2. Business Math, Eighth Edition Cleaves/Hobbs © 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved 2.1 Fractions Learning Objectives Identify types of fractions Convert an improper fraction to a whole or mixed number Convert a whole or mixed number to an improper fraction Reduce a fraction to lowest terms Raise a fraction to highest terms

3. Business Math, Eighth Edition Cleaves/Hobbs © 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved 2.1.1. Identify Types of Fractions A fraction is used to identify parts of a whole. It describes the relationship between the part and the whole. There are four parts: and one is shaded white or 1 in 4 which is ¼.

4. Business Math, Eighth Edition Cleaves/Hobbs © 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved Key Terms Denominator - the number appearing below the fraction line. Numerator - the number appearing above the fraction line. Fraction line - horizontal line dividing numerator and denominator. Proper fraction - a fraction has a value than is less than “1” (⅔, for example.)

5. Business Math, Eighth Edition Cleaves/Hobbs © 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved Look at the fraction ⅔ 2 is the numerator 3 is the denominator Is it a proper fraction? Yes, because the value of the fraction is less than “1”.

6. Business Math, Eighth Edition Cleaves/Hobbs © 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved What part of the area is shaded blue? The fraction is 3/7. ¾ Identify the fraction

7. Business Math, Eighth Edition Cleaves/Hobbs © 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved Improper fraction The numerator is a greater value than the denominator, and therefore is greater than “1”. Proper or improper? 10/4 6/7 9/8

8. Business Math, Eighth Edition Cleaves/Hobbs © 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved Convert an improper fraction to a whole or mixed number Divide the numerator or the improper fraction by the denominator. If the remainder is zero, the quotient is a whole number. If the remainder is not zero, the improper fraction will be expressed as a mixed number.

9. Business Math, Eighth Edition Cleaves/Hobbs © 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved Try these examples 120/10 12 119/3 39 ⅔ 135/4 33 ¾

10. Business Math, Eighth Edition Cleaves/Hobbs © 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved Write a mixed number as an improper fraction Find the numerator of the improper fraction. Multiply the denominator of the mixed number by the whole number part. Add the product from the previous step to the numerator of the mixed number. Use the denominator of the mixed number.

11. Business Math, Eighth Edition Cleaves/Hobbs © 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved Look at this example Convert 10 ¾ to an improper fraction The numerator of the fraction is “3.” Multiply the whole number, which is “10” by the denominator which is “4”; the result is 40. Add the numerator to product; 40 + 3 = 43. Retain the same denominator. 43/4 is the improper fraction equivalent.

12. Business Math, Eighth Edition Cleaves/Hobbs © 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved Reduce a fraction to lowest terms Inspect the numerator and denominator to find any whole number by which both can be evenly divided. Carry out the operation until there is no one number that both can be evenly divided by. Tip: Check if the denominator can be divided by the numerator: 3/15, for example, can be reduced to 1/5 when 3 is divided into 15.

13. Business Math, Eighth Edition Cleaves/Hobbs © 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved Reduce to lowest terms 24/40 3/5 27/63 3/7 21/147 1/7

14. Business Math, Eighth Edition Cleaves/Hobbs © 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved Find the greatest common divisor of two numbers The most direct way to reduce a fraction to lowest terms is to use the GCD. The GCD is the largest number by which the denominator and the numerator can be evenly divided. For example, the GCD of 15 and 20 is 5. Any number greater than 5 would result in a quotient with a remainder.

15. Business Math, Eighth Edition Cleaves/Hobbs © 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved How to find the GCD For example: find the GCD of 42 and 28. Divide the larger number by the smaller number: 42 divided by 28 = 1 R 14 Divide the divisor by the remainder of the previous operation (28) by (14) 28 divided by 14 = 2 R 0. When the R = 0, the divisor from that operation (14, in this case) is the GCD.

16. Business Math, Eighth Edition Cleaves/Hobbs © 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved Try these examples 30, 36 GCD = 6 30, 125 GCD = 5 17, 51 GCD =17

17. Business Math, Eighth Edition Cleaves/Hobbs © 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved Raise a fraction to higher terms ¾ is equal to ? 8 Look at the two denominators and divide. “4” goes into 8 two times. Multiply “3” by “2” to get the equivalent numerator. ¾ = 6/8

18. Business Math, Eighth Edition Cleaves/Hobbs © 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved Try these examples Determine the equivalent fraction in higher terms: 4/5 = ?/25 20/25 7/8 = ?/40 35/40 3/5 = ?/60 36/60

19. Business Math, Eighth Edition Cleaves/Hobbs © 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved 2.2. Adding and Subtracting Fractions To add fractions with like denominators: Add the numerators The denominator remains the same Convert an improper fraction to a mixed number, if necessary ¼ + ¾ + ¼ = 5/4 or 1 ¼

20. Business Math, Eighth Edition Cleaves/Hobbs © 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved Adding fractions with different denominators You must first find the lowest common denominator (LCD). Smallest number that can be divided evenly by each original denominator. For example: ¾ and ⅝ [using inspection] Convert ¾ to an equivalent fraction in eighths and then add.

21. Business Math, Eighth Edition Cleaves/Hobbs © 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved Find the LCD for 4/5, 1/2 and 1/8. It is not as apparent which number might be the LCD given the denominators of 5, 2 and 8. You can use prime numbers to find the LCD Prime number: a number greater than 1 that can be divided evenly by only itself and 1 Adding fractions with different denominators

22. Business Math, Eighth Edition Cleaves/Hobbs © 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved Find the LCD using prime numbers Denominators Prime numbers 528 2 514 2 512 2 511 5 111

23. Business Math, Eighth Edition Cleaves/Hobbs © 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved Find the LCD Multiply the prime numbers from the first column together (2x2x2x5) to get the LCD. The LCD is 40. Convert the fractions to the equivalent using 40 as the denominator. 4/5 becomes 32/40. ½ becomes 20/40. 1/8 becomes 5/40.

24. Business Math, Eighth Edition Cleaves/Hobbs © 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved Add the numerators 32/40 + 20/40 + 5/40 = 57/40 If the numerator is greater than the denominator, it is an improper fraction and can be expressed as a mixed number. It would be 1 17/40 Inspect the fraction to determine if it is expressed in lowest terms.

25. Business Math, Eighth Edition Cleaves/Hobbs © 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved Subtracting fractions with like denominators Subtract the smaller numerator from the greater one. The denominator remains the same. 5/8 – 3/8 = 2/8 Reduce to lowest terms, if necessary. 2/8 = 1/4

26. Business Math, Eighth Edition Cleaves/Hobbs © 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved Subtracting fractions with different denominators As in addition, to subtract fractions you must have a common denominator. Use the same methods of inspection or prime numbers to determine the LCD. Carry out the operation. Reduce to lowest terms as needed.

27. Business Math, Eighth Edition Cleaves/Hobbs © 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved 5/12 -1/3 = ? Find the LCD, which is 12. Change 1/3 to an equivalent fraction. 1/3 = 4/12 Carry out the operation: 5/12- 4/12 = 1/12 Reduce to lowest terms, if needed. Subtracting fractions with different denominators

28. Business Math, Eighth Edition Cleaves/Hobbs © 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved Try these examples 7/8 – 2/3 = 5/24 2/3 – 1/5 = 7/15 4/5 -1/6 = 19/30

29. Business Math, Eighth Edition Cleaves/Hobbs © 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved Subtracting mixed numbers 10 ⅛ – 7 ½ = Convert the fraction portion of each mixed number to equivalent fractions. 10 1/8 - 7 4/8 = Borrow “1” from the whole number to carry out the operation. 9 9/8 – 7 4/8 = 2 5/8 Reduce to lowest terms, if necessary.

30. Business Math, Eighth Edition Cleaves/Hobbs © 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved Try these examples Maria has 6 ⅛ cups of flour, but only needs 4 ¼ cups for her recipe. How much will she have left? 1 ⅞ Julia needs 3 ⅔ yards of tape to finish a display. Bob brought her a 5 ⅞ yard piece from the supply room. How much will be left? 2 and 5/24

31. Business Math, Eighth Edition Cleaves/Hobbs © 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved 2.3 Multiplying and Dividing Fractions Multiply fractions and mixed numbers Divide fractions and mixed numbers 1/4 divided by 2/3 = ? 4/5 x 5/8 = ?

32. Business Math, Eighth Edition Cleaves/Hobbs © 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved Multiply fractions and mixed numbers Find the numerator of the product: multiply the numerators of the fractions. Find the denominator of the product: multiply the denominators of the fractions. Reduce to lowest terms

33. Business Math, Eighth Edition Cleaves/Hobbs © 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved Look at this example ⅓ x ⅞ = 1 x 7 = 7 3 x 8 = 24 The product is 7/24. Can this fraction be reduced further? NO!

34. Business Math, Eighth Edition Cleaves/Hobbs © 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved Tip! To keep things simple, if possible, reduce before multiplying. ⅓ x ¾ = ? The “3” in the denominator in the first fraction and the “3” in the numerator in the second fraction cancel each other out and become “1”. The answer is ¼.

35. Business Math, Eighth Edition Cleaves/Hobbs © 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved Multiply mixed numbers and whole numbers Write the mixed numbers and whole numbers as improper fractions. Reduce numerators and denominators as appropriate. Multiply the fractions. Reduce to lowest terms and / or write as a whole number or mixed number.

36. Business Math, Eighth Edition Cleaves/Hobbs © 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved Try this example 1 ⅔ x 3 ¾ = ? 1 2/3 becomes 5/3 3 ¾ becomes 15/4 5/3 x 15/4 = ? The “3” can be reduced to “1” and the “15” to “5” before multiplying. Multiply: 25/4. Convert to a mixed number. 6 ¼

37. Business Math, Eighth Edition Cleaves/Hobbs © 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved No. When the multiplier is a proper fraction, the product is less than the original number. 5 x 3/5 = 3 This is also true when the multiplicand is a whole number, fraction or mixed number. 2½ x ½ = 1¼ Are products always larger than their factors?

38. Business Math, Eighth Edition Cleaves/Hobbs © 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved Reciprocals The relationship between multiplying and dividing fractions involves a concept called reciprocals. Two numbers are reciprocals if their product is equal to 1. 2 is the reciprocal of ½. What is the reciprocal of ⅓? 3

39. Business Math, Eighth Edition Cleaves/Hobbs © 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved Divide fractions or mixed numbers Write numbers as fractions. Find the reciprocal of the divisor. Multiply the dividend by the reciprocal of the divisor. Reduce to lowest terms, and/or write as a whole or mixed number.

40. Business Math, Eighth Edition Cleaves/Hobbs © 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved Here’s an example 3 ¼ ÷ ⅔ = ? To carry out this operation,  Convert 3 ¼ to an improper fraction  Change ⅔ to its reciprocal which is 3/2  Change from division to multiplication 13/4 x 3/2 = 39/8 39/8 = 4 ⅞

41. Business Math, Eighth Edition Cleaves/Hobbs © 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved Try this problem Madison Duke makes appliqués. A customer has ordered five appliqués. Madison has ¾ yard of fabric and each appliqué requires 1/6 of a yard. Does she need more fabric? ¾ ÷ 1/6 becomes ¾ x 6 Simplify by dividing 4 and 6 by 2. Multiply 3/2 x 3. The answer is 4 ½; therefore she can only make 4 appliqués and she needs more fabric.

42. Business Math, Eighth Edition Cleaves/Hobbs © 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved Try this problem A home goods store is stacking decorative boxes on shelves. If each box is 6 ⅔ inches tall, and the shelf space is 45 inches, how many boxes will fit on each shelf? Six