Least common multiple/denominator & Greatest common factor.

1. Least common multiple/denominator & Greatest common factor. The least common multiple is the smallest common multiple of two or more numbers. Find the LCM of 6, 24, 18, 10, 50 First decompose each number into its prime factors. 6= 2 * 3 24 = 2 * 2 * 2 * 3 18 = 3 * 3 * 2 10 = 2 * 5 50 = 2 * 5 * 5 The LCM is the product of the highest occurrence 2’s, 3’s and 5’s of the set. Observe that the highest occurrence of 2 is 3 times, 3 occurs the most twice and the highest occurrence of 5 is twice. So the LCM is 2*2*2 * 3*3*5*5 = 8 * 9 * 25 = 1800 The greatest common factor(GCF) is the largest common factor of two or more numbers. For example to find the GCF of 180 and 126, list the prime factors of 180= 2*3*3*7 and 126 = 2*2*3*3*5, the GCF is the product of the underlined numbers: GCF = 2*3*3 = 18

2. Fractions A fraction represents part of a whole or it can represent the number of equal parts of a whole. a Each part of a fraction has a name, in the following fraction ----- b A is called the numerator and b is called the denominator. A proper fraction is a fraction less than one. The numerator is smaller than the denominator. For instance 2 is a proper fraction. 3 An improper fraction is a fraction greater than or equal to 1. The numerator of an improper fraction is greater than or equal to the denominator. 4 is an improper fraction. 3 A mixed fraction is a fraction greater than 1 with a whole-number part and a fractional part. 2 3 ------- is a mixed fraction (read 3 and 2 third) 3

3. Equivalent fractions and reduced fractions Equal fraction with different denominators are called equivalent fractions. 2/3 and 10/15 are equivalent fractions. A fraction is reduced to its simplest form when there are no common factors in the numerator and the denominator. ¾ is reduced to its lowest term while 15/35 is not reduced to its lowest term. To write an equivalent fraction to a/b, we multiply the denominator and numerator by the same number. To reduce a/b to its lowest term, Case I)divide the numerator into the denominator if a > b. Case II) if a < b. Divide the numerator and the denominator by their GCF. a and b are nonzero numbers.

4. Writing equivalent fractions and reducing • Write an equivalent fraction for each of the following: • 2 = ----- b) 3 = ---- c) 7 = ----- • 3 12 2 8 8 24 • Answers: a) 12  3 x 2 = 8/12 b) 8 2 x 3 = 12 /8 • 12 8 • c) 24  8 x 7 = 21/24 • 24 • Reduce the following fractions to their lowest terms: • 21/35 b) 65/45 c) 18/72 d) 5/3 • a)Divide each part by their GCF 7, we have 7/5 or 1 and 2/5. • b) 65 into 45 to get 1 and 4/9 c) divide each part by 18 to get ¼ • d) Divide 5 into 3 get 1 and 2/3.

5. To convert an improper fraction to a mixed fraction, divide the numerator into the denominator. Rewrite the whole number plus the remainder over the divisor. To convert a mixed fraction to an improper fraction, multiply the denominator by the whole number part and add it to the numerator. Examples: a)Convert 23/ 5 and 28/9 to improper fractions: Observe that 5 goes into 23, 4 times with a remainder of 3. So we write 4 as a whole number, 3 goes in the numerator and the divisor remains 5. Convert to a mixed fraction 3 5 7X 3 + 5 23/5 = 4 ---- Convert to an improper fraction 3------ = -------------- = 26/7 5 7 7 Convert to a mixed fraction 1 4 b)28/9 =3---- Convert to an improper fraction 3 ------= 19/5 9 5 Convert improper fractions to mixed fractions and mixed fractions to improper fractions

6. Adding/Subtracting fractions To add/subtract fractions with a common denominator, we add the numerators and reduce if necessary.To add/subtract fractions with a different denominators, we must find the LCD. Using the concept described in the previous slides the LCD is 72. The denominator of each fraction divides 72 evenly and multiply each numerator as follows to get A) 7 5 1 7 72 18 X 7 + 72  24 X 5 + 72  6 X 1 + 72  12 X 7 28 - 15 + 12 + 42 67 --- - ---- + --- + ---- = ----------------------------------------------------------------------- = ----------------------------= --------- 18 24 6 12 72 72 72  . MULTIPLY / DIVIDE AS INDICATED: First, we reduce by cancellation, Observe that the GCF of 52 and 65 is 13, 13 goes into 52, 4 times and 13 goes into 65, 5 times. The gcf of 55 and 75 is 5. 52 55 4 11 44 a) --- X ---- = ----- X --------- = --------- 75 65 15 5 45 To divide fractions, we inverse multiply the second term. 45 27 4 5 60 5 10 50 5 b) --- ÷ ---- = ----- X --------= ------- X --------- = ------ = 5----- 18 60 18 27 3 3 9 9 3 3 1 c)Add/subtract 2 –1 ---- = (2 –1) - ¾ = ¼ d) Simplify 2-- + --- = 2+(1/4 + ¾) = 3 4 4 4

7. Applications of fractions A)One-third of the players elected to the baseball Hall fame were pitchers. If 183 players are in the hall of fame, how many were pitchers and how many were not pitchers? Answer: 1/3 of 183 = 1/3 X 183 =61 pitchers B)John works 4 and half hour on Monday, 8 and one third of an hour on Tuesday, 7 and 2/3 of an hour on Wednesday, 6 and ¾ of an hour on on Thursday and 3 and a1/4 of an hour on Friday. i))How many hours did John work for the week? Answer: (4 + 8 + 7+ 6 + 3) + ( ½ +1/3 + 2/3 + ¾ 1/4) = 30.5 hours or 30hours & 30 minutes ii) If John’s hourly rate is 6/hr, What is his salary for the week? Answer: 6 X 30.5 = $183 C) 1 1 2--- inch piece is cut from a 6--- inch board. 4 4 How much of the board is left? Answer: we need to subtract 2 and ¼ from 6 and 1/4 to get 6 1/4 - 2 ¼ = ( 6 –2 ) + ( 1/4 – ¼) = 4 inches