47 Numerator Denominator


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1 JH WEEKLIES ISSUE # Mathematics Fractions Mathematicians often have to deal with numbers that are not whole numbers (1, 2, 3 etc.). The preferred way to represent these partial numbers (rational numbers) is with the fraction. I. The Fraction Before we begin our discussion of fractions, there is some terminology that you should be familiar with: 47 Numerator Denominator The center bar in a vertical fraction such as this is formally called the vinculum (informally, the fraction bar). Common fractions include numbers such as 1 / 2, 3 / 4, and 2 / 5. Simply put, fractions represent division: we divide the numerator into the number of parts specified by the denominator. This means that the denominator in a fraction cannot ever be zero because it is prohibited by mathematics to divide something into zero parts (there is always at least one part to every whole).
2 II. Comparison of Fractions When you have two fractions that do not have the same denominator, it can occasionally be difficult to compare them to figure out which is larger or smaller. One eas y trick involves multiplication. Example II.1: Comparison of Fractions Which fraction is larger: 35 or 1564? While we could go ahead and find the decimal representation (see VI ) to compare the fractions, or we could the multiply comparison method. By multiplying the numerator of the first fraction by the denominator of the second, and the numerator of the second by the denominator of the first, we can compare the fractions Because we know that 192 is greater than 75, we also know that the first fraction 35 is larger than the second fraction The dot that we use between numbers above is another way of representing multiplication, and will be used throughout this document. We could also simplify the fractions or find a common denominator. III. Simplification and Equivalence of Fractions You can have an infinite number of fractions that all represent the same decimal number. Example III.1: Equivalence of Fractions Represent the number 0.5 with a fraction. Of course, the simplest fraction that we could use would be 1 / 2. We could also use 2 /4, or 3 /6, 4 /8, 5 /10, 70 /140, / or an infinite number of other fractions. This multiplicity of fractions can get kind of confusing; what if you have two fractions, but they both look like that last example ( / )? How could you tell that they are equivalent? We can simplify fractions by factoring the numerator and denominators and looking for common factors. For most problems, you will be asked to give your answer in simplified form or in lowest terms.
3 Example III.2: Simplifying Fractions by Factoring Simplify the fraction We can start simplifying this fraction by factoring the numerator into simpler numbers. First, we recognize that 60 is an even number this means that we can factor a 2 out of it. 60=30 2 Again, 30 is an even number, so we can factor a 2 out of it. 60=30 2= We know that 15 is equal to =30 2=15 2 2= All four of these factors are prime numbers, which means that we cannot factor them further. We now turn to the denominator, recognizing similarly that 84 is an even number, so we can factor out a 2. 84=2 42 Similarly: 84=2 42=2 2 21= Reassembling our fraction from these factored components 6084= We know from multiplication that we can factor out those three common factors as separate fractions 6084= = For more on fraction multiplication, see V.3. We know that any number divided by itself is equal to 1, so we have 6084= = = =57 Thus, we have simplified the fraction 6084 to 57. Another way to look at this problem is repeated division of the numerator and denominator to remove common factors: 6084= =3042= =1521=3 53 7=57
4 IV. Proper and Improper Fractions, Mixed Numbers Mathematics states that fractions are proper when the numerator is less than the denominator. Example IV.1: Proper Fractions Proper fractions include fractions like 3 /8, 15 /35, and 2 /5. When the numerator of a fraction is greater than its denominator, then the fraction is called improper. Example IV.2: Improper Fractions Improper fractions include fractions like 4 /3, 7 /2, and 19 /4. Even though they are called improper, improper fractions still represent numbers they simply represent numbers that are greater than 1. In order to make improper fractions acceptable to mathematics, we need to separate the whole number part from the fractional part. Example IV.3: Separating Improper Fractions Separate 133 into a whole number and a fraction. By separating the numerator into added factors, we can rewrite the fraction 133= = For more information on adding fractions, please see V.1. number divided by itself is 1, we can rewrite again Now, because any 133= =4+13 Notice now that we have a whole number and a proper fraction: We can now rewrite our separated improper fraction as a mixed number, which is a whole number and fraction pair. They are often used for cooking (i.e. 2 1 /2 cups of flour).
5 Example IV.4: Building a Mixed Number Using the result from IV.3, create a mixed number. This is very simple; we just remove the plus sign and back the fraction and the whole number up next to eachother. 4+13=413 Note that our resulting mixed number symbolizes adding 4 to 1 / 3 not multiplying them. Alternatively, we could use the division method with remainders to create our mixed number. Example IV.5: Building a Mixed Number by Division Rewrite 13 /3 as a mixed number. We divide 13 by 3 with remainders: 04r13) Thus, we see that 3 goes into 13 four times with 1 remaining. Therefore, our mixed number is 4 1 /3, which agrees with our result of example IV.4. V. Adding, Subtracting, Multiplying, and Dividing Fractions For adding and subtracting, you must have a common denominator. It is not necessary to have a common denominator for multiplying and dividing. 1. Adding Fractions If two fractions have a common denominator, then it is very easy to add them together you simply add the numerators and leave the denominator the same. Example V.1.i: Adding Fractions with a Common Denominator Add the fractions 4 /9 and 3 /9. We proceed as detailed above: 49+39=4+39=79 4 /9 + 3 /9 equals 7 /9. However, if two fractions do not have the same denominator, we have to find equivalent fractions so the denominators are the same. We can go about this by three different methods: a. If one denominator is a factor of another (such as 3 is of 9), then you can easily convert one fraction to the other s denominator.
6 Example V.1.ii: Adding Fractions with Similar Denominators Add the fractions 4 /9 and 1 /3. We recognize that 3 is a factor of 9 (3 3 = 9), so we can multiply 1 /3 by 3 /3 to get a denominator of =39 1 /3 is equivalent to 3 /9. We can perform the multiplication above due to the multiplicative identity property (any number multiplied by 1 is itself), and the fact that any number divided by itself is equal to 1. Alternatively, the least common multiple of 3 and 9 is 9 please see V.1.c for more information. The problem then becomes adding 4 / 9 and 3 / 9, which we have already done in example V.1.i. b. We can multiply each fraction by an equivalent of 1 based on the other s denominator (see example) and then simplify; this trick is typically good when the denominators are less than 15. Example V.1.iii: Adding Fractions with Different Denominators Add the fractions 6 / 7 and 2 / 3, giving your answer as a simplified improper fraction. We proceed by multiplying 6 /7 by 3 /3, and 2 /3 by 7 /7: 67 33=1821; 23 77=1421 Both fractions now have the denominator of 21, which is the least common multiple ( V.1.c). We can proceed: =3221 As it turns out, there are no common factors between 32 and 21 (besides 1), and we are asked to leave our answer as an improper fraction. We are done: 6 /7 + 2 /3 = 32 /21. c. We can find the least common multiple (LCM) of the denominators, which is the smallest whole number of which both denominators are divisors. This can be done in two different ways: the simpler is presented here, the more complex in the appendix
7 because it requires more explanation. The LCM methods take the longest, but are more useful when you are adding more than two fractions or when the denominators are larger numbers. To find the LCM, list multiples of the denominators, looking for the smallest number that is common between the two lists. That will be your common denominator. Start by listing a few of the larger denominator s multiples. Be careful not to confuse multiples with factors ; factors are component parts of a number where as multiples are where you take the number and add it to itself (see example). It may help you to remember that multiple sounds like multiply. Example V.1.iv: Adding Fractions by LCM Add 1 /2 and 4 /23. We start by listing multiples of each of the denominators, looking for the LCM: Multiples of 2: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50 Multiples of 23: 23, 46, 69 The lowest common multiple is 46. Thus, we want to convert our fractions so they have a common denominator of 46. Making the conversions, we have = =3146 Coincidently, we would have come to this result directly by the multiplication method listed in (b) above.
8 2. Subtracting Fractions When subtracting fractions, all of the logic applied above in the adding fractions section still applies: we can only subtract fractions that have a common denominator. The only wrinkle in subtracting fractions is that we can have negative answers. Example V.2.i: Subtracting Fractions Perform the operation 6 /7 3 /5, giving your answer in lowest terms. First, we use the techniques listed in V.1 to convert our fractions so they have common denominators. 67=3035; 35=2135 Now we perform the operation 6735= = =935 Our answer ( 9 / 35 ) is as simplified as we can make it it is in lowest terms so we are done. Example V.2.ii: Subtracting Fractions Perform the operation 3 /4 8 /9, giving your answer in lowest terms. Again, we use the techniques in V.1 to convert our fractions so they have common denominators: 34=2736; 89=3236 We now perform the operation 4489= = =536 Our answer ( 5 /36) is in lowest terms, so we are done. Example V.2.iii: Subtracting Fractions Perform the operation 11 /15 2 /5, giving your answer in simplified form. We begin by finding equivalent fractions with the same denominator for both fractions: 1115=1115; 25=615 We can now perform the operation =515
9 Now we simplify our result to put it in lowest terms. 515=13 3. Multiplying Fractions The result is now in simplified form: 11 /15 2 /5 = 1 /3 Multiplying is one of the easiest operations to perform on fractions because you can do as much or as little simplification as you would like at the beginning to reduce the numbers you need to carry. This is accomplished by factoring and canceling numbers out of the numerators and denominators of your participating fractions. Otherwise, you simply multiply numerators and multiply denominators, then simplify. Example V.3.i: Multiplying Fractions Perform the operation 3 /4 2 /3, giving your answer in lowest terms. We proceed with straight multiplication =3 24 3=612 Simplifying by pulling out a factor of 6 from both the numerator and the denominator, we get 612=66 12=1 12=12 Our answer is 1 /2. Alternatively, we could have performed the simplification ahead of time by canceling a 2 from the denominator of 3 / 4 with the numerator of 2 / 3, and a 3 from the numerator of 3 /4 and the denominator of 2 /3: 34 23= = =12 Again, our answer is 1 /2. Example V.3.ii: Multiplying Fractions Perform the operation 168 / /216, giving your answer in lowest terms. We begin by simplifying each of the two fractions: = =3 78 7=38
10 24216= =33 9=19 We can further simplify the fractions when we multiply them together: 3819= =124 Our answer, in lowest terms, is  1 / Dividing Fractions It is similarly easy to divide fractions because of a very handy trick. Division of fraction A by fraction B is directly equivalent to the multiplication of fraction A and the reciprocal of fraction B. Once you have your new multiplication pair, simply proceed as with multiplication. To get the reciprocal of a fraction, you simply swap the numerator and the denominator. Therefore, the reciprocal of 2 / 5 is 5 / 2. Please note that if you are dividing by a fraction where the numerator is 0, then you cannot perform the operation because the reciprocal of that fraction would be dividing by zero. Example V.4.i: Dividing Fractions Perform the operation 3 /7 30 /21, giving your answer in lowest terms Formally, the operation calls for this: Such a fraction is in the form of a complex fraction, and is directly equivalent to 3 /7 30 /21. But because we have our reciprocal trick, we know the following: = By simplifying, we find that our problem reduces to = =310 Our answer 3 /10 is in lowest terms, so we are finished. VI. Converting Fractions to Decimals T he most common method for converting the fraction into a decimal is to go through long division, dividing the numerator by the denominator. We can do this because a fraction is nothing more than a division equation: Example VI.1: Fractions to Decimals by Long Division Convert 35 to decimal form.
11 Because 35=3 5, we begin by setting up a division form. 5)3 5 does not go evenly into 3, so we add a decimal zero. 0.5)3.0 5 does evenly go into 30, so we proceed with our normal division: 0.65) The decimal representation of 35 is 0.6 This works fine; it can be slightly tedious and eat up a lot of time, however, especially for more complex fractions. Luckily, there are several tricks that can be used to speed up your computation if the denominator can be easily manipulated. If you can manipulate the fraction so the denominator is a power of 10, then you can easily compute the decimal from the fraction by simply moving around the decimal point in the numerator. Example VI.2: Fractions to Decimals by Manipulation Convert 35 to decimal form. This time, we recognize that the denominator of the fraction (5) is exactly half of 10. This means that if we multiply both the denominator and the numerator by 2, we can get a fraction where the denominator is =3 25 2=610 We are allowed to do this because of the multiplicative property any number times 1 is equal to that number and because 2 / 2 is equal to 1. For more on fraction multiplication, please see V.3. Now that we have our fraction with a denominator of 10, we know from division that we can simply move the decimal point in the numerator one place to the left to represent division by 10. Therefore, our answer is 0.6, which agrees with our answer by long division. Similarly, if we were to get our fraction in terms of 100, we would move the decimal point in the numerator twice over to the left. Further, if you can manipulate your fraction into a fraction that is recognizable, the decimals are easy to compute. For this purpose, we recommend memorizing a table of common decimal values (included in the appendix) after all, if you can reduce a fraction to 5 / 8 but do not remember the decimal equivalent, you will be stuck computing the decimal by long division (for reference, 5 /8 = 0.625).
12 VII. Converting Decimals to Fractions 1. Terminating Decimals to Fractions When a decimal value has a specific number of decimal places ( i.e , 0.5), then it is very easy to convert the decimal value into a fraction. Simply count up the number of decimal places ( i.e has 3, 0.25 has 2), multiply your decimal by a one followed by that many zeros (for 3 decimal places, use 1000; for 2 decimal places, use 100); this is your numerator. Your denominator is the power of ten that you just multiplied with. Simplify your fraction, and you are done. Example VII.1.i: Converting Terminating Decimals to Fractions Convert to a fraction in lowest terms. The decimal value has three decimal places, so we multiply by =525 Therefore, 525 is our numerator. Our denominator is simply the number we just multiplied with, Ergo, our unsimplified fraction is: Simplifying using techniques in this document, we find that our final fraction is 2140; 0.525= =2140. Terminating decimals are known as rational numbers ; they can be represented in a fractional form. 2. Repeating Decimals to Fractions When a decimal value has an infinite number of decimal places typically represented by an ellipses ( ) it is possible to find a fractional representation if the decimals repeat (like or ). Such repeating decimals are occasionally represented by a vinculum over repeating decimals, such as 0.23= or = The technique for finding the fractional representation of repeating decimals formally goes through some algebra; here, we present the technique in a similar manner to finding the representation of a terminating decimal. Cut your repeating decimal off when the digits begin to repeat. Then count up the number of repeating decimal values and multiply the repeating value by a one followed by that many zeros; this is your numerator. Your denominator is the power of ten that you just multiplied with minus one. Simplify your fraction, and you are done. This technique takes advantage of the repeating property of nines.
13 Example VII.2.i: Converting Repeating Decimals to Fractions Convert 0.3 to a fraction in lowest terms. We begin by identifying the repeating element, which in this case is 3. This one value repeats an infinite number of times. Therefore, we cut off our decimal at 0.3 and multiply 0.3 by =3 This 3 is our numerator. Our denominator, therefore, is 10 1, or 9. We assemble our fraction and then simplify: 39=13 Therefore, 1 /3 is equal to Example VII.2.ii: Converting Repeating Decimals to Fractions Convert to a fraction in lowest terms. We begin by identifying the repeating element 265. There are three repeating values in this decimal. Therefore, we cut off our decimal at 0.265, and multiply by 1000: =265 This 265 is our numerator. Our denominator, therefore, is , or 999. Then, we simplify Turns out that 265 / 999 is a fraction in lowest terms. Our work is done : = Like terminating decimals, repeating decimals are known as rational numbers ; they can be represented in a fractional form. 3. NonRepeating NonTerminating Decimal Values When a decimal value never repeats and has an infinite number of places, then that number is called irrational it cannot be represented as a fraction. Common irrational numbers include 2, π, and e.
14 EXERCISES Exercise I: Comparison of Fractions Given each set of fractions, rank them from greatest to least , 35, , 34, 49, , 13, , 38, 1112, 37, 33, 910 Exercise II: Simplification of Fractions Simplify each of the following fractions to lowest terms Exercise III: Proper and Improper Fractions, Mixed Numbers For the following set of fractions, identify those that are proper and those that are improper. If the fraction is improper, convert it to a mixed number Exercise IV: Operations on Fractions Perform the following operations, giving your answer in improper reduced form =? =? =? =? =? =? =? =? Exercise V: Fractions to Decimals Convert the following fractions into decimal form Exercise VI: Decimals to Fractions Convert the following decimal numbers into fractions ANSWERS TO EXERCISES
15 Exercise I 1. 35,1123, , 12, , 34, 49, , 33, 1112, 910, 38 Exercise II Exercise III 1. proper proper Exercise IV Exercise V Exercise VI
16 APPENDIX (1/2) Least Common Multiple Method 2 This method for finding the LCM is very useful when numbers start to get large or when you are dealing with more than two fractions. 1. Factor the denominators of the subject fractions down to prime factors. 2. Count the number of times each prime factor appears in the factorizations. a. If a factor appears in more than one denominator s factorization, the count should be for the denominator where the factor appears the most. b. For example, if the factor 2 appears three times in the factorization of A and twice in the factorization of B, then the count for the factor 2 is three. 3. Multiply all of the unique factors that appeared in the factorizations together; each unique factor should be multiplied in the number of the count for that factor. 4. This resulting number is your least common multiple. Proceed with your operation. It is helpful to look at an example. Example Appendix: LCM by Prime Factors Find the least common multiple between 16, 17, and 18. We need to find the prime factorization of the denominators 6, 7, and 8. 6=2 3 7=7 8=2 2 2 The counts for each of the prime factors are 2: three times, 3: one time, 7: one time. Therefore, the LCM is =168.
17 Table of Fractions, Decimals, and Percentages * APPENDIX CONTINUED (2/2) Fraction Decimal Percent 1/ % 1/ % 2/ % 1/ % 3/ % 1/ % 2/ % 3/ % 4/ % 1/ % 5/ % 1/ % 3/ % 5/ % 7/ % 1/ % 2/ % 4/ % 5/ % 7/ % 8/ % 1/ % 3/ % 7/ % 9/ % 1/ % 1/ % *We know that percentages were not covered in this issue of Weeklies, but felt it prudent to include them for reference. You can convert a decimal to a percentage by moving the decimal point twice to the left. Further, this table is incomplete these are the fractiondecimal conversions that we recommend that you memorize. It is always helpful to have others as well.
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