Introduction to Fractions Some people seem frightened by the word "fraction." Fractions are nothing to be scared of; in fact, we use them almost everyday. A quarter is an example of a fraction we use quite often. Why is it called a quarter anyway? A quarter is 1/4 of a dollar. In other words, it takes 4 quarters, or 4/4 of a dollar to make 1 dollar. If you had 5 quarters you would have 5/4 of 1 dollar, which is actually more than a dollar! Some of this may seem a bit confusing, but as you read on, you'll see it's quite simple. First, here's a little information on fractions. A fraction is simply another way of writing a division problem. Normally we use fractions to represent "remainders" or numbers that are smaller than "whole numbers." Whole numbers are numbers like 1, 2, 3, and 4. They do not contain any fractions. You probably already know that numbers just keep going and going. There is no end to the number line. Did you also know that fractions make up the vast majority of these numbers? Think about this: just counting from 1 to 2, there are an endless amount of fractions between these two numbers. There are just as many fractions between 1 and 2 as there are whole numbers all together. Both are endless! Mind boggling isn't it? 1/1,000,000,000 is an example of a fraction between 0 and 1. Now that you know a fraction is simply a division problem, here is a more detailed explanation. The top part of a fraction is called the "numerator" and the bottom part is called a "denominator." The numerator is divided by the denominator. When the numerator is larger than the denominator, you can divide out at least one "whole number." This is known as an "improper fraction." When the numerator is smaller than the denominator, you have a "proper fraction." With all this said, let's slow down a bit and start trying to get a better understanding of the basics. Examine the picture below to see several ways of describing a fraction. Now that we have had a little introduction into fractions, let's begin our lesson by looking at a piece of pie cut into 8 pieces. Before any of the pieces are eaten, we have a "whole" pie, or all 8 pieces. If we were to write this number as a fraction, it would be written as 8/8. Normally we would just write it as 1 pie, but if you are really getting into this fraction thing, think of it as 8 pieces left out of the 8 pieces needed to make a "whole" pie. Again, that's 8/8 as a fraction. Now let's invite some hungry friends over. 4 of your friends quickly grab 4 pieces of the pie. Pieces 5,6,7, and 8 are now gone! Now we only have 4/8 of a pie left. How did we get this "fraction" of a pie? Remember we started with 8 pieces, making a "whole" pie. 4 of the 8 pieces, ( or 4/8 ), are now gone. All that remains are 4 pieces of the 8 original pieces, ( also 4/8 ). About this time mom comes into the room and makes everyone put their pieces of pie back, telling them that they must wait until after supper to eat pie! Oh no! Now how will we handle this new "fraction" of a pie? Well fortunately the pie is no longer a "fraction." Once again it is a "whole" pie, although we could still write it as a fraction if we really wanted to. "Whole" numbers can also be expressed as fractions, as we have already seen. Here is what happened to the pie when everyone put the pieces back. 4 pieces out of 8 were taken, ( 4/8 of the pie ). 4 pieces out of 8 were left, ( also 4/8 of the pie ). When we added the 4 taken to the 4 left, we got the 8 original pieces all back as a whole pie, ( 8/8 of the pie, or 1 "whole" pie ). How do we know that 8/8 of a pie is the same as 1 "whole" pie? Because the pieces left, ( numerator or top number ), equals the pieces needed to make a "whole" pie, ( denominator or bottom number ). Anytime the numerator equals the denominator you have 1. In this example it was 1 pie. Another way of looking at it is to remember that fractions are "division" problems; and 8 divided by 8 is equal to 1, just as 4 divided by 4, ( or 4/4 ), would equal 1. What if we had cut the pie into 4 pieces instead of 8? How about 2 pieces for those of us that really love mom's pies! Take a look at the picture below and see if you notice the relationship between each of them. See how they are the same? 4/8 is equal to 2/4, and both are equal to 1/2 of the pie because they all represent the same amount of pie. The only difference is that the pie is being cut into fewer pieces each time. Below is another relationship between 2/8 and 1/4. Study it carefully and see if you are beginning to notice how certain fractions are equal to each other. What would the fraction look like if all of the pie were gone? Anytime the numerator is 0 the fraction will equal 0. Let's look at some "fractional" math problems now. Suppose mom had not interrupted our previous pie party. Let's say we still have 4/8 of the pie left. ( 4 pieces left of the 8 original ). 2 other friends take 2 more pieces of pie. This time another 2/8 of the "whole" pie is taken, ( 2 of the 8 original pieces of pie ). Since we were already down to 4/8 of the pie, we now only have 2/8 of the pie left! ( 2 pieces left out of the 8 original ). How are we able to determine these fractions so easily? Here is a little secret. As long as the denominators are the same, you can "add" and "subtract" fractions as easily as you normally do with "whole" numbers. Our denominator, ( or bottom number ), in both fractions is 8. When the denominators are the same we can leave them alone. Here the denominator simply represents how many pieces the pie was originally cut into. The numerators, ( or top numbers ), are 4 and 2. We already know 4-2=2, so we can easily determine that 4/8 - 2/8 = 2/8, as shown above. What if we wanted to check our previous answer to see if we were indeed correct? Remember that to check subtraction problems, we must add our answer with what we subtracted. When we subtracted 4-2=2 above, we would have taken our answer of 2 and added the 2 back to what we subtracted to get 4. Checking fractions is just as easy. Remember that the denominators MUST be the same though, or our addition and subtraction won't work. Since we were able to check 4-2=2 by adding 2+2=4, we can do the same thing to this fraction. 2/8 + 2/8 = 4/8. Now that we have what we started with, our answer must have been correct. Earlier I mentioned "proper" and "improper" fractions. Let's examine these two fractions a bit further. So far we have mostly been working with "proper fractions" where the numerator, ( or top number ), is smaller than the denominator, ( or bottom number ). What if we had an empty box with 9 slots? "Fractionally" we could write this number as 0/9, since all slots, out of the 9 are empty. In other words 0/9 is equal to 0 because we have zero items in our box so far. Looking through our pog collection, we find 6 pogs that we want to box. We could express this as enough pogs to fill 6/9 of the box, since 6 pogs will fill 6 out of the 9 empty slots. Notice in the picture above I stated that the box was 2/3 full instead of 6/9 full. Either answer would have actually been correct, since 2/3 and 6/9 of the box represent the same exact amount of area. How can we tell this? Looking at the box again, imagine what would happen if we took all the dividers out of the box and just divided it into 3 rows. In each of the 3 rows we could put 3 pogs. Since we only have 6 pogs all together, we are only going to be able to fill 2 of the rows. We will have one row left empty. So out of 3 possible rows, only 2 are full. In other words, 2/3 of the box is full. * Compare this picture with the one before * Now let's discuss "improper fractions." Improper fractions simply have a larger numerator, ( top number ), than the denominator, ( bottom number ). Since fractions are "division" problems, this means that when you divide the numerator by the denominator, you're going to get at least one non-fractional number. Think about dividing 4 by 2. 4 divided by 2 equals 2. So how does this relate to fractions? 4/2 is the "improper" fractional expression for 4 divided by 2. In other words, the fraction 4/2 equals 2, just as it did when we divided. What about a fraction like 5/2? The fraction 5/2 will give us a "whole" number and a "remainder." Since 2 will only go into 5 two times, we have 1 left. What do we do with it? Because 5 divided by 2 equals 2 with a "remainder of 1, that 1 "remaining" is still divided by 2. Since 1 won't divide by 2 and leave a "whole" number, we are left with only a "fraction" of a "whole." We must write the result as 1/2, respecting that we still had something "remaining," being divided by 2. If things aren't quite clear at this point, try taking a close look at the example below. Study it carefully, and then we'll see what's going on. We started with an empty box capable of holding 9 items. The problem is that we have 12 items! What will we do? 12 items will not fit into 9 slots. At this point, we can look at the 12 items as the fraction 12/9, with the denominator 9 representing the available slots. Dividing 12 by 9 gives us 1 with a remainder of 3. Since 1 box is capable of holding 9 of the 12 items, this box has 9 out of 9 of its slots filled, ( 9/9 of a full box ). In other words, 1 box is completely full, ( 9/9=1 ). What about the 3 remaining items? We need another box! Fortunately this other box is exactly the same size as the first and has 9 empty slots, ( 0/9 ). We place the "remaining" 3 items into 3 of the 9 empty slots and now have 3/9 of another box filled. So all together we have 1 "whole" box filled plus another 3/9 of a box filled. Remembering the idea of sectioning the box into 3 rows as we did before, we could also think of the 3/9 full box as being 1/3 full. We have determined now that the "improper" fraction 12/9 is actually the same as 12/9 of a full box, ( or 1 and 1/3 of a full box ). Now that we have a fairly good understanding of fractions, let's look at a problem where the denominators are not the same. If you look closely at the example below, you can probably guess the answer is going to be 1 "whole." How did you determine that? Was it because the fractions represented the same area? Remember why they did? The areas are simply cut into different numbers of slices. There is still the same amount of pie in each area. The only difference between the two areas is the number of pieces they are cut into. Try to imagine slicing a pie. The first cut is down the center, giving you 2 pieces. Another cut will give you 4 pieces. Yet another cut will yield 8 pieces. Now think about what would happen if you took away 1/2 of the 8 pieces. Since 4 is half of 8, you would have 4/8 of the pie left. Wait! I thought we took 1/2 of the pie, NOT 4/8! We did. Now are you beginning to better understand the relationship between 4/8 and 1/2? Can you also see that 2/4 and 1/2 represent the same area? Hopefully, now you have a better understanding of fractions and won't let them scare you like some people do. I would recommend looking over the pictures above several more times until you completely understand all the ground we have covered. Remember that fractions are simply "division" problems. Also remember that you can add or subtract numerators, ( top numbers ), just like you normally do as long as the denominators are the same. A fraction of a pie is only a "fraction" of a "whole" pie. Please don't try to make them into anything any harder. Think about it: fractions are just as easy as counting quarters! :-)

QUIZ

 Question 1:1/2 + 1/2 = _____ Answer: A:  1/2 B:  2/4 C:  2/2 D:  Both A and B Question 2:1/4 + 1/4 = _____ Answer: A:  2/8 B:  2/4 C:  2/2 D:  None of the above Question 3:2/3 + 1/3 = _____ Answer: A:  3/3 B:  3/6 C:  1 D:  Both A and C Question 4:2/2 - 1/2 = _____ Answer: A:  1/2 B:  2 C:  1 D:  1/4 Question 5:6/8 - 2/8 = _____ Answer: A:  4 B:  4/8 C:  4/6 D:  4/0 Question 6:9/8 + 1/8 = _____ Answer: A:  10/16 B:  1 and 2/8 C:  10/8 D:  Both B and C Question 7:4/2 - 3/2 = _____ Answer: A:  1/3 B:  1/4 C:  1/2 D:  1/0 Question 8:2/4 + 1/2 = _____ Answer: A:  1 B:  3/4 C:  4/4 D:  Both A and C Question 9:3/2 - 2/4 = _____ Answer: A:  1/4 B:  1/2 C:  4/4 D:  1 and 1/4 Question 10:1/16 + 16/16 = _____ Answer: A:  17/16 B:  1 and 1/16 C:  17/32 D:  Both A and B

~ Other Links ~

Fractions - First Glance

GCSE Maths - Fractions

Fraction Man

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Ask Dr. Math - Fractions

Harcourt E-Lab

Harcourt Math Glossary 