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Fractions Review Name:______________________________________ Equivalent Fractions: to create equivalent fractions, just multiply the numerator (top) and the denominator (bottom) by the same number. WHY? Example: 1 3 3 1 , 3 3 9 3 , 1 3 All three of these answers and one third are equivalent fractions! Try: Write three equivalent fractions for each of the fractions below. 1 4 2 5 3 8 Reducing Fractions: You should reduce all fraction answers as much as possible. Reducing fractions means that the top and the bottom don’t share any factors. One way to do this is to completely factor the top and bottom then cross off any matches. Example: 6 3 2 because there is a 2 on the top and the bottom we can cancel it out. WHY? 10 5 2 We’re left with 3 6 which is the reduced form of . After you do this for awhile you can start to see a 5 10 shortcut of just dividing the top and bottom of the fraction by their shared factor. Try to reduce the following fractions. If you need to list all the factors, do it then cross off any matches. 8 10 30 35 24 90 40 72 Challenge: 2940 3150 Fractions Review Name:______________________________________ Mixed Fractions vs. Improper Fractions: Both of these fractions are seen in math. You should be able to between the two, however, improper fractions are typically more useful and easiest to deal with. So we’re going to focus on how to change mixed fractions to improper ones. What we mean by mixed fraction: a mixed fraction is a whole number and a fraction together—like 2 12 . It means you have 2 wholes and an additional half. What we mean by improper fraction: an improper fraction is a fraction where the numerator is larger than the denominator. For example, 15 . 2 It is typically easier to do the four arithmetic operations (adding, subtracting, multiplying, and dividing) with improper fractions. So, how do you changed mixed fractions to improper ones? Multiply the denominator by the whole number, then add the numerator. The answer is your numerator and the denominator is what it was in the mixed number. Example: Change 2 12 to an improper fraction. Denominator = 2, Whole number = 2, Numerator = 1. 5 2 x 2 = 4, 4+1 = 5, so 2 12 2 Try to convert the following mixed fractions to improper fractions: 2 163 6 52 **For the rest of this review, we’ll assume that you should change any mixed fractions to improper fractions before you do any operations.** Multiplying Fractions: This is the easiest operation to do with fractions. You just multiply the two numerators together and the two denominators together. You should also reduce your final answer. Examples: 2 4 2 4 8 Can we reduce? What if you aren’t sure, what can you do? 3 15 3 15 45 2 18 36 Can we reduce? 9 4 36 Try to multiply the following fractions: 4 5 21 12 6 3 7 8 Sometimes we can take a shortcut if you have larger numbers: 3 21 7 17 Fractions Review Name:______________________________________ Dividing Fractions: Dividing is also not that bad because you don’t need to have common denominators. You just need to know that dividing by a number is the same thing as multiplying by the reciprocal. (An easy example to think about is asking how many halves are there in 4? Is the same as doing 4 times 2. One half and 2 are reciprocals.) Any time you are doing a division problem that has fractions in it, just change it to a multiplication problem, then it’s not that bad. Example: 5 6 5 3 9 3 4 12 Can we reduce this answer? 5 4 5 9 45 5 6 5 1 5 1 6 5 Try dividing the following fractions: 2 3 3 8 5 10 7 11 Common Denominators: Now that we’ve dealt with the “easy” operations to do with fractions we need to talk about why adding and subtracting are harder—they require that the fractions being added or subtracted have common (the same) denominators. Why? We already know how to find equivalent fractions, now we need to focus on how to pick the denominator we want to be the “common” one. Example: What common denominator should we use to be able to compare 2 3 and ? Looking at the 3 4 denominators of the fractions, you need to ask yourself what number has factors of 3 and 4? 12 has both 3 and 4 as a factor. (What if you don’t find the smallest common denominator? That’s ok, you’ll just have more reducing to do at the end.) So we’re going to make equivalent fractions for each that have 12 as a denominator: 2 , what do we need to multiply the denominator by to get 12? Remember that to get an equivalent 3 2 4 8 fraction we have to multiply the top and the bottom by the same number. So, . Is it ok that it’s 3 4 12 For not reduced right now? YES!!! 3 , what do we need to multiply the denominator by to get 12? Remember that to get an equivalent 4 3 3 9 fraction we have to multiply the top and the bottom by the same number. So, . 4 3 12 For Try to make the following pairs of fractions have common denominators. (Do not reduce your answers!) 1 2 and 4 5 5 3 and 12 4 Fractions Review Name:______________________________________ Adding Fractions: When we want to add fractions, we are really trying to add pieces of whole numbers together. We need the pieces to be of the same size to be able to put them together in a meaningful way. That’s why we need to make sure that before we add fractions together, we have common denominators. 2 3 . First we need them to have common denominators. We just 3 4 8 9 . did that on the previous page. Once we change them to have common denominators we have 12 12 17 So we have 8 12ths and 9 12ths, how many total 12ths do we have? . 12 Examples: Let’s say we want to do You do NOT add the denominators! Now, let’s try 7 1 . First, what common denominator should we use? What number has 8 and 6 as a 8 6 factor? 48 has both as a factor, but you might know that 24 also has both as a factor. It’s fine to use either one, but using the smaller one means less reducing later. Change both fractions so they have 24 as a denominator: 21 4 25 . Now add the numerators to get an answer of . Can you reduce? 24 24 24 Subtracting Fractions: Just like adding fractions, subtracting fractions requires that you have common denominators. Once you have common denominators, simply subtract the numerators and place that result over the common denominator. Example: 5 7 First we need a common denominator. What number has factors of 6 and 10? 60 6 10 does. Anything smaller? 30 also does—we can use either as long as you make sure you reduce your 5 we need to multiply the top and bottom by 5, 6 25 7 21 making it . To change we need to multiply the top and bottom by 3, making it . 10 30 30 25 21 . How many 30ths are there if we take 25 of them and subtract 21? So, now the problem is 30 30 4 2 There are left after the subtraction. Can we reduce? YES! What is the final answer? . 15 30 answer as much as possible. We’ll use 30. To change Try to find the most reduced answer possible for the following problems: 7 9 15 20 4 5 21 12 9 7 20 15 1 2 4 7