Fractions are Hard!

3.9 Converting units

Today, March 6, 2016, the price of gasoline in my home town in Arizona is \(\$1.53\frac{9}{10}\) dollars per gallon. My father, living in Adelaide, Australia, tells me that the price of petrol there currently \(99.8\) cents per litre. (Australians call petroleum “petrol” and Americans call it “gas.”)

Who is currently paying the higher price for gas?


Note: I am quoting the Arizonan price in US dollars and cents, my father is quoting the Adelaidian price in Australian dollars and cents. The exchange rate today is $US 1 = $AUS 0.74. Also, one gallon is equivalent to 3.785 litres.


Before we tackle this challenge, let’s examine a more straightforward conversion problem.


Floogles and woogles each have uniform weights. If \(31\) floogles on a scale balance perfectly with \(42\) woogles, in terms of woogles, how much do \(700\) floogles weigh?


We have



Double the number of floogles balances with double the number of woogles: \(62 floogles \longleftrightarrow 84 woogles\).

Triple the number of floogles balances with triple the number of woogles: \(93 floogles \longleftrightarrow 126 woogles\).

Three-fifths of the number of floogles balances with three-fifths of the number of woogles: \(18.6 floogles \longleftrightarrow 25.2 woogles\).

And so on.

We can adjust the given relation by multiplying (and dividing) by any factors of our choosing.


We want to examine \(700\) floogles. Since we want to focus on floogles, let’s compute the weight of one floogle by dividing through by 31.


Now multiply through by 700.


That’s it! Seven hundred floogles have weight \(\dfrac{42}{31}\times 700 \approx 948.4\) woogles.


This example illustrates the basic principle behind converting between different units of measure: once you are told one pair of equivalent amounts, one can adjust those numbers by any multiplicative factor to one’s advantage.



We have

\(0.74\) $US \(\longleftrightarrow 1\) $AUS

\(1\) gallon \(\longleftrightarrow 3.785\) litres.


The price of gas in Arizona is \(1.539\) $US per 1 gallon and the price of gas in Adelaide is \(0.998\) $AUS per 1 litre.

Let’s convert the Australian quantities into American ones.


From the currency relation we see:

\(0.74\times 0.998\) $US \(\longleftrightarrow 1\times 0.998\) $AUS

\(0.739\) $US \(\longleftrightarrow 0.998\) $AUS


So Australians are paying \(0.739\) $US per \(1\) litre of gas.


From the volume relation we see:

\(\dfrac{1}{3.785}\) gallons \(\longleftrightarrow 1\) litre

\(0.262\) gallons \(\longleftrightarrow 1\) litres.


So Australians are paying \(0.739\) $US per \(0.262\) gallons of gas.


We want to know the number of US dollars Australians are paying for 1 gallon of gas. At present we have, for Australians:


\(0.739\) $US  \(\longleftrightarrow 0.262\) gallons.


Divide through by \(0.262\) to see:


\(2.821\) $US \(\longleftrightarrow 1\) gallon.


Australians are currently paying $1.28 (US) more per gallon of gas than Americans. That is about \(83\%\) more.


Question: Science classes often teach a quick method for converting units. This method looks like the Key Fraction Property at play. Is it? Why does the “quick” method from science class work?
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