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In Math for Grownups

How Far? Estimating metric distances

Earlier this month, I showed you how to convert currencies, when given the exchange rate. When you’re not using an online calculator, that process involves proportions, which are pretty simple to use, but do require a little figuring on paper. This same process works for any conversions, including miles to kilometers, liters to ounces, etc.

But while being exact with your money is pretty important, estimating how far you have to drive or walk is usually good enough. So instead of going into details about metric-to-traditional measurement conversions, let’s look at how you can find these distances with a little mental math. First, you’ll need to know a few facts:

1. A mile is longer than a kilometer. So, when you convert miles to kilometers, the answer  will be larger than the original amount. (mi –> km = larger answer)

2. A kilometer is shorter than a mile. So, when you convert kilometers to miles, the answer will be smaller than the original amount (km –> mi = smaller answer)

2. In fact, 1 mile equals 1.61 kilometers. And 1 kilometer equals 0.625 mile.

3. Those values are pretty darned close to 1.5 kilometers and 0.5 mile.

Remember, we’re estimating here, so you’re not looking for an exact answer. Forget what your middle school math teacher said about the precision of math. You don’t always need to getan exact answer. But there’s another fact you’ll need to consider:

4. The larger the value that you’re converting, the less precise your answer will be.

If you depend on the estimate 1 mi = 1.5 km and you’re converting 15 mi to km, your answer will be pretty close. BUT if you’re converting 1,468 mi to km, your estimate will be a lot lower than the actual answer.

Look, estimating is no big deal. In fact it’s a really, really powerful tool that can make your life much easier. You do need to know when estimation is in your best interests and when you should pull out the calculator. (See? Math really isn’t all that black and white!)

Let’s look at an example. Zoe has finally made it to London! She’s spending the summer studying Shakespeare and working part-time as a docent at the Tate Modern. And she’ll have some time to roam around Europe a bit. She’s rented a car so that she can chart her own path, and next Friday afternoon, she’s going to cross the channel to France, where she hopes to spend four days winding her way down to Paris and back.

But how long will it take her to get there? According to her map, the distance is 454 km. Since Zoe is used to miles, she’d like to convert the distance so that it makes more sense to her. She’s okay with a rough estimate, especially since she has no firm schedule. So she decides that knowing there are about 1.5 km in a mile is good enough.

To make the math even easier, she decides to round the distance as well: 450 is pretty close to 454. Now she can easily do the math in her head, but we’ll get to that in a minute. Let’s write it out first.

Because she’s converting kilometers (shorter) to miles (longer), her answer will be smaller than the original amount. That means she’ll need to divide.

450 km ÷ 1.5 = 300 mi

So she’ll travel about 300 miles to get from London to Paris — not a huge distance!

But how could she do this in her head? For that, she’ll need to remember a few things about fractions.

1.5 = 3/2

450 ÷ 1.5 = 450 ÷ 3/2

450 ÷ 3/2 = 450 • 2/3

(That’s because when you divide by a fraction, it’s the same thing as multiplying by its reciprocal — or the same fraction upside down.)

So in order to convert kilometers to miles in her head, she’ll need to multiply the value by 2 and then divide by 3 (which is the same as multiplying the value by 2/3. In other words:

450 • 2/3 = (450 • 2) ÷ 3 = 900 ÷ 3 = 300

Whew!

But once Zoe remembers this little trick, she can estimate these conversions quickly and easily.

30 km = ? mi

30 km • 2 = 60

60 ÷ 3 = 20

30 km = 20 mi (approximately)

Make sense? Try it for yourself: convert 75 km to mi and then use an online calculator to check your answer. Remember, if you’re using the process above, you’ll get an estimate, not an exact value!

So take a guess: If you’re converting mi to km, what process would you use? See if you can figure it out and then offer your explanation in the comments section. Feel free to choose a value to convert, if it’s easier to explain that way.

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In Basic Math Review

The Metric System: What’s the big deal with bases?

I’m vacationing this week in sunny Radford, Virginia,and ike most parts of the United States, the metric system is not used here (to mark distances, anyway). But if you cross the border into another part of the world, there’s little doubt that you’ll be measuring kilometers rather than miles and grams instead of pounds. That’s because most of the world has embraced the metric system. (In fact, only two other countries — Burma and Liberia — have resisted the change along with the U.S.)

Ask any scientist or mathematician: the metric system is infinitely more intuitive and much, much easier to remember and understand. But why? The answer is simple: Base 10. What this means is even simpler: in base 10 the foundational number is 10. Take a look:

10 • 1 = 10

10 • 10 = 100

10 • 100 = 1,000

and so on…

Each time you add a digit in our number system, you are effectively multiplying by 10. That means that 99 is the last two-digit number in base ten, and 999 is the last three-digit number. In fact our entire decimal system is base ten. (But it wasn’t always like that.)

But here’s the thing — you don’t care (and you shouldn’t really care). We are so used to base 10 that we don’t even think about it any more. It’s like knowing how to ride a bicycle or drive a car; once you learn it, you don’t even give it a second thought, but if you’re asked about it, it’s hard (or impossible) to explain.

When you were in school, you probably were asked to convert numbers into different base systems — and this was probably pretty darned confusing. We’re not going to do that here for one simple reason: You don’t need to know how to do this. BUT it is important to know that different base systems are useful in a variety of situations and professions. For example, computers function in base 2 (or binary), which is simply a system of zeros and ones. Computer graphics depend on a hexadecimal system or base 16 (0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F). Crazy, huh? Even less technical situations depend on a variety of bases — dozens and grosses are base 12 (one dozen is 1 • 12 and one gross is 12 • 12).

Compared to these other bases, base 10 is pretty darned easy, right? And that’s why so many mathy folks don’t understand why our country hasn’t embraced the metric system. Yep, unlike traditional measure systems, the metric system is base ten. Let’s compare:

Traditional system: 12 inches = 1 foot

Metric system: 100 centimeters = 1 meter

Traditional system: 5,280 feet = 1 mile

Metric system: 1,000 meters = 1 kilometers

Just a glance at these conversions and even the most math-phobic person would probably agree: the metric system is much easier to maneuver.

But agreeing that the metric system is easier doesn’t help you with conversions when you’re traveling, does it? On Wednesday, we’ll take a look at those conversions. I’ll show you some really easy ways to estimate the conversions. Because who wants to do math on vacation?

What other bases can you think of? How do you use them in your everyday life? Share your ideas in the comments section.

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In Math for Grownups

Keeping Current: Using proportions to convert currency

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.

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In Math for Grownups

Keeping Current: Converting currency right

You’ve booked that trip to Paris. Vive 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!

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