Last Friday, we looked at exchanging currency — how far will your money go in another country? In that post, I introduced you to online currency conversion calculators and helped you assess whether or not your answer made sense. Today, we’re going to look at doing these conversions by hand.

Out of every basic math skill I know and have taught, proportions are the most useful — and most often forgotten. You can use them to shrink photos proportionally (so that the Eiffel Tower doesn’t look squat and fat or that mime doesn’t resemble a human hericot vert), alter a recipe to feed an army or find unit price. With proportions, you don’t need to remember whether to multiply or divide. Get the numbers in the right place, cross multiply, solve for x, and you’re good to go.

But let’s back up for a second. What is a proportion? It’s simple, really. A proportion is merely two equivalent ratios. (Remember, a ratio is a way to compare two numbers, often written as a fraction.)

[pmath size=12]1/2[/pmath]= [pmath size=12]2/4[/pmath]

The two fractions (ratios) in the above statement are equivalent: 1 out of 2 is the same thing as 2 out of 4. But that’s just an example. The key to setting up currency exchange proportions is knowing where each part goes.

There are four parts: the original currency (\$1USD, for example), the currency exchange rate (the value of \$1USD in the other currency), the value you are converting, and the value after the conversion (the answer or x). You want to be sure that all of your parts are in the right place.

But there is more than one right place! So, I suggest being consistent with these parts. That way, you can always, always use the same proportion for each conversion that you do.

[pmath size=12](\$1USD)/(euro exchange rate)[/pmath] = [pmath size=12](USD value)/(euro value)[/pmath]

That looks a little clunky, but it’s not really difficult to dissect. Look at it carefully, and you’ll notice a few things:

1. The \$USD amounts are in the numerators of the ratios.
2. The € amounts are in the denominators of the ratios.
3. The conversion exchange (\$1USD to €) is in the first ratio, while the actual values are in the second ratio.

To use this proportion, you need three of the four values found in this proportion. What do you think they will be? One of them will always be 1, because it’s the base value of the currency exchange. If you’re converting \$USD to €, you’ll use \$1USD. If you’re converting € to \$USD, you’ll use 1€. The second known value will be the currency rate. Last Friday, we used \$1USD = 0.794921€, so let’s stick with that, making the second value 0.794921. The third value will always be the value you’re converting.

Let’s look at an example. You spy a gorgeous pair of boots in Paris for only 324€. You have \$500USD budgeted for a special splurge. Are these special boots within your budget? Plug things into the proportion to see:

[pmath size=12]1/0.794921[/pmath] = [pmath size=12]x/324[/pmath]

Before you let your nerves get the best of you, look at this proportion carefully. Which values have gone where? Now, do you think there is another way to set up this proportion? (Psst… the answer is yes.)

[pmath size=12]0.794921/1[/pmath] = [pmath size=12]324/x[/pmath]

Or even:

[pmath size=12]1/x[/pmath] = [pmath size=12]0.794921/324[/pmath]

Notice that while the numbers themselves have changed places, their relative positions have not. The \$USD values (1 and x) are still related (either in the same ratio or in the numerator or denominator), and the € values (0.794921 and 324) are still related (either in the same ratio of in the numerator or denominator).

But how do you solve this proportion? (In other words, “Holy crap! There’s an x in there, and it freaks me out!”) Take a deep breath and cross multiply. Choose one of the proportions above (I’m going with the first one), and picture a giant X on top of it. One segment of the X lies on top of the numerator of the first ratio and the denominator of the second ratio (the 1 and the 324). The other segment of the X lies on top of the denominator of the first ratio and the numerator of the second ratio (the 0.794921 and the x). Multiply the connected values, like this:

1 • 324 = x • 0.794921

Now you can simplify and solve for x.

324 = 0.794921x

Divide each side of the equation by 0.794921 (in order to get the x by itself).

324 ÷ 0.794921 = x

407.587672 = x

You’ve just discovered that 324€ is equal to \$407.59USD. That’s within your budget, so you’re good to go!

Now, try the other conversions to show that they work, too. See? Flexibility in math! (Who knew?)

What did you think of this process? Scary? Easy? Too hard? Stupid, because you can always use a calculator? Do you have another way to convert currency (besides proportions and using a calculator)? Share your ideas in the comments section.