#color(purple)("I have not shown how to deal with the units of measurement"#
#color(purple)("as it may distract you from the numerical solution ")#
Let the unknown count of gallons be #x#
Let total cost be #t = $19.25#
There are at least two approaches to solving this.
#color(blue)("Method 1 - Direct calculation")#
#color(brown)("initial condition")#
1 gallon cost $3.50
Total cost = total count of gallons x cost for each gallon
#color(brown)("Determine the value of "x)#
#t=x xx $3.50#
but #t=$19.25# so we have:
#$19.25 = x xx$3.50#
Divide both sides by $3.50 giving:
#(19.25)/(3.50) = x#
#x = (cancel(19.25)^5.5)/(cancel(3.5)^1) = 5.5/1 = 5.5" gallons "-> 5 1/2 "gallons"#
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Method 2 - Ratio method")#
Note that the ratio method is the same as the direct calculation method. It just looks different.
method:
step 1#->#determine the gallon count for $1
step 2#->#multiply up the gallons for $1 to gallons for $19.25
Using fractional form of ratio
#color(brown)("Initial condition")#
#("quantity")/("cost") -> 1/($3.50) -= x/($19.25)#
For multiply or divide in ratios what yo do to the bottom you do to the top to maintain the correct proportions.
Conversions done in stages so you can follow what is going on.
#color(brown)("Step 1 - quantity for $1")#
#("quantity")/("cost") -> 1/($3.50) -= (1-:3.50)/($3.50-:3.50) = color(green)((1-:3.50)/1)#
#color(brown)("Step 2 -quantity for $19.25")#
#("quantity")/("cost") ->color(green)( (1-:3.5)/1) -=(color(green)(1-:3.50)xx19.25)/(color(green)(" 1 ")xx19.25) = (5.5" gallons")/19.25#
.............................................................................................
Done in 1 stage this would look like:
#(1xx19.25/3.5)/(3.5xx19.25/3.5) = 5.5/19.25#
