Business Math Chapter 2 Fractions 1 2 1

Business Math Chapter 2: Fractions 1

2. 1 Fractions Learning Objectives n Identify types of fractions n Convert an improper fraction to a whole or mixed number n Convert a whole or mixed number to an improper fraction n Reduce a fraction to lowest terms n Raise a fraction to highest terms 2 Cleaves/Hobbs: Business Math, 7 e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved

2. 1. 1. Identify types of fractions n A fraction is used to identify parts of a whole. It describes the relationship between the part and the whole. n There are four parts: and one is shaded or 1 in 4 which is ¼. 3 Cleaves/Hobbs: Business Math, 7 e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved

Key Terms n Denominator-the number appearing below the fraction line. n Numerator- the number appearing above the fraction line. n Fraction line- horizontal line dividing numerator and denominator. n Proper fraction- a fraction has a value than is less than “ 1” (⅔, for example. ) 4 Cleaves/Hobbs: Business Math, 7 e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved

Look at the fraction ⅔ n n n 2 is the numerator 3 is the denominator Is it a proper fraction? Yes, because the value of the fraction is less than “ 1”. 5 Cleaves/Hobbs: Business Math, 7 e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved

Identify the fraction n¾ n What part of the area is shaded? n The fraction is 3/7. 6 Cleaves/Hobbs: Business Math, 7 e Copyright 2005 by 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”. n n Proper or improper? 10/4 6/7 9/8 7 Cleaves/Hobbs: Business Math, 7 e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved

Convert an improper fraction to a whole or mixed number n Divide the numerator or the improper fraction by the denominator. n If the remainder is zero, the quotient is a whole number. n If the remainder is not zero, the improper fraction will be expressed as a mixed number. 8 Cleaves/Hobbs: Business Math, 7 e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved

Try these examples n 140/10 n n n 14 260/3 n 86 ⅔ n 33 ¾ 135/4 9 Cleaves/Hobbs: Business Math, 7 e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved

Write a mixed number as an improper fraction n Find the numerator of the improper fraction. n Multiply the denominator of the mixed number by the whole number part. n Add the product from the previous step to the numerator of the mixed number. n Use the denominator of the mixed number. 10 Cleaves/Hobbs: Business Math, 7 e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved

Look at this example. Convert 10 ¾ to an improper fraction n The numerator of the fraction is “ 3. ” n Multiply the whole number, which is “ 10” by the denominator which is “ 4”; the result is 40. n Add the numerator to product; 40 + 3 = 43. n Retain the same denominator. n 43/4 is the improper fraction equivalent. 11 Cleaves/Hobbs: Business Math, 7 e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved

Reduce a fraction to lowest terms n Inspect the numerator and denominator to find any whole number by which both can be evenly divided. n Carry out the operation until there is no one number that both can be evenly divided by. n 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. 12 Cleaves/Hobbs: Business Math, 7 e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved

Reduce to lowest terms n n n 18/ 30 n 3/5 n 3/7 n 1/7 27/63 21/147 13 Cleaves/Hobbs: Business Math, 7 e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved

Find the greatest common divisor of two numbers n The most direct way to reduce a fraction to lowest terms is to use the GCD. n The GCD is the largest number by which the denominator and the numerator can be evenly divided. n For example, the GCD of 15 and 20 is 5. Any number greater than 5 would result in a quotient with a remainder. 14 Cleaves/Hobbs: Business Math, 7 e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved

How to find the GCD n For example: find the GCD of 42 and 28. n Divide the larger number by the smaller number: 42 divided by 28 = 1 R 14 n Divide the divisor by the remainder of the previous operation (28) by (14) 28 divided by 14 = 2 R 0. n When the R = 0, the divisor from that operation (14, in this case) is the GCD. 15 Cleaves/Hobbs: Business Math, 7 e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved

Try these examples. n n n 30, 36 n GCD = 5 30, 125 17, 85 n GCD =17 16 Cleaves/Hobbs: Business Math, 7 e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved

Raise a fraction to higher terms ¾ is equal to ? n 8 Look at the two denominators and divide. n “ 4” goes into 8 two times. n Multiply “ 3” by “ 2” to get the equivalent numerator. n ¾ = 6/8 17 Cleaves/Hobbs: Business Math, 7 e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved

Try these examples. Determine the equivalent fraction in higher terms: 4/5 = ? /25 n 20/25 n 35/40 n 36/60 7/8 = ? /40 3/5 = ? /60 18 Cleaves/Hobbs: Business Math, 7 e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved

2. 2. Adding and subtracting fractions To add fractions with like denominators: n Add the numerators n The denominator remains the same n Convert an improper fraction to a mixed number, if necessary n ¼ + ¾ + ¼ = 5/4 or 1 ¼ 19 Cleaves/Hobbs: Business Math, 7 e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved

Adding fractions with different denominators n You must first find the lowest common denominator (LCD). n Smallest number that can be divided evenly by each original denominator. n For example: ¾ and ⅝ [using inspection] n Convert ¾ to an equivalent fraction in eighths and then add. 20 Cleaves/Hobbs: Business Math, 7 e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved

Adding fractions with different denominators n Find the LCD for 4/5, 1/2 and 1/8. n It is not as apparent which number might be the LCD given the denominators of 5, 2 and 8. n You can use prime numbers to find the LCD n Prime number: a number greater than 1 that can be divided evenly by only itself and 1 21 Cleaves/Hobbs: Business Math, 7 e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved

Find the LCD using prime numbers Denominators 5 2 8 2 5 1 4 2 5 1 2 2 5 1 1 1 Prime numbers 22 Cleaves/Hobbs: Business Math, 7 e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved

Find the LCD n Multiply the prime numbers from the first column together (2 x 2 x 2 x 5) to get the LCD. n The LCD is 40. n Convert the fractions to the equivalent using 40 as the denominator. n 4/5 becomes 32/40. n ½ becomes 20/40. n 1/8 becomes 5/40. 23 Cleaves/Hobbs: Business Math, 7 e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved

Add the numerators n 32/40 + 20/40 + 5/40 = 57/40 n If the numerator is greater than the denominator, it is an improper fraction and can be expressed as a mixed number. n It would be 1 17/40 n Inspect the fraction to determine if it is expressed in lowest terms. 24 Cleaves/Hobbs: Business Math, 7 e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved

Subtracting fractions with like denominators n Subtract the smaller numerator from the greater one. n The denominator remains the same. n 5/8 – 3/8 = 2/8 n Reduce to lowest terms, if necessary. n 2/8 = 1/4 25 Cleaves/Hobbs: Business Math, 7 e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved

Subtracting fractions with different denominators n As in addition, to subtract fractions you must have a common denominator. n Use the same methods of inspection or prime numbers to determine the LCD. n Carry out the operation. n Reduce to lowest terms as needed. 26 Cleaves/Hobbs: Business Math, 7 e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved

Subtracting fractions with different denominators n 5/12 -1/3 = ? n Find the LCD, which is 12. n Change 1/3 to an equivalent fraction. n 1/3 = 4/12 n Carry out the operation: n 5/12 - 4/12 = 1/12 n Reduce to lowest terms, if needed. 27 Cleaves/Hobbs: Business Math, 7 e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved

Try these examples n 7/8 – ½ = n n 2/3 – 1/5 = n n 3/8 7/15 4/5 -1/6 = n 19/30 28 Cleaves/Hobbs: Business Math, 7 e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved

Subtracting mixed numbers n 10 ⅛ – 7 ½ = n Convert the fraction portion of each mixed number to equivalent fractions. 10 1/8 -7 4/8 = n Borrow “ 1” from the whole number to carry out the operation. 9 9/8 – 7 4/8 = 2 5/8 n Reduce to lowest terms, if necessary. 29 Cleaves/Hobbs: Business Math, 7 e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved

Try these examples n Maria has 6 ⅛ cups of flour, but only needs 4 ¼ cups for her recipe. How much will she have left? n 1⅞ n 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? n 2 and 5/24 30 Cleaves/Hobbs: Business Math, 7 e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved

2. 3 Multiplying and Dividing Fractions n Multiply fractions and mixed numbers n Divide fractions and mixed numbers 1/2 divided by 1/3 = ? 3/5 x 7/8 = ? 31 Cleaves/Hobbs: Business Math, 7 e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved

Multiply fractions and mixed numbers n Find the numerator of the product: multiply the numerators of the fractions. n Find the denominator of the product: multiply the denominators of the fractions. n Reduce to lowest terms 32 Cleaves/Hobbs: Business Math, 7 e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved

Look at this example. n n n ⅓x⅞= 1 x 7=7 3 x 8 = 24 The product is 7/24. Reduce to lowest terms, if necessary. 2/3 x 3/4 = ? 33 Cleaves/Hobbs: Business Math, 7 e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved

Tip! n To keep things simple, if possible, reduce before multiplying. n ⅓x¾=? n 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”. n The answer is ¼. 34 Cleaves/Hobbs: Business Math, 7 e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved

Multiply mixed numbers and whole numbers n Write the mixed numbers and whole numbers as improper fractions. n Reduce numerators and denominators as appropriate. n Multiply the fractions. n Reduce to lowest terms and / or write as a whole number or mixed number. 35 Cleaves/Hobbs: Business Math, 7 e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved

Try this example. n n n n 1⅔x 3¾=? 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. 6¼ 36 Cleaves/Hobbs: Business Math, 7 e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved

Are products always larger than their factors? n No. When the multiplier is a proper fraction, the product is less than the original number. 5 x 3/5 = 3 n This is also true when the multiplicand is a whole number, fraction or mixed number. 2½ x ½ = 1¼ 37 Cleaves/Hobbs: Business Math, 7 e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved

Reciprocals n The relationship between multiplying and dividing fractions involves a concept called reciprocals. n Two numbers are reciprocals if their product is equal to 1. n 2 is the reciprocal of ½. n What is the reciprocal of ⅓? n 3 38 Cleaves/Hobbs: Business Math, 7 e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved

Divide fractions or mixed numbers n Write numbers as fractions. n Find the reciprocal of the divisor. n Multiply the dividend by the reciprocal of the divisor. n Reduce to lowest terms, and/or write as a whole or mixed number. 39 Cleaves/Hobbs: Business Math, 7 e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved

Here’s an example. n 3¼ ÷ ⅔=? n 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 n 13/4 x 3/2 = 39/8 n 39/8 = 4 ⅞ 40 Cleaves/Hobbs: Business Math, 7 e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved

Try this problem. n Madison Duke makes appliques. A customer has ordered five appliques. Madison has ¾ yard of fabric and each applique requires 1/6 of a yard. Does she need more fabric? n ¾ ÷ 1/6 becomes ¾ x 6 n Simplify by dividing 4 and 6 by 2. n Multiply 3/2 x 3. n The answer is 4 ½; therefore she can only make 4 appliques and she needs more fabric. 41 Cleaves/Hobbs: Business Math, 7 e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved

Try this problem n 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? n Six 42 Cleaves/Hobbs: Business Math, 7 e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved