Dimensional Analysis


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Dimensional Analysis is a powerful way to solve problems. The power comes from the simplicity of the approach. It avoids memorization and only requires some observation of where you start and where you want to end up.
	TRADITIONAL APPROACH: This is the traditional approach to doing this kind of math problem. We want to find out how many feet are in 24 inches. First we have to know that 12 inches equal one foot. The next step is to see how many times 12 inches fit into 24 inches because each time it does, that means another foot. This is one meaning of division, so dividing 24 by 12 finds the answer.
	Instead of dividing by 12, could we use multiplication instead? Yes, if we multiply by (1/12). When multiplying by fractions, the denominator gets divided into the numerator and we still get our answer of 2. The bottom setup implies multiplication and will make it faster to write out longer problems.
	DIMENSIONAL ANALYSIS: This approach analyzes the dimensions of the problem and sets up a problem so that the starting dimension is changed to the answer by using a conversion fraction. Here the starting amount is 24 inches and the answer wanted is in feet. The conversion is 1 foot = 12 inches. We write the inches on the bottom so that the inches will cancel leaving only feet.
	In dimensional analysis you only set up the problem as multiplication. If division is needed, the number is placed in the denominator. This will be apparent because placing the value in the denominator will cancel out the dimension (unit of measure) that you want to get rid of.
	How many inches in 2 miles? START & END: Draw a horizontal line and put the starting amount of 2 miles on the left. Put the end amount of inches on the right. Draw a couple of vertical lines. CONVERSIONS: I don't know how many inches are in a mile, but I do remember the high school track was 440 yards around and that was equal to a 1/4 mile. Let's use that to convert miles to yards. We write the yards on top and miles on the bottom so that units of miles will cancel. Now use the fact that 36 inches equals a yard to cancel the unit of yards. Finally, I multiply the 2 x 440 x 36 inches and divide by 1/4 (0.25) to get the answer of 126,720 inches.
	Let Dimensions Guide You: The beauty of this approach is that you don't have to think about whether to multiply or divide, just let the units (dimensions) tell you what to do. If you place them correctly, the units you start with and the ones used for conversions will cancel, and you will only be left with the units of your answer. In this case, it was inches.
	Why equal? Remember the beginning miles and the final inches have to be the same length. The length doesn't change; it's just measured with different units or dimensions (inches instead of miles). When we write 440 yards over 1/4 miles, those two lengths are the same, so in essence we are just multiplying by one because an amount divided into the same amount always equals one.
	It would not have mattered if we used 1,760 yards over 1 mile because those lengths are equal. Also, we could have used 1 inch over 1/36 of a yard because they are equal. Our answer would still be the same 126,720 inches.
	How many feet per second are we traveling if we are going 60 miles per hour? Whenever you hear "per," set it up as a fraction. So 60 miles per hour is 60 miles over 1 hour. As before put the starting amount on the left and the end amount on the right. Like before, we can get from miles to feet with two multiplications. To turn hours to seconds, we also take two steps. Like always, the units are placed so that you cancel out what you don't want to keep. After all the canceling our answer of 88 feet per second is the same speed of 60 mph, just in different units.
	Most of the problems you encounter will involve metric conversions. So let's do some. How many millimeters (mm) are in two kilometers? When you look at the beginning and end amounts, they both have the word, "meters" in it, so that stays. We only need to cancel "kilo" and add "milli". Again, the conversion fractions have to be equal. So "kilo" is written on the bottom so "kilo" will cancel. "kilo" means 1000 so that's placed on top. "milli" is on top because that's in the final units. The equivalent to "milli" is one thousandth (0.001). All that's left is to multiply 2 x 1000 and divide by 0.001 to give us 2 million millimeters.
	Let's say a recipe called for a concentration of 0.2 kilograms of salt per liter. *What is that concentration in milligrams of salt per milliliter?* Remember "per" means to set up the concentration as a fraction. We write the given concentration on the left and the requested equivalent concentration on the right. When we examine the beginning and final units, we see that kilo has to be canceled and that we need "milli" (thousandths) versions of grams and liters. The units of "grams" and "liter" is in both beginning and final units, so we don't have to do anything with those. To add "milli" to both, just put milli on top of milli since those are equal. After multiplying we see that 0.2 kilograms per liter is the same concentration as 200 milligrams per milliliter.
	In algebra, the approach to solving problems revolves around solving equations. For example, in this problem, you start with a given formula for distance using velocity x time. Since the question is asking for "How fast," you solve for velocity. After solving for "v" you plug in the 300 miles and the 5 hours and get the velocity of 60 miles per hour. In dimensional analysis, you don't bother much with formulas nor do you need them. The dimensions guide the way. Now lets see how it's done with dimensional analysis...
	The problem is asking for how fast you are traveling, which is assumed to be in miles per hour. So that's our final amount, which is written on the right. The dimensions of miles per hour has miles on top and hour on the bottom. That means the starting amount would have 300 miles on top and the 5 hours on the bottom. There's no need for any conversion fractions here. Simply divide the 300 miles by 5 hours to get 60 miles per (1) hour. So if you didn't do well in algebra, that's OK with dimensional analysis.
	The following prices rates are equivalent: "$5 a pound" "$5 per pound" "$5 for every pound" and "$5 for each pound." They should all be set up as a ratio, which a fraction can represent.
Converting Volume to Mass or Mass to Volume 10.0 mL x 19.3 g = 193 g 1 mL 50.0 g x 1 mL = 2.59 mL 19.3 g	In the above examples we converted distance in miles to distance in inches. We also converted speed in mph to feet per second. With density problems we can convert mass to volume or volume to mass, which is a new trick. The density of a substance gives us a way to make that conversion. For example, gold's density is 19.3 grams per mL. If starting with milliliters, we use 19.3g/mL to convert to grams. If starting with grams, we use 1 mL/19.3g to convert to milliliters. As usual, we just set it up so that the starting units cancel and we end up with the unit we want.
	It would be great if you are able to learn to do these kind of calculations using a spreadsheet program like Microsoft Office's Excel or the free one from OpenOffice called Calc. You at least need to do them using a scientific calculator.

Below is a typical layout of a spreadsheet. Across the top are the columns indicated by letters. On the left are the rows indicated with numbers. The boxes where you type are called cells. The location of a cell in defined by its column letter and row number. For example, the cell where the number "19.3" is located is D1. To do dimensional analysis in a spreadsheet just type the number in one cell and the dimension next to it. The spreadsheet lets you layout the problem so you can see all the units. You can even color the units that cancel and the ones that will remain. That helps you see you have it set up correctly.

	A	B	C	D	E	F	G	H
1	355	mL	x	19.3	g	=	?	grams
2				1	mL

By the way, it's possible to turn off the grid lines to make the setup look more like a standard equation

	A	B	C	D	E	F	G	H
1	355	mL	x	19.3 g		=	?	grams
2				1 mL

To get the spreadsheet to calculate your answer, you would go to G1 and type an equal sign "=". That tells the spreadsheet that you are putting in a formula. Then click on A1 where the 355 is sitting. The spreadsheet will insert "A1" into the formula. Then type an asterisk "*" which means to multiply. Then you click on "19.3" and "D1" will appear in the formula, which will now read "=A1*D1". Then press the Return (Enter) key, and the answer will then appear because it's been told to multiply A1 by D1. You can then type different values for the volume (A1) or density (D1) and after typing a new value press Return key or click out of that cell and the new gram amount will automatically be recalculated. So the beauty of a spreadsheet is that after you set up one problem. It can be used to calculate different values of volume or density without going through the calculations again. It also keeps it all organized so you don't make mistakes.

The below table is like the top one but I've set it up so you can type in values for the volume and density and it will calculate the grams automatically. Initial values are for 10mL of water, which has a density of 1.0 grams per mL. Try changing the density to 19.3 (density of gold) and the volume to 355 (volume of soda can). Click the "D Calculation" button to have it do the calculations. On a Web page we need a button to cause it to calculate. Then try some other volumes and densities.

	A	B	C	D	E	F	G	H
1		mL	x		g	=		grams
2				1	mL

Using spreadsheets is especially helpful when you have a long series of calculations because you are more prone to make mistakes doing it with a calculator. Let's do the conversion of miles per hour into centimeters per second. The first 3 conversion fractions is to convert miles to centimeters. The next 2 is to change hours into seconds. I put all units (dimensions) that cancel in red and those that don't in green. Again, units cancel if one is in numerator and one is in denominator. To calculate the answer, we multiply all numerator and divide by each denominator. So go to cell S1 and type an equal sign "=". Now click on the "2" next to miles. Now type "*" to indicate multiplying. Now click the "440" followed by typing "*". Now click "36" followed by typing "*". Then click "2.54". No need to click on the ones. Now type "/" to indicate you want division. Now click the "0.25" followed by typing "/". Now click "60" followed by typing "/". End by clicking the other "60". At this point the formula will read "=A1*D1*G1*J1/D2/M2/P2". Press Return key and the answer will appear. The beauty of this again is that now all you have to do is change the number of miles and you automatically get the new centimeters per second.

miles

440

yards

inches

2.54

min

0.25

mile

yard

inch

min

sec

In the below spreadsheet, I put in an input text field for the miles per hour. When you click the Do Calculation button, it will do all the calculations to get centimeters per second. So it shows how easy it is to find answers to a long series of calculations. It also keeps you organized and reduces dumb mistakes.
Searching the web for the conversion of miles per hour to centimeters per second can be found as one conversion factor (it's 1 mph =44.704 cm/sec); however, there will be other calculations where you can't look up a shortcut and will have a long series of conversions.

Start velocity

miles to yards

yards to inches

inches to cm

per hr to per min

per min to per sec

desired units