# You were able to purchase 5 gallons of gas for $19.85. How many gallons did you buy if you spent 847.64? ##### 1 Answer Nov 17, 2016 $213.51 \text{ gallons to 2 decimal places}$#### Explanation: $\textcolor{b l u e}{\text{Preamble:}}$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; $\textcolor{b r o w n}{\text{Initial condition}}$$\left(\text{gallons")/("cost in $}\right) \to \frac{5}{19.85}$
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$\textcolor{b r o w n}{\text{Solving for total fuel}}$

Let the unknown count of gallons be $x$

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

$\implies \left(\text{gallons")/("cost in$}\right) \to \frac{5}{19.85} \equiv \frac{x}{847.64}$Don't like decimals so lets get rid of them: Multiply both sides by $\frac{1}{100}$$\implies \left(\text{gallons")/("cost in $}\right) \to \frac{5}{1985} \equiv \frac{x}{84764}$
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$\textcolor{b l u e}{\text{Shortcut method}}$

Multiply both sides by 84764

$\text{ } \frac{5 \times 84764}{1985} = x \times \frac{84764}{84764}$

but $\frac{84764}{84764} = 1$

" "color(red)(x=213.51" gallons to 2 decimal places")
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$\textcolor{b l u e}{\text{Maintaining the ratio method}}$

$\implies \left(\text{gallons")/("cost in$}\right) \to \frac{5}{19.85} \equiv \frac{x}{847.64}$Change the denominator of 19.85 into 847.64 and apply the same process to the numerator of 5 $= \frac{5 \div 19.85}{19.85 \div 19.85} = \frac{5 \div 19.85}{1} = \frac{\textcolor{g r e e n}{5 \div 19.85 \times 847.64}}{1 \times 847.64}$$= \frac{213.511}{847.64}$" "color(red)(213.51" gallons to 2 decimal places") ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ $\textcolor{b l u e}{\text{Foot note}}$Notice that $\text{ } \textcolor{g r e e n}{5 \div 19.85 \times 847.64} = \frac{5 \times 84764}{1985}\$

Which is the same as in the shortcut.