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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.)

1/2= 2/4

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.

($1USD)/(euro exchange rate) = (USD value)/(euro value)

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:

1/0.794921 = x/324

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.)

0.794921/1 = 324/x

Or even:

1/x = 0.794921/324

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.

You’ve booked that trip to ParisVive les vacances! Now that your credit card has borne the brunt of your plane tickets and hotel reservations, with just enough space for a couple of fantastic meals, it’s time to turn to the cash. How much should you bring — and even more importantly, how far will it go?

When traveling out of country, you need to consider the currency exchange rate. Only very rarely is this exchange equal. (In other words, one Euro almost never equals one U.S. dollar.) That means, you’ll need to use a conversion to find out how far your cash will actually go.

There are actually three things to talk about here: using an online conversion calculator, doing the conversions by hand and checking your answer to see if it’s reasonable. Remember, math is infinitely flexible, so there’s no reason you have to do this in one particular way. Next Wednesday, we’ll look at doing conversions with paper and pencil. Today, it’s all about online calculators and checking your answer.

First, the conversion calculators. Go ahead and use them! If nothing else, a reliable online calculator will give you the most up-to-date conversion rate with the click of a button. For example, using the XE currency conversion calculator, I found that $1USD is equal to 0.794921€ (as of Monday, July 2, 2:05 p.m.).  This means that one U.S. dollar is worth a little more than 75 percent of a Euro.

If you know the exchange rate, it’s really easy to exchange values of 10, 100 or 1000. In these cases, you can simply move the decimal point.

$10USD = 7.94921€

$100USD = 79.4921€

$1000USD = 794.921€

Notice that when there is one zero (as in 10), you move the decimal point one place to the right. When there are two zeros (as in 100), you move the decimal point two places to the right. And when there are three zeros (as in 1000), you move the decimal point three places to the right.

Of course, if you want to convert $237.50USD to Euros, that trick won’t work. In that case, you can plug $237.50 into the online calculator. If you have $237.50USD in your pocket, that’s 188.717€.

XE also has iPhone and Droid apps, so you can take the online calculator on the road with you. (Note: I don’t have any relationship with XE. It just looks like a good, reliable online currency calculator. Want to recommend something different? Feel free to respond in the comments section.)

The thing about online calculators is that they’re only as good as the information that you put in. If you think you’re converting $USD to €, but you’re actually doing it the other way around, well, your fancy pants calculator is not going to spit out the answer you were looking for. You have to know how to assess whether your answer is correct.

I’m the first to admit that I get this very confused. I have to stop and think really hard to be sure that I’ve done the conversions correctly. (And to be honest, this is one of the reasons I prefer to do it by hand.) But there are some simple rules you can consider that will help:

  • If the conversion rate is less than 1, the conversion will be less than the original amount.
  • If the conversion rate is greater than 1, the conversion will be greater than the original amount.

Let’s say that $1USD equals $1.26SGD (Singapore dollar). If you convert $USD to $SGD, will your answer be greater or less than the original amount? If you said greater — you’re right! But if you convert $SGD to $USD, the answer will be less than the original amount. Make sense?

The good news is that you can figure this out before you leave. Write it down or keep a note on your phone. Then you will always be able to check to see if your answer makes sense. Because the worst thing is to come home from a relaxing vacation to find that you’ve spent way too much.

Be sure to come back next Wednesday to get the deets on how to do these conversions by hand. It really isn’t that difficult — and the process is applicable in so many other situations, so it’s worth learning.

Where are you traveling this summer? Share your plans in the comments section below!