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Fraction Basics Simplifying Fractions Knowing your multiplication facts and/or divisibility rules helps when figuring out what both the numerator and denominator of a fraction have in common and then reducing it to lowest terms. Example 1: 24 24 ÷ 2 12 12 ÷ 2 6 6÷2 3 = = = = = = 56 56 ÷ 2 28 28 ÷ 2 14 14 ÷ 2 7 24 24 ÷ 8 3 Example 2: = = € € 56 €56 ÷ 8€ 7 € € € Notice that the second example used knowledge of the multiplication facts to simplify in less steps than the first example but that both € €the fraction € answers are the same. Equivalent Fractions To find the missing number, first figure out what the given numerator or denominator was multiplied by to get its corresponding numerator or denominator, then multiply that number to get the missing number. Example: 5 5×6 5 30 = = = 12 72 12 × 6 72 12 72 Always remember that whatever you multiply the top (numerator) by you must€multiply € the€bottom €(denominator) € € by! Prime Factorization Use a factor tree to break down a number into its prime numbers, then rewrite the prime factorization using exponents in ascending order. Example: 90 = 2 × 32 × 5 90 9 3 10 3 2 The exponent tells how many of the base number are multiplied together. 3 × 3 = 32 5 × 5 × 5 = 53 2 × 2 × 2 × 2 × 2 = 25 5 Writing Improper Fractions as Mixed Numbers What is nice is that the fraction tells you how to do this as it is another form for writing a division problem. Example: €25 6 4 25 25 1 6) 25 6) 25 =4 6 6 6 -24 1 € numerator € divided by denominator € Notice that the mixed number is written with the quotient as the whole number and with the remainder over the divisor as the fraction. € Writing Mixed Numbers as Improper Fractions Multiply the whole number by the denominator, add the numerator, then put it over the denominator. Example: € 6 2 2 20 6×3+2 6 = 3 3 3 € GCF (Greatest Common Factor) GCF and LCM are often confused with each other. If you focus on the last word of each, factor and multiple, then you will remember that factors are a part of a product (smaller) and multiples are more of a product (bigger). There are three different methods to finding the GCF, but the Cake Method is the easiest especially when three numbers are involved. Method 1: List the factors. 12 = 1, 2, 3, 4, 6, 12 16 = 1, 2, 4, 8, 16 GCF = 4 Notice they have both 2 and 4 in common, but 4 is the greatest. Method 2: Prime Factorization 12 16 3 4 2 4 2 2 12 = 2 × 2 × 3 16 = 2 × 2 × 2 × 2 GCF = 2 × 2 = 4 4 2 2 2 Method 3: Cake Method It is a kind of upside-down division where you divide both numbers by the same common number and keep doing so until the only number in common is 1. 2 | 12 16 2 | 6 8 3 4 Only multiply the numbers on the side to get the GCF. GCF = 2 × 2 = 4 There is nothing in common with 3 and 4 except 1. LCM (Least Common Multiple) There are also three different methods to finding the LCM, but in this case not one method is better than the rest. It all depends on the numbers involved. Listing the Multiples is easiest for smaller numbers, Prime Factorization is easiest for larger numbers, and the Cake Method is the easiest when there are only two numbers involved because it doesn’t work with three. Method 1: List the Multiples 12 = 12, 24, 36 18 = 18, 36 LCM = 36 In truth, I really don’t list out all of the multiples. I start with the larger of the two numbers and ask myself if the smaller goes into it evenly. If that doesn’t work, then I work through its multiples. 18 Does 12 go evenly into 18? No 18 × 2 = 36 Does 12 go evenly into 36? Yes, LCM = 36. Method 2: Prime Factorization Use the factor tree, then circle all of the prime numbers with the highest prime numbers. 12 = 22 x 3 18 = 2 × 32 LCM = 22 x 32 = 36 Method 3: Cake Method 3 | 12 18 2| 4 6 2 3 LCM = 3 × 2 × 2 × 3 = 36 Notice for the LCM that all of the numbers outside of the upside-down division signs are multiplied together. Comparing Fractions Again, this is a case where the fractions dictate which method to use is best. Method 1: Make Equivalent Fractions 9 16 5 9 8 16 5 × 2 10 9 5 < = 8 × 2 16 16 8 Method 2: Cross Multiply € € € 4 9 Ordering Fractions € € € × 9 €20 2 4 5 9 € ×5 2 4 5 9 € € € € 18 2 4 2 > 5 9 5 € € Give all of the fractions a common denominator, then order them. LCM is very helpful here. 1. The LCM of 6, 12, and 4 is 12, so the 5 7 3 common denominator is 12. Example: 6 12 4 2. Make equivalent fractions. 3. Now it is easy to see which fraction is the 5×2 7 3× 3 least and which is the greatest. 6 × 2 12 4 × 3 € € € 4. Order the fractions using the original fractions. 10 7 9 12 12 12 € € € 7 3 5 Therefore the order from least to greatest is 12 4 6 € € € € € €