as in the shortcut.