Introduction to Fractions A fraction is a quantity defined by a numerator and a denominator. For example, in the fraction ½, the numerator is 1 and the denominator is 2. The denominator designates how many equally sized pieces each whole unit is divided into. Thus, the denominator dictates the piece size. The numerator designates how many of those pieces you have to make up your quantity. Example: 2/6 of a pie eaten. In the figure below, we see that our pie is cut up into 6 equal pieces since the denominator is 6. We choose 2 of those pieces since the numerator is 2. CONTINUED ON NEXT PAGE!
Example: There are 8 people that each want a piece of pie. They all want pieces that are 1/6 th of a pie in size. What fraction is this and how can we illustrate the total amount of pie eaten? What we want here is a fraction with a piece size of 6, so the denominator is 6. Since we want 8 pieces, the numerator is 8. This makes a fraction of 8/6. But as you can see below, 8/6 = 1 + 2/6, which we write as 1 2/6. Also note that since 2/6 is a third of a pie, our total amount may be written as 1 1/3 Example: What fraction of a pie has been eaten in the illustration below? Here, the pie is divided into 9 equal pieces. 6 of the pieces have been eaten, so 6/9 of a pie has been eaten. As you can see, 6/9 makes up 2/3 of the pie eaten. Continued on Next Page!
Reducing Fractions As you saw in the previous example, the fraction 2/6 reduced to what appeared to be 1/3. We could show this is true by dividing both the numerator and the denominator by 2. In the other pie example, 6/9 appeared to reduce to 2/3. We can show this is true by dividing 6 by 3 to get 2 and also dividing 9 by 3 to get 3. In general, a fraction is reduced by dividing both the numerator and the denominator by the same number. Example: Reduce 12/48 as much as possible. If we divide both 12 and 48 by 2, we get the fraction 6/24. If we divide both 6 and 24 by 2, we get 3/12. If we then divide both 3 and 12 by 3, we get ¼. We would say that 12/48. reduced to lowest terms, is equal to ¼. Notice that we could have divided both 12 and 48 by 12 and gotten our completely reduced answer in one step? To reduce a fraction to lowest terms in one step, divide both numerator and denominator by the Greatest Common Factor of the numerator and denominator. Example: Reduce 100/200 to lowest terms by dividing numerator and denominator by the greatest common factor. The greatest common factor of 100 and 200 will be 100, since 100 divides into both numbers evenly and there is no other common factor that is bigger. If we divide both 100 and 200 by 100, we get a result of ½. CONTINUED ON NEXT PAGE
Proper Fractions, Improper Fractions, and Mixed Numbers A proper fraction is one where the numerator is less than the denominator. For example, ¾ is a proper fraction. Proper fractions of positive numbers always have a value less than 1. An improper fraction is one where the numerator is greater than or equal to the denominator. Improper fractions of positive numbers always have a value of 1 or more. In the second pie example, 8/6 was an improper fraction. And we showed that 8/6 was equal to one whole pie (6/6) added to 2/6 of a second pie. A mixed number is the result of expressing an improper fraction in an equivalent manner as a whole number added to a proper fraction. In the second pie example, we were able to express 8/6 as 1 added to 2/6. We write this sum of 1 + 2/6 without the plus sign as 1 2/6. To convert an improper fraction to a mixed number, divide the numerator by the denominator. The quotient (without the remainder) will equal the whole number part. The remainder will equal the numerator of the proper fraction part. Example: Convert 30/4 to a mixed number and reduce. We divide out 30 by 4, as shown below. The quotient is 7 and remainder is 2 so we conclude that 30/4 = 7 2/4. Furthermore, since 2/4 reduces to ½, we can reduce our answer to 7 ½. This process is illustrated below. Each 4 pieces represent one whole unit. We want to know how many whole units there are so we divide the total number of pieces, 30, by 4 to find that there are 7 whole units. The remainder tells us how many leftover pieces there are. Continued on Next Page
Example: Write 110/3 as a mixed number. 110 3 = 36 r2 so 110/3 = 36 2/3 Example: Write 40/4 as a mixed number. When we divide 40 by 4 we get 10 with zero remainder. This means that 40/4 is equal to 10. Also, if we reduce 40/4 by dividing both 40 and 4 by 4, we get 10/1 which is equal to 10. To convert a mixed number back into an improper fraction, we would multiply our whole number part by the denominator and then add the numerator of the fraction part. This sum would go over the denominator of the fraction part. Example: Write 6 ¼ as an improper fraction. 6 x 4 = 24, 24 + 1 = 25, Answer is 25/4 We would multiply the whole number 6 by 4 (the piece size) to get 24 (fourths) and then we would add 1 to this to get 25 (fourths). This is shown below. CONTINUED ON NEXT PAGE
Rewriting Fractions and Comparing Fractions In order to compare fractions or add or subtract fractions, we must have the same size pieces. This means the fractions must have the same denominators. Example: How much pie do we get when we add 1/3 of a pie to 1/2 of a pie? These are not the same piece size. And if we were to simply add the numerators together and the denominators together to get 2/5, we would have an answer that is less than half of a pie! We must rewrite these two fractions with common denominators. What we can do here, is take the pie that is cut in 3 equal pieces and divide each of these pieces in half to make 1/6 size pieces. We would then have 2 out of 6 equally sized pieces. We could also take the pie that is cut in two and divide each of these in thirds to make 1/6 size pieces. We would then have 3 out of 6 equally sized pieces. We then would add the 2 sixth-size pieces to the 3 sixth-size pieces to get 5 sixth-size pieces, or 5/6. This is shown below. CONTINUED ON NEXT PAGE
Rewriting Fractions With Common Denominators In the previous example, we changed the problem 1/3 + 1/2 into the problem 2/6 + 3/6 by subdividing the pie pieces so we had the same size piece in each pie. In one fraction, we doubled numerator and denominator and in the other fraction, we tripled each numerator and denominator. Mathematically, we did this: In general, you may build a fraction into an equivalent fraction with a larger denominator by multiplying both its numerator and denominator by the same number. Example: Write each of the fractions below with the common denominator of 60. Then list the fractions from smallest to largest. 11/20, 18/30, 8/15 Since 60 divided by 20 = 3, we multiply numerator & denominator of 11/20 by 3. Since 60 divided by 30 = 2, we multiply numerator & denominator of 18/30 by 2. Since 60 divided by 15 = 4, we multiply numerator & denominator of 8/15 by 4. After writing all with the same denominator of 60, we can compare the same size pieces and conclude that 8/15 is the smallest and 18/30 is the largest.