CHAPTER # : Fractions Definition The word fraction comes from the Latin word meaning to break into pieces. If you divided an apple pie up between you and one other person equally, there would be pieces of the pie. And if you eat one of the pieces then you have eaten of pieces of pie. As a fraction we would write that you ate or ½ of the pie. There are three parts to a fraction. The top number is called the numerator. The bottom number is called the denominator. The line that separates the numerator from the denominator is called the fraction bar or division bar. Apple Pie is the official state pie of Vermont. YUM! Let s go back to our example of the pie that you ate. The is the numerator and that tells us how many of the pieces you ate. The is the denominator and it tells us how many total pieces were available to be eaten. So fractions allow us to compare the number of pieces to the total number of pieces available. FIGURE. shows a pie divided into 0 pieces. Three of those 0 pieces are colored red. So 0 is the fraction that represents how many pieces are red while 0 is the fraction that represents how many pieces are not red. FIGURE.
Now, take a look at the EXAMPLE. and see if you can come up with two fractions that represent the pie being shown. EXAMPLE. Write two fractions for the following pie chart. One fraction needs to represent the number of green pieces while the other needs to represent the number of white pieces. Our numerator for the first fraction needs to represent the number of green pieces which is and our denominator needs to represent the total number of pieces which is. So would be the first fraction. The numerator of the second fraction needs to represent the number of white pieces which is and our denominator needs to be for the total number of pieces. So would be the second fraction. EXAMPLE. Write a fraction that represents how many girls are present when there are 0 boys present in a group of people. Since there are people total this will be our denominator. To find the number of girls you need to subtract 0 from to get girls and this will be our numerator. So is our final answer. If you see a fraction in a book or something else that you are reading you need to pronounce the fraction in a certain way. For 0 we would say three-tenths and for we would say three-fourths. In general, we would add a -ths to the end of the denominator. There are some denominators that do not follow this pattern. Denominators of and have a different word that is used when we say the fraction. For we would say three-halves while for we would say two-thirds.
Equal Fractions Take a look at the following FIGURE. You should see two different fractions represented. FIGURE. Even though these two fractions of pies look different they are actually equal. So, even though 6 they have different numerators and denominators these two fractions are equal. Being able to recognize equal fractions will make working with fractions a lot easier on you. So why is? They are equal because they represent the same amount of a whole. 6 FIGURE. shows this where the bold lines indicate the pie divided into thirds and the thinner lines indicate a pie divided into sixths. FIGURE. Mathematically, we can see that these two fractions are equal because the same number was multiplied to the numerator and the denominator. 6
As long as you multiply the same number to both the numerator and denominator you will not change the value of the fraction. The fraction will look different but it represents the same part as the other fraction. This next paragraph is going to be a preview of the next section of text. So if this paragraph does not make sense to you right now, go ahead and work through the next section and then come back to this paragraph. If we take a closer look at the previous equivalent fraction statement we will see that we are really only multiplying by one. Remember that when one is multiplied to anything the value does not change. Here s that relationship again. We can separate out the middle fraction as a multiplication of two fractions. 6 The fraction so we are only multiplying the fraction one-third by one. This is why the two fractions are equal. Multiplication of Fractions Scientists use a great deal of math in their research and study of the world in which we live. One of the most common math operations performed with fraction is multiplying. Let s say we wanted to multiply / and / together. Another way to say the exact same problem would be to find the product of / and /. A third way to say the exact same problem would be to take / of /. Anytime you see the word of in a math problem you can automatically start to think of multiplication. So taking / of / we are essentially taking a fraction of a fraction. FIGURE. shows a picture representing what happens when one-half is multiplied by four-fifths. FIGURE. X
When the / fraction is overlaid on the / fraction we can see that the common area of overlapping is the purple squares. So since there are purple squares out of a total of 0 then our new fraction would be /0. In equation form, FIGURE. would be represented by this equation. Remember, red and blue make purple! The general rule when multiplying fractions is to multiply the numerators together and then to multiply the denominators together. So in solving the above multiplication problem we would actually multiply like this. 0 0 a. What is the product of / and /? EXAMPLE. Remember to multiply the numerators and then to multiply the denominators together. b. What is /8 of /? When the word of appears in a math problem we can assume that we are dealing with multiplication. So we will multiply the numerators and denominators together. 8 8 6 Sometimes we can make the multiplication simpler by canceling numbers that appear in the numerator and the denominator within the same problem. If we multiply these two fractions together five s will cancel each other out. / / But not only can we cancel the same number but we can cancel multiples of one another. Let s look at this problem again
One of the nice things about multiplying is that we can switch the order of multiplying without changing the final value. For example, x is the same as x. If I do that to the above problem I will get the following fraction. 0 So all I did was switch around the four and the one. In order to show you something I am going to separate out that last fraction into a normal multiplication problem again. / / We can cancel the four and the two because they are multiples of one another. The answer of is equal to 0. So when you are multiplying fractions look for opportunities to cancel numbers and it will simplify the problem for you. A lot of fraction work in science deals with increasing or decreasing an amount of a measurement. Let s take a look at this EXAMPLE here. SCIENCE EXAMPLE. A scientist calculates that the depth of concrete used as a base for constructing a tower needs to only be seven-tenths of the original measurement of 0 ft. What is the new measurement of the depth of the base? To solve this problem we need to multiply 0 by seven-tenths. If it makes more sense to you to turn 0 into a fraction then go ahead and do so by making it 0/. Then we can multiply the numerators and denominators together. 0 0 0 We can cancel like terms by canceling 0 and the 0 leaving a in the numerator. 0 0 ft deep 0
Division of Fractions When dividing fractions there is rule you must follow. But before we actually get to the rule, let s look at an EXAMPLE of dividing fractions that have common denominators. Let s look at this 9 problem? 8 8 With words we would say nine-eighths divided by three-eighths. Another way to say this would be to say how many three-eighths would fit into nine-eighths. Since the denominators are the same then all we really only need to ask is how many threes fit into nine. The answer would be three because ++ 9. Well, the same works for fractions. Three fractions of three-eighths can fit into nine-eights so the answer is. Imagine having a pie divided up into eight pieces like FIGURE.. FIGURE. FIGURE.6 shows a diagram of this division problem. Now we want to know how many of those red pieces (/8) would we need to make 9/8. Well, another /8 piece (marked in blue) would make 6/8 total and then another /8 piece (marked in green) would make our 9/8. FIGURE.6
As long as the denominators are the same then we can do the math with just the numerators. The harder problems are when the denominators don t match. Well, now it s time to introduce the rule we will follow when dividing fractions. Dividing fractions is accomplished by multiplying by the reciprocal. What do we mean by reciprocal? Most people say flip the fraction upside down. Well, we aren t literally flipping the fraction upside down. But what you will do is switch the numerator and denominator. So /8 will become 8/ and that is how we take the reciprocal of a fraction. Here are some EXAMPLEs of dividing with fractions. 9 a. What is? 8 8 EXAMPLE. To solve this problem, we take the reciprocal of /8 which is 8/ and multiply that to the 9/8. 9 9 8/ 9 8 8 8/ b. What is? We need to find the reciprocal of / which is /. Now we can do the multiplying. / / Addition/Subtraction of Fractions The rules that you just learned for multiplication and division need to stay with those respective operations. Undoubtedly, through your high school and quite possibly your college years there will come a time when you try to apply the rules for multiplication and division of fractions to an addition or subtraction problem. But I cannot keep you from making those mistakes; however, I can show you how these rules do not apply to addition or subtraction with the following scenario and figure.
Try to visualize adding one-fourth of a pie to another pie pan that contains one-half of a pie. What do you see? Hopefully, you see the same thing that is in the FIGURE.. FIGURE. So one-fourth added to one-half equals three-fourths. Let s look at this problem in equation form. + But looking at only the left side of the equation, doesn t it look like the answer should be two-sixths? + + + You see, what I just did there was apply the rule for multiplication to an addition problem. I just showed you graphically what the answer would be but most of the time we do not take the time to visualize the answers to math problems. So please take the time to make sure your answers seem correct before going on the next problem. The rule for addition and subtraction of fractions is that you can only add fractions together that have the same denominators. Take a look at the following example problems. a. +? EXAMPLE. To solve this problem we are going to add the numerators together and then keep the denominator the same. + + b.? 8 8 6
To solve this problem we are going to subtract the numerators and then keep the denominator the same. 8 8 8 8 Now, what if the fractions do not have the same denominator? Can we still add or subtract them? Sure, we can but we have to get the denominators the same first. We are going to do this by finding the least common multiple of each denominator. Let s use the first addition problem of this section as our example again. We took one-fourth and added one-half together. + In order to add these two fractions together we need for them to have the same denominators. So looking at and, the least common multiple would be. So we want the second fraction to have a denominator of four.? Now all we need to do is find the numerator of our new fraction. Well, to get to four it looks like we multiplied the original denominator by. Anything you do to the denominator you have to do to the numerator as well that way the fraction will be equal to the original. What we are really doing is the opposite of reducing. So now we can replace the one-half with our new fraction that has a common denominator. And we can now finish the addition of these two fractions. + + + + + That is where the three-fourths came from in FIGURE. that you hopefully visualized at the beginning of this section. Let s take a look at few more examples of adding fractions with unequal denominators.
EXAMPLE.6 a. +? 9 Our first step is to find a common denominator. Since will go into 9 three times then we will choose 9 as the common denominator. Now we need to find the fraction that is equivalent to twothirds but has nine as the denominator.? 9 Since we multiplied the denominator by then we must do the same to the numerator. To solve this problem we are going to substitute six-ninths in for two-thirds and add the numerators together and then keep the denominator the same. 6 9 + 9 6 + 9 6 9 9 b.? 8 6 Our first step again is to find a common denominator. Since 8 will go into 6 two times then we should choose 6 as our common denominator. So now we need to find a fraction that is equivalent to seven-eighths but has 6 as the denominator.? 8 6 We had to multiply the 8 by to get 6 so we need to do the same to the numerator. 8 8 To solve this problem we will substitute this equivalent fraction into the problem and subtract the numerators and then keep the denominator the same. 6 9 6 6 6 6
Notice in the problems we just worked in EXAMPLE.6 that we only had to change one of the fractions in order to get a common denominator. Well, sometimes you just may have to change both fractions. For example, imagine if the denominators of two fractions we were adding were and. Two does not go into three and three does not go into two. So what are we going to do? Well, the math skill we are using here is called the least common multiple. Numbers do not have to be multiples of each other to have a least common multiple. We just need to find the smallest number that AND will go into. A number that comes to mind for this would be 6. Two goes into six times while three goes into six times. Anytime you have denominators of and you can use 6 as the least common multiple. Let s solve a couple more of these harder problems. a. +? EXAMPLE. Since and are not the same then we need to find a common denominator before we can add. Also, you should notice that does not go into and does not go into evenly. This means we are going to have to change both fractions to a completely different common denominator. and will both go into but we should take a moment to determine if this is the least common multiple. The only other multiple of that is smaller than is and will not go into so we are safe to choose as our least common multiple. So these are the two fractions we are now allowed to add. b.? 6 9 Remember: Prime numbers have no factors. + + 9 We need the least common multiple of 6 and 9. 6 and 9 both go into so we could use that as our denominator but we need to check to make sure that it is the least common multiple of these two numbers. To double-check we should run through the multiples of 9 to see if any of them are also multiples of 6. First, check 9x which is 8. Is 8 a multiple of 6? Yes, it is and it is smaller than. So multiplying the two denominators together may not always give you the LEAST common multiple. We will use 8 as our new denominator.
6 6 8 9 9 8 Here are the two fractions we will now subtract in order to get the answer to the problem. 8 8 8 8