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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.

We'll be doing a lot more of these dimensional analysis problems, but this is a good introduction.

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