Fractions seem to "fracture" the average person. Some CBEST test takers cringe at the thought of doing anything with fractions. Forget the anxiety. Fractions are just numbers that behave the same as others. All it takes is basic math skills.
     Fraction means a "part or parts of a whole." We all know what "a half" means. Something cut in two parts. And we all know what "a fourth" means. Something cut into four parts. Whatever number is on the bottom, called the denominator, is how many total parts something has been cut into. The number on top, the numerator, tells us how many of those parts we have. 2/5 means we have cut something up into 5 equal pieces and have 2 of those pieces. 5/12 means we have cut something up into 12 equal pieces and have 5 of them. The first thing to realize is that the more pieces something is cut into, the smaller the piece is. 12 people eating the same pizza will get a very small piece compared to if only 5 people were eating the same pizza.
     Any number over itself is equal to one (or one whole.) 3/3 means we have cut something up into 3 pieces and we have all three, therefore we have the whole thing. Similarly with 2/2, 5/5, 12/12, etc.
     If the number on top is larger than the number on the bottom, we have an improper fraction. That is, it is worth more than one (or a whole one.) 13/10 means we have 10/10, one whole, and 3/10 left over. We can write that as 1 3/10. The simple rule is to ask yourself how many times the bottom number will divide into the top. That is our whole number. The remainder is what is left as a fraction over the original bottom number. This new form with whole numbers and a fraction is called a mixed fraction.
     Example: 14/5
5 goes into 14 two times (5 x 2 = 10). This leaves 4 left over (14 - 10). Written 2 4/5, or two and four fifths.
     Sometimes a fraction needs to be "reduced." Every CBEST answer will be in the most reduced form. That means no improper fractions and any fraction will be completely reduced. We have explained how to change an improper fraction to a mixed fraction, so we will just talk about reducing. For this, you do need to know your times tables. The easiest way to reduce a fraction is to think of the largest number that will divide into (without leaving a remainder) the numerator and denominator. How many times does this number go into each number in the fraction? These answers are the new numbers in your new fraction. If you cannot divide the numerator and denominator by any number (other than one), it is reduced as far as it can be.
     Example: Reduce 12/20
The Largest number that will go into 12 and 20 is 4. 4 goes into 12 three times, and 4 goes into 20 five times. That is our new fraction: 3/5.  No number will go into 3 and five except 1, so it is completely reduced.
     Example: Reduce 14/6
The first thing to do is realize we have an improper fraction. 6 goes into 14 two times, 6 x 2 = 12, 14 - 12 = 2, so we have a mixed number now: 2 2/6, two and two sixths. Can we reduce the new fraction? 2 will go into 2 and 6. 2 divided by 2 equals 1, 6 divided by 2 equals 3. Our new and completely reduced fraction is 2 1/3, two and one third.

Getting a Feel for Fractions

     One of the main ways to get comfortable for the CBEST test is to become a math person. Or, at least try and think like one. This is a recurring theme throughout these lessons. Very important if you have some type of math or test phobia. Let's do some things with fractions. First, any fraction (not improper and not mixed) is always worth less than one (or a whole.) 4/5, 2/3, 7/9, etc. are all worth less than 1. Similarly, adding any two fractions like this will give you an answer less than two. 2/3 + 4/5 is less than 2. So is 6/7 + 9/13, 11/20 + 12/29, etc. So, if you need to add any two fractions like these, rest assured the answer will be less than 2.
     Remember we talked about cutting things in half? This works good for fractions. We know that 1/2 + 1/2 will equal one. Suppose we have 1/4 + 1/5. 1/4 and 1/5 are each less than 1/2. Therefore, when adding these we must get an answer LESS than one. What is we have 2 fractions greater than 1/2, say 3/4 + 4/7. We know that these are both greater than 1/2. How? 2 is half of 4, but we have 3/4. 3.5 is half of 7, but we have 4. (I hope you know that 3.5 is the same as 3 1/2. More on that later.) So 3/4 + 4/7 is greater than 1/2 + 1/2. Or, greater than one. Likewise, from the previous paragraph, we know these 2 fractions also add up to less than two. Therefore, 3/4 + 4/7 is between 1 and 2. But both of these fractions are not equal to 1/2. This leads to the conclusion that 3/4 + 4/7 is greater than 1 and less than 2. If there is only one answer choice that fits this limit, that MUST be the answer. Get good at recognizing if a fraction is more, equal, or less than one half.
     Being familiar with similar fractions is also beneficial. 1/2 = 2/4 = 4/8 = 5/10, etc. 1/4 = 2/8 = 4/12, etc. The list is endless.
     Also, recognize what fractions add up to 1 (one whole.) 1/4 + 3/4 = 1, 1/2 + 1/2 = 1, 7/10 + 3/10 = 1, etc.
     And, recognize what 1 minus a fraction is. (The reverse of the above.) 1 - 4/5 = 1/5, 1 - 3/4 = 1/4,etc.
     Know how to tell how close a fraction is to a common one. Common fractions are 1/2, 1/4, 1/3, etc. For example, 24 out of 50 would be roughly 1/2 because 25/50 = 1/2. Likewise 13/36 is about 1/3 because 12/36 = 1/3. And sometimes the bottom is close as in 12/23 is very close to 1/2 because 12/24 = 1/2. The more you are familiar with fractions, the more confident and competent you become. It is also a step towards being a math person. How can this be helpful? If you were asked to find an equivalent fraction to 24/50, you would know that you do not have 25/50, just a little less. This would allow you to eliminate all fractions in the answers that were one half or more. Remember, my technique is finding right answers without doing too much arithmetic.
 

Adding Fractions

     Okay, now that you are familiar with fractions, we will start adding them. If fractions have the same bottom number, just add the top numbers and put that total over the bottom number as one fraction, Reduce if necessary. For example, 3/7 + 2/7 = 5/7. Okay, all of you know how to do that. We will deal with adding fractions with unlike denominators. If you know how to do it the old-fashioned-fourth-grade way, go ahead and skip this part. If you want a straight forward, almost no-brainer way, then take notes. Forget about the way you did it before. Suppose we have the following problem:
3/4 + 7/9 = ?
The way to do this with little thinking is as follows. Multiply the 2 numbers on the bottom, 4 x 9 = 36. This is your new denominator. Now multiply the top of one of the original fractions by the bottom of the other one. In this case we'll do 3 x 9 because 3 is on top of the 4, and 9 is under the 7. 3 x 9 = 27. Keep this number off to the side. Now do the same thing with the other fraction first. That is, multiply 7 x 4 = 28. Now add this 28 to the 27 we got in the previous step to get 28 + 27 = 55. This is our new top number. We will put this 55 over our new denominator of 36. The new fraction is 55/36. Yes, we need to reduce. 36 goes into 55 once. This means we have a whole number of 1. The reminder, 55 - 36, is 19. The answer is 1 19/36. We cannot reduce any further. Follow this pattern and you should have no trouble.
     Try the following problem the same way.
2/3 + 5/12 = ?
Multiply the 2 bottom numbers: 3 x 12 = 36, our new denominator.
Then do (2 x 12) + ( 5 x 3) = 24 + 15 = 39, our new numerator.
The answer is 39/36. 36 divides into 39 once, leaving a remainder of 3. We now have 1 3/36. But 3/36 can be reduced. 3 goes into 3 once, and into 36 twelve times. Our new and reduced answer is 1 1/12.

Now for a big CBEST tip. Try and follow this one. Whenever you add fractions with unlike denominators, your answer will have a denominator that is a factor of what you used for your new denominator when starting the problem. Below are two detailed examples.
Example #1
5/8 + 4/7 = ?
Using the technique above, our new denominator will be 7 x 8 or 56. What numbers go into 56? 1,2,4,7,8,56. We van always exclude 1. Any reduction in your new fraction will need a denominator of one of these numbers. So, any answer that does not have those numbers as a denominator is out. But from the previous information in this section, we also know 5/8 + 4/7 is more than 1, but less than 2. And now we also know what are possibilities for the new denominator. Chances are, there will be only one answer on the list that fits the bill. And that again is my technique. Finding the answer by doing as little math as possible!
Example #2
2/5 + 1/20 = ?
5 x 20 = 100, so our new denominator is 100. (Yes, I know that there is a way to use 20. If you use 20, then you obviously have no trouble doing these problems anyway.) We know that our reduced fraction for the answer will have 100,50,25,20,10,5,4, or 2 in the denominator. We also know that the answer will be less than 1 by our knowing that both fractions are less than 1/2.

Short cut!
Looking at the previous example, you should recognize that 5 will go into 20 an even number of times. We can still multiply 5 x 20 to get a new denominator of 100. But if we remember that 5 will go into 20, we know that our answer will have a reduced fraction with a denominator of 20 or less.

What about mixed numbers? When adding mixed fractions, simply add the whole numbers together and put them off to the side. Ignore them and work on the fractions.

Example: 4 3/5 + 8 3/8 = ?

Here the two whole numbers add up to 12. That means our answer is more than 12 because we have two fractions to add on. Any answers that are 12 or less are wrong. You will be surprised at how many answers this actually eliminates. And sometimes, there is only one or two left! 3/5 is a little more than half, and 3/8 is a little less. Logically, we know they add up to something around one. This would mean the answer is around 13. But the two fractions don't add up to 1, so an answer of exactly 13 is out. 5 x 8 = 40 for our new denominator. We know the new reduced denominator will have a 40,20,10,8,5,4, or 2. We may be able to pick the answer from the information we have so far. And you always want to be checking answers after each step. At worst we can simply do (3 x 8) + (5 x 3) = 24 + 15 = 49. Our new fraction is 49/40 which reduces to 1 9/40. But we need to add that one whole number to the 12 we got earlier. Our answer is 13 9/40.

Any mixed fractions can be added in a similar way.

Subtracting Fractions

     Okay, subtracting is the hard part. But if you remember things from above, it will help. Chances are, there will be no subtracting problems on the CBEST that are not mixed numbers. So, we will concentrate on mixed numbers only.
Example #1
8 2/3 - 5 1/2 = ?
For some of you, it is a task you don't want to do. I can hear "to borrow or not to borrow." Forget it. We are going to get rid of the need to borrow. But you need to be an expert on what adds up to one and what one minus a simple fraction is.
     As stated before, 1/2 + 1/2 = 1. The reverse is 1 - 1/2 = 1/2. Another one, 1/3 + 2/3 = 1, 1 - 2/3 = 1/3. Do you get the idea? 5/8 + 3/8 = 1, and 1 - 5/8 = 3/8. And so on. Now let's get back to our example problem.
     First of all, subtract 1 from the first number 8 - 1 = 7. Now subtract the second whole number from that. Here it is a 5, so we do 7 - 5 = 2. That means our answer is at least 2 and we will set that 2 aside and go back to what we have left. We now have a new problem 1 2/3 - 1/2. We can take the 1 from the mixed fraction and subtract the second fraction. 1 - 1/2 = 1/2. That means, we are left with and addition problem! 2/3 + 1/2 will be the answer added to the 2 we have set aside. And you can do 2/3 + 1/2 with no sweat, right? 2 x 3 is our new denominator. (2 x 2) + (3 x 2) = 4 + 6 = 10. We now have 10/6 which reduces to 1 4/6. That also reduces to 1 2/3. We add that to our 2 to get 2 2/3. No borrowing!

Example #2
7 1/4 - 3 2/3 = ?
We are going to do 6 - 3 first to get 3. Set that aside. We now have a new problem 1 1/4 - 2/3. 1 - 2/3 - 1/3. We just need to add 1/3 + 1/4, add it to our 3 and we have the answer!

Example #3
13 2/7 - 8 2/5 = ?
Do 12 - 8 to get 4. Set that aside. Our new problem is 1 2/7 - 2/5. 1 - 2/5 = 3/5. Our answer will be (3/5 + 2/7) + 4.

More detailed example #4
16 1/4 - 9 5/7 = ?
First do 15 - 9 to get 6. Set that aside. We now have 1 1/4 - 5/7. Do 1 - 5/7 to get 2/7. Add 2/7 + 1/4. Our denominator will be 7 x 4 = 28. Our top number will be ( 2 x 4) + ( 7 x 1) = 8 + 7 = 15. New fraction is 15/28. Tack on our whole number of 6 and we have an answer of 6 15/28. Cannot be reduced. Now that was too easy.

Comparing Fractions

     Which fraction is larger? Which is smaller? Very simple technique to figure this out.
Example:
Which is larger, 2/5 or 4/9 ?
To do this multiply the top of the first by the bottom of the second and write your answer above the first. That is, 2 x 9 = 18, put the 18  above the 2/5. Similarly, multiply the top of the second by the bottom of the first. Here that is 5 x 4 = 20. Write that above the second fraction. Which is larger 18 or 20? 20 is, and, 20 is larger than 18, so the side that 20 is on will be the side of the larger fraction. 4/9 is larger than 2/5. Remember, when you multiply the 2 numbers, put your answer above the top number you used.
Example:
Which is larger, 5/8 or 7/12 ? Again, 5 x 12 = 60. Put that above the 5. 7 x 8 = 56. Put that above the 7. Since 60 is larger than 56, 5/8 is the larger fraction.
     The CBEST test may ask you to use the inequality signs, <, >, or even = in between the 2 fractions or numbers. How do you remember which way? The "<", ">" will always point to the smaller number. Small point/small number.
Example:
Put <,>, or = between the fractions.
2/7 ___ 3/10
2 x 10 = 20, that goes with the 2/7. 3 x 7 = 21, that goes with the 3/10. 3/10 is larger. The sign will point to 2/7 as follows:
2/7 < 3/10

     Just to prove this works, let's try some that are equal. If you get equal numbers, the fractions are equal.
How does 5/10 compare to 7/14
5 x 14 = 70, and 7 x 10 = 70. Equal numbers! The fractions are equal and we would use an equal sign.
5/10 = 7/14.

 More on camparing fractions

*If fractions have the same denominator, you can compare the numbers on top. Larger number, larger fraction.
Examples: 1/6 is smaller than 2/6, which is smaller than 3/6, etc.
* Adding the same number to the top and bottom of a fraction will result in a larger fraction.
Examples: 3/4 is smaller than 4/5 (adding 1 to top and bottom), which is smaller than 10/11 (adding 6 to each one),etc. This allows us to know right away that 7/8 is smaller than 17/18 because 10 was added to the 7 as well as 8. These two little tricks can come in handy.

     You should now be an expert on fractions!