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

#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.
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Using ratio but in fractional format;

#color(brown)("Initial condition")#

#("gallons")/("cost in $") -> 5/19.85#
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#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#
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#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")#
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#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")#
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#color(blue)("Foot note")#

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

Which is the same as in the shortcut.