This supplementary section is a review of rules of working with fractions. It is fairly basic and if you are rusty with fractions it is probably a good idea to practice some examples and problems in order to get better at them. Knowing how to add, multiply, and divide fractional values without having to spend tons of time on them is a very good time-saving device.
As usual, it is best to illustrate the addition of fractions with an example. There are many ways to work with fractions to simplify your life, but here I will just give you simple procedures that work.
Example: Suppose you were asked to add the numbers 1/8 and 3/4. This is probably simple enough to do that you might not even be worried about it, but we can go throught the procedure in order to review.
1. Look at the two fractions and stare at the denominators. Remember that the denominator is what appears in the bottom of the fraction. Now, pick the lowest common denominator between the two values. This means that you take the lowest integer value which can divide the two numbers and give an integer. Thus, between 4 and 8 the lowest value that can divide each one and give an integer is 8. So, the common denominator is 8.
2. Take the common denominator and divide it by each denominator:
8÷4 = 2
8÷8 = 1
3. Take each of the values from step 2 and multiply them by the corresponding numerator (what was on the top part of the fractions):
2 x 3 = 6
1 x 1 = 1
4. Add the two results from step 3, i.e., 6+1=7.
5. The sum of the two fractions is the number in step 4 divided by the lowest common denominator: 7/8.
Thus: 3/4 + 1/8 = 7/8.
The above example is particularly simple and nice. The denominators 4 and 8 have a lot in common. One denominator divides the other and the lowest common denominator is simply one of them. However, the lowest common denominator is not always so nice. For instance, if you were asked to add 1/5 to 1/7, the lowest common denomintor is actually 35 (which is equal to 5 x 7). When the two integers in the denominator have no divisible number in common their lowest common denominator is usually just the numbers multiplied by each other.
This procedure is really not that difficult. All you do is multiply the top parts together and multiply the bottom parts together. Here is a simple example to illustrate this.
Example: Multiply 3/16 by 30/41.
Procedure: 3/16 x 29/41 = (3 x 30)/(16 x 41) = 90/656
Simplify: 90 and 656 are both divisible by 2, so we can simplify our answer by factoring out this 2 from both top and bottom and they will cancel. Thus: 90/656 = (2x45)/(2x328) = 45/328
Note that the factors of 2 cancelled out from top and bottom; this is generally true, and anytime you see numbers in a mulitiplication that are common to top and bottom you can cancel them This last step is not necessary but it usually is nice to have the simplest answer written down. It looks less confusing and allws you to go further with the problem without having to carry around unncessary numbers.
Multiplying fractions is really no different than multiplying numbers. You can even give a final decimal form for the answer. For instance, for the above example, the final answer of 45/328 can be written out in decimal form by actually dividing 45 by 328, which gives: 0.137195 ...
Sometimes - and this specially holds true when you are working with units and dimensional analysis - you will be faced with the situation of dividing two ratios (or two fractions). The procedure for this is to do the following:
Step 1: Multiply the numerator of the top fraction by the denominator of the bottom fraction. This is the numerator of your result.
Step 2: Multiply the denominator of the top fraction with the numerator of the bottom fraction. This is the denominator of your result.
Example: Divide 3/7 by 2/9.
Step 1: Numerator of top x Denominator of bottom = 3 x 9 = 27
Step 2: Denominator of top x Numerator of bottom = 7 x 2 = 14
Thus, the result is: 3/7÷2/9 = 27/14
The above results can be summarized into more general terms. In astronomy (and in physics) you will encounter situations where you are working not with numbers but with variables. Recall that we label variables with alphabetical letters, and they are genereal in the sense that you either don't know them or that they can take on a set of values that is not specified. Thus, you keep such measurable quantities in variable notation (for instance x, y, T, I, ...). So, in working with equations, and in manipulating units, you will have to multiply and divide variable quantities. The rules for working with variables are no different than with numbers, except you keep using the notation of variables.
a/b + c/d = (a x d + c x b)/(b x d)
a/b x c/d = (a x c)/(b x d)
a/b ÷ c/d = (a x d)/(b x c)
Notice that these are no different than the rules that were applied to the cases with numbers in each of the examples above. Review those examples and convince yourself that these rules written in terms of general variables work for the above examples.