Unit Conversion

UNIT CONVERSION (c) Dept. of Physics, University of Guelph

The conversion of numbers from one system of units to another often puzzles students, but if it is treated as just another problem in arithmetic using arithmetic's rules the problems disappear. For example: Knowing that there are 2.54 cm in 1.0 inch, how many cm are there in 15 inches? Of course the answer is simple

15 in = 15.0 X 2.54 = 38.1 cm

But what actually was being done here? The complete solution is as follows:

Notice that the units were treated just like arithmetic quantities and the "in" were canceled. Let's do one that is not quite as obvious! How many (mm)2 are there in 4.0 (in)2? The solution is:

Notice that the quantities 2.54 cm/in and 10 mm/cm were used but, because the units had to be squared, then the numbers that accompanied them had also to be squared.

Another example: Convert 30 mi/hr into m/s. This is a common conversion. It is necessary to know that there are 1.6 km in 1 mile and 3600 s in 1 hr (60 X 60).

Notice that the only tricky part here is the time conversion 3600 s/hr which is the wrong way up for our conversion but of course it is equally true that there are (1/3600) hr/s. So long as the relevant relations between the quantities are assembled in advance, then any conversion can be performed using these strict arithmetic rules.

Try this more complicated conversion: An old (and ridiculous) unit of thermal conductivity sometimes still encountered in building materials is Btu.hr-1.in.F-1.ft-2 where 1 Btu = 1054.8 J 1 in = 2.54 cm 1 ft2 = 0.0929 m2 9F=5C What is 1 (Btu.hr-1.in.F-1.ft-2) in proper units (W.m-1.C-1)?

Go to answer: The solution is:

Just take the quantities one at a time: 1. Convert Btu to J and cancel Btu 2. Convert in to cm and then m canceling in 3. Convert ft2 to m2 and cancel ft2 4. Convert F to C and cancel F 5. Convert hr to s and cancel hr If you’re in a car moving at 60 miles an hour, is that faster or slower than 60 feet per second? If you want to convert 60 miles to kilometers, the conversion factor is 1.61, but do you multiply or divide? Everybody finds these questions confusing, not just students. Yet if you use a little algebra, they are easy to answer. You can do any conversion, quickly and reliably. This article shows you how. This page is about converting units yourself. If you’ve just come here to do a particular conversion and you don’t care about the method, I recommend Google Calculator, which lets you type an expression using the “in” operator and get the conversion. Here’s a simple form:

The Big Secret The method for converting units comes right from one simple principle:

Numbers with units, like 16.2 meters or 32 ft/sec², are treated exactly the same as coefficients with variables, like 16.2x or 32y/z². Once you grasp this, you see at once why the laws of units work as they do. You can’t add 32 ft to 32 ft/sec, any more than you can add 32x to 32x/y. And when you divide 32 miles by 4 hours to get 8 miles/hour, that’s exactly the same as dividing 32x by 4y to get 8x/y.

Multiply by 1 To convert units, there’s only one other thing you need to bear in mind:

You can multiply anything by 1 and not change its value. Obvious, right? “So why mention it?” I hear you ask. Because multiplying by 1 — a carefully chosen form of 1 — is the key to converting units. Let me illustrate with an example. I’ll deliberately pick an easy one, one you could probably do in your head, so that I can show the steps excruciatingly clearly. Suppose you want to convert four and a half hours to minutes. Of course 60 minutes you know that = 1 hour Now divide both sides by 1 hour. (Remember you can do this because 60 minutes you treat the unit “hour” just like a variable. If you had 60x = 1y, you could certainly divide both sides by 1y.) After dividing, you have ---------= 1 1 hour Why did I do that? Because if (60 min)/(1 hr) = 1, then I can multiply any measurement by that fraction and not change its value. (I’ll explain a little later why I divided through by 1 hr and not 60 min.) So go back to the 4½ hours that we wanted to convert to minutes. To do the conversion, simply multiply by that well-chosen form of 1: 4.5 hr × 1 which is the same as

6 0 min 4.5 hr × -----1 hr

Now, x times y/z is the same as xy/z, so our units expression is the same 4.5 hr × as 60 min -------------1 hr Notice that you have hours (hr) in both top and bottom. Just as you would divide through by x when x was in both top and bottom, so you can divide through by the “variable” hr:

4.5 × 60 × min

------------1 which multiplies out to 270 min Summary: to convert units, construct a fraction that is equal to 1, multiply the original measurement by that fraction, and simplify.

Is This Really Multiplying by 1? “But wait a minute!” I hear you say. “You started with 4.5 and ended up with 270. How is that multiplying by 1?” The answer is that we didn’t start with a “dimensionless” pure number 4.5, but with 4.5 hours; and we didn’t end up with a pure number 270 but with 270 minutes. You should be able to convince yourself that if you bake a turkey for 270 minutes or 4½ hours, either way you wait the same length of time for dinner. A number with units is different from a number without units or with different units, just as 8x is different from both 8 and 8y. Think of it this way: 3.27 dollars or euros is the same as 327 cents, when you multiply by the “carefully chosen form of 1”, 100 cents/dollar or 100 cents/euro. If the top and bottom of the fraction are equal, the fraction equals 1 and the value after multiplying is the same as the value before multiplying — but expressed in different units, which of course is the whole point. You might be asking yourself, “Why all the fuss? Anybody knows that to convert hours to minutes you have to multiply by 60.” Well, yes, that’s true. But I deliberately chose a simple example to show the method. I’ll try to use more realistic (i.e., harder) examples from here on.

Be Reasonable! If you follow the procedures on this page, it will be impossible for you to multiply by a conversion factor where you should divide, or vice versa. But we all make careless mistakes, so it’s good to have a rough and ready check on your work. You can always apply this rule: “if the containers are small, you need more of them to hold the same total.” For example, an hour is longer than a minute, so you expect 4.5 hours to convert to some larger number of minutes. If you ended up with 0.075 minutes, you would know you had made a mistake. In any math work, it’s always best to work a problem two different ways, to guard against careless errors. But second best is to work the problem carefully and apply some test for reasonableness, like this one.

How to Pick a “1” You might be wondering how I knew to pick that particular fraction that was equal to 1. There are just two simple steps: 1. Find a conversion factor between the given units and the desired units, and write it as an equation.

Example: whether you have miles and need kilometers, or you have kilometers and need miles, you can use either conversion factor between miles and kilometers, namely 1 mi = 1.61 km or 1 km = 0.621 mi. Either equation will work equally well. 2. Convert that equation to a fraction with the desired units on top and the given units on the bottom. More formally, divide both sides by the value of the side that contains the given units. (Actually, this rule is oversimplified, as we’ll see below.) Example: To convert from miles to kilometers, you need a fraction with the desired units (kilometers) on top and the given units (miles) on the bottom. Based on the above conversion factors, that fraction must be either 1.61 km -------

1 km or

1 mi

-------0.621 mi

Those fractions look different, but remember that they’re both equal to 1 and therefore they are just different forms of the same fraction. Either one will work just fine for the conversion. Once you’ve selected a useful fractional form of 1, multiply the original measurement by the fraction, and simplify. Example: If the original measure was 15.7 miles, you would multiply by either of the above fractions and obtain 25.3 km.

Where to Find Conversion Factors I don’t just pull the conversion factors out of my hat. Many books have tables of conversions, including the venerable Handbook of Chemistry and Physics. There are also several good sources on line. My favorite is at the US National Institute of Standards and Technology, , though the sheer mass of information can be overwhelming. Looking at such references, you may note that this article uses common abbreviations like sec (seconds) and hr (hours), rather than the official abbreviations (s and h, respectively). That is deliberate, since most students are more familiar with the longer forms. In scientific work, you’d be expected to use the official forms.

Chaining Conversions If you can remember some conversions, you may be able to combine them to avoid looking up a specific conversion. If you have a calculator handy, it can be faster to do extra arithmetic than to go to a reference and look up a single conversion factor. For example, how many meters are in the 440-yard dash? To convert 440 yards to meters, you could look up the conversion factor between yards and meters. But if you happen to remember that 1 in = 2.54 cm and 36 in = 1 yd, it’s probably faster just to

use those (plus 100 cm = 1 m) than to look up the single conversion factor. This means you multiply by three different forms of 1: 36 in

2.54 cm

1 m

440 yd × ----- × ------- × -----1 yd

1 in

100 cm

and collect terms to 440 × 36 × 2.54 yd in cm m -------------------------100 yd in cm Doing the arithmetic, and dividing top and bottom by yd, in, and cm, you have the answer, 402 m. Having started with 440 yd and multiplied by 1×1×1, you know that the initial value equals the final value: 440 yd = 402 m Your own example might be different: you may happen to remember different conversions than I do. But if you commit a few factors to memory, you will find that by combining them you can avoid looking up a whole lot of conversions.

Compound Units What about more complex units, like converting miles per hour to kilometers per hour, or even miles per hour to feet (or meters) per second? You use the same technique and multiply by a well-chosen fraction that equals 1, only you need to do it for each unit to be converted. It’s just a more general form of chaining, which you already know how to do. The following examples take you through progressively more complicated situations: 1. mi/hr ==> km/hr illustrates a straight conversion with the “per hour” units unchanged. 2. mi/sec ==> mi/hr is another single conversion, but this time the units to be converted are in the denominator so step 2 in picking a fraction is a little different. 3. km/hr ==> m/sec shows how to do two conversions on the same quantity. 4. sq ft ==> sq m shows what to do when units are raised to a power.

Example 1: miles per hour to kilometers per hour This problem can be solved using either 1 mi = 1.61 km or 1 km = 0.621 mi. I’ll work it both ways, in parallel. To start, write the original measurement as a fraction:

11.6 mi ------hr Going from mi/hr to km/hr, you see that you end up with the same denominator you started with, so only the numerator has to change units. In other words, this is just our old friend miles ==> kilometers, with the “per hour” tagging along unchanged. So the conversion is the same one you’ve done before. Simply pick a fraction with the desired units (km) on top and the given units (mi) on the bottom: 11.6 mi

1.61 km

------- × ------hr

11.6 mi or

1 km

------- × --------

1 mi

hr

0.621 mi

As you see, you can use either conversion factor, miles to kilometers or kilometers to miles. It doesn’t matter because, by forming a fraction equal to 1, you automatically make the right choice between dividing and multiplying. Going on to simplify the fractions, you have 11.6 × 1.61 mi km -----------------

11.6 mi km or

hr mi

----------0.621 hr mi

Either way, divide top and bottom by mi and you have 11.6 × 1.61 km --------------

11.6 km or

hr

-------0.621 hr

Do the arithmetic to get 18.7 km/hr either way.

Example 2: miles per second to miles per hour Escape velocity from the earth’s surface is about 7.0 mi/sec. What is that in mi/hr? Here again, you’re converting only one unit, seconds to hours (1 hr = 3600 sec), and the “miles per” is just along for the ride. But what’s different here is that the units you’re converting are in the denominator of the fraction, not in the numerator. Look what happens if you apply the old rule of desired units on the top:

7.0 mi 1 hr -----× -------

sec 3600 sec and you end up with

7.0 mi hr ----------3600 sec sec

This is no good: you can’t simplify this fraction to remove the seconds (sec). Rule 2 in picking a fraction, as originally stated, only applied when the units to be converted were in the top of the fraction (or not in a fraction at all). Here’s the more general form of Rule 2, the form that will always work: when picking your fraction that equals 1, put the given units in the opposite position from where they are in the original measurement. If the original measurement had the given units on the top, your 1-fraction will have them on the bottom; but if the given units are on the bottom of the original measurement then your 1-fraction must have them in the top. Do this so that you can divide top and bottom by the given units when simplifying. Let’s come back to the example, 7.0 mi/sec converted to mi/hr, and do it the right way: Write the conversion equation:

1 hr = 3600 sec

The given units (sec) are in the denominator (bottom) of the original measurement, so sec must be in the numerator (top) of the conversion sec fraction:

3600

1 = -------1 Multiply the original measurement by 1:

7.0 mi 3600 sec ------ × -------sec hr

Ah, that’s much better! Now you can divide top and bottom by sec:

7.0 × 3600 mi

-----------hr Multiply 7 by 3600 to get your final answer, to two significant figures:

about 25,000 mi/hr

Example 3: kilometers/hour to meters per second A race car has a top speed of 310 km/hr. What is that in m/sec? For this example you’ll combine chaining multiple conversions with the new and more general form of Rule 2 for picking the fraction. You have two conversions to do, kilometers to meters and hours to seconds. You know the conversion factors:

1 hr = 3600 sec 1 km = 1000 m

In converting km to m, the given units (km) are on top, so in the conversion fraction km will be on the bottom. By contrast, in converting 310 km 1000 m hr to sec, the given units (hr) are on the bottom so the conversion fraction will have hr on the top. To do two conversions, you multiply by 1 hr two fractions (1 × 1): ------ × ------ × -------hr 1 km 3600 sec Now divide top and bottom by hours (hr) and by kilometers (km):

310 × 1000 m ----------3600 sec

and do the arithmetic to obtain the answer:

310 km/hr = 86 m/sec

Example 4: square feet to square meters Sometimes you have to deal with squared units. In the US, you often see them with a “sq” prefix. But they are actually easier to manipulate if you treat them just like variables (again!) and use the ² sign.

I correspond with a friend outside the US, and we are describing our homes to each other. If my apartment measures 850 square feet, what is that in square meters? In other words, convert 850 ft² to m². Solution: I need a fraction equal to 1, with m² on the top and ft² on the bottom. The way to obtain that is to form a fraction equal to 1 with plain m on the top and plain ft on the bottom, and then square it (since 1² = 1). As it happens, I don’t remember the conversion from feet to meters, 1 ft = 12.00 but I do remember the conversions between both of them and inches: in

So I construct my fraction in two steps:

1 m in

= 39.37

1 = ×

1 1

1 m 12.00 in 1 = -------× -------39.37 in 1 ft

12.00 m 1 = -------39.37 ft

0.3048 m 1 = -------ft Now remember that the original measurement is in ft². Therefore I ( must multiply the original measurement, 850 ft², by the square of the above fraction, to get ft² in the denominator and match the ft² in the 0.3048 m )² original measurement: 850 ft² × ( -------- ) ( ft

)

When a fraction is squared, that’s the same as squaring the top and squaring the bottom, including units:

850 × 0.3048² ft² m² ------------------ft²

Divide through by ft² top and bottom, and do the arithmetic to get the 850 ft² = 79 answer: m² What about cubic measure? How many cubic feet is 12 cubic yards? It’s exactly the same deal, except that you’ll need to cube your well-chosen form of 1 to do the conversion. 3 ft

Start with 1 yard = 3 feet, so your fraction is (3 ft)/(1 yd):

12 yd³ × ( ---- )³ 1 yd This simplifies to

12 × 3³ yd³ ft³ --------------1 yd³

12 × 27 ft³

12 yd³ = 324 ft³

Recap of the Procedure In a nutshell, do all conversions of units by multiplying the original measurement by a well-chosen form of the number 1. A bit less briefly: 1. Find the conversion factor for the given and desired units, and write it as a fraction with the given units in the opposite position from the original measurement. (If the original measurement has the given units in the numerator, the conversion fraction needs them in the denominator, and vice versa.) The value of that fraction is 1, since the top and bottom are equal. 2. If the given units are raised to a power, raise the conversion fraction to that same power.

3. Multiply the original measurement by the conversion fraction, and simplify.

But My Calculator Has a “Convert” Key! So? sooner or later you’ll have to convert a measurement with units that aren’t in your calculator. At that point a lot of students start to guess, and the more complex the units the more likely they’ll guess wrong. If you understand what you’re doing, using the techniques on this page, you’ll have smooth sailing. You can also use these same techniques to do currency conversions, which are probably not on your calculator because the rates fluctuate. See the practice problems. (The currency convertor at xe.com is a great tool for when you’re on line.)

Some Troubles Not every conversion can be done using the techniques on this page.

Temperatures Converting between temperatures in Fahrenheit and Celsius (sometimes called “Centigrade”), you cannot just multiply by a carefully selected form of 1. The reason is that the two measures have different zero points. What do I mean by that? Well, with pretty much every other measurement you’re likely to meet, you’re converting between two sets of units where the zero point is the same. 0 pounds equals 0 kilograms, 0 liters equals 0 cubic centimeters, and so on. But with temperature this is not true: 0 degrees C is a different temperature from 0 degrees F. You could apply the techniques on this page to convert temperatures after relating them to a common zero point, but it’s probably a lot easier just to remember the standard formula as a special case: F = 1.8C + 32. You may recognize this as the slope-intercept form of the equation of a straight line. With other conversions, the intercept is 0 because the conversion line passes through (0,0); but with temperature there’s a nonzero intercept because 0 degrees in one measure is not equivalent to 0 degrees in another.

Impossible Conversions Temperatures take special caution because of differing zero points, but at least temperatures can be converted. However, some conversions are completely impossible, not just impossible using the techniques on this page but impossible by any means at all. For instance, you can’t convert gallons to square feet (or liters to square centimeters) using any techniques. Why is that? Because gallons and liters measure volume, and square feet or square centimeters measure area. It’s like converting x³ to x²: it’s just not meaningful. You can use dimensional analysis to show this in a formal way, but informally just remember that area is two dimensions of length and volume is three dimensions of length, and measurements you convert must always have the same number of dimensions. One day I may write a page on dimensional analysis, but for now you can

look at Dimensional Analysis from the University of Guelph Department of Physics if you’re interested in this topic.

Practice Problems Here are some problems to practice on, with the conversion factors you need and my answers. You should be able to do all of them easily by using the techniques on this page. Remember not to make your answers more precise than the original measurements! If you run into trouble, or if you get a different answer and after careful checking it still looks right to you, you might post a note to the newsgroup alt.algebra.help. Don’t just post the problem, but show how you tried to solve it. That way you’ll get the most specific, focused help. 1. The Introduction asks which is faster, 60 miles an hour or 60 feet a second. Well? (1 mi = 5280 ft; 1 hr = 3600 sec.) Answer. 60 mi/hr = 88 ft/sec, which is faster than 60 ft/sec. 2. How much does a 2-liter bottle of soda pop weigh in pounds? (Assume that the pop has the density of water, namely 1 kg/liter, and that the weight of the bottle itself is negligible.) (1 kg = 2.2 lb.) Answer. 4.4 pounds. 3. An Englishman returning home from Norway has 860 kroner of pocket money that he never spent. How much is that in pounds? (Assume an exchange of NOK 12.32 = £ 1.00.) Answer. £ 69.81. 4. In book IV of The Lord of the Rings, Frodo and Sam rappel down a cliff, using a rope 30 ells long. How high was the cliff, if the rope nearly reached the bottom? (1 ell = 45 in; 12 in = 1 ft; 1 m = 39.37 cm.) Answers. About 110 feet or 35 meters. (Since we know only that the rope “nearly” reached the bottom, and don’t know exactly how much was tied to an anchor at the top, the second figure in each of those answers is a bit mushy. We might say 100-110 feet or 30-35 meters.) 5. A Canadian vacationing in the States pays $1.689 a gallon for gasoline. What would be the equivalent price at home? (Gasoline is sold by the liter in Canada, as in most countries.) (1 US gal = 3.785 liter; assume a conversion rate of C$ 1.60 to the US dollar.) Answer. The price is equivalent to 71 Canadian cents a liter. 6. You buy a 750 ml bottle of rum. How many rum-and-Cokes can you make, using an ounce and a half of rum in each drink? (1 US fluid ounce = 29.57 ml.) Answer. About seventeen drinks.

7. What is 65 degrees in radian measure? (pi radians = 180°.) Answer. About 0.361×pi, or 1.134 radians. Comment: Technically an angle in radians is simply a pure number, and radians are not units. When you say “2 radians”, that is identical to the unitless number 2. And though degrees are units, they are dimensionless, like percentages. However, the technique presented on this page works just fine for converting between degrees and radian measure. 8. How many cubic meters are there in a cubic mile? (1 mi = 1609.344 m.) Answer. 1 mi³ = 4.168182×10^9 (about 4168 million) m³. 9. My 1967 Encyclopædia Britannica says that Lake Erie has a surface area of 9930 square miles and an average depth of 58 feet. How much water does it hold, in cubic miles? in liters? (1 mi = 5280 ft; 1 liter = 0.001 m³, and use the answer to the previous problem.) Answers. 109 mi³, 4.55×10^14 (455 million million) liters. 10. Lake Erie has a surface area of 9930 square miles. If an inch of rain falls on the lake one day, how many gallons have been added to its volume? How many liters? (1 mi = 5280 ft; 1 ft = 12 in; 1 US gal = 231 in³; 1 US gal = 3.785 liters.) Answers. About 1.73×10^11 (173,000 million) gallons, 6.53×10^11 (653,000 million) liters.