CHAPTER 2 – Fractions Instructor: Dr.Gehan Shanmuganathan

Business Math, Ninth Edition Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel © 2012 Pearson Education, Inc. Upper Saddle River, NJ All Rights Reserved Learning Outcomes 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 higher terms. 2-1

Business Math, Ninth Edition Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel © 2012 Pearson Education, Inc. Upper Saddle River, NJ All Rights Reserved A fraction is used to identify parts of a whole. –It describes the relationship between the part and the whole. There are four parts. –One is shaded white, or 1 in 4 which is. Identify types of fractions Fractions Section 2-1

Business Math, Ninth Edition Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel © 2012 Pearson Education, Inc. Upper Saddle River, NJ All Rights Reserved Denominator –The number appearing below the fraction line. Numerator –The number appearing above the fraction line. Fraction line –Horizontal line separating numerator and denominator. Proper fraction –A fraction with a value less than “1” (ex., ) Key Terms… FractionsSection 2-1

Business Math, Ninth Edition Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel © 2012 Pearson Education, Inc. Upper Saddle River, NJ All Rights Reserved Look at the fraction Is it a proper fraction? FractionsSection is the numerator. 3 is the denominator. Yes, because the value of the fraction is less than “1”. HOW TO:

Business Math, Ninth Edition Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel © 2012 Pearson Education, Inc. Upper Saddle River, NJ All Rights Reserved What part of the area is shaded blue? Identify the fraction FractionsSection 2-1 HOW TO: MORE

Business Math, Ninth Edition Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel © 2012 Pearson Education, Inc. Upper Saddle River, NJ All Rights Reserved What part of the area is shaded blue? Identify the fraction FractionsSection 2-1 HOW TO:

Business Math, Ninth Edition Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel © 2012 Pearson Education, Inc. Upper Saddle River, NJ All Rights Reserved The numerator is greater than or equal to the denominator. –Therefore the fraction is greater than or equal to “1”. Proper or improper? Improper fraction FractionsSection 2-1 Proper Improper HOW TO:

Business Math, Ninth Edition Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel © 2012 Pearson Education, Inc. Upper Saddle River, NJ All Rights Reserved Divide the numerator of 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. FractionsSection 2-1 Convert an improper fraction to a whole or mixed number = 12 Examples = = 2-1-2

Business Math, Ninth Edition Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel © 2012 Pearson Education, Inc. Upper Saddle River, NJ All Rights Reserved Convert a whole or mixed number to an 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. –This is the numerator of the improper fraction. Use the denominator of the mixed number as the denominator of the improper fraction. FractionsSection

Business Math, Ninth Edition Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel © 2012 Pearson Education, Inc. Upper Saddle River, NJ All Rights Reserved An Example… FractionsSection 2-1 Convert to an improper fraction. 10 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 the product: = 43. Retain the same denominator. The improper fraction equivalent is

Business Math, Ninth Edition Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel © 2012 Pearson Education, Inc. Upper Saddle River, NJ All Rights Reserved 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. Reduce a fraction to lowest terms FractionsSection 2-1 TIP: Check if the denominator can be divided by the numerator: Example: can be reduced to when 3 is divided into

Business Math, Ninth Edition Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel © 2012 Pearson Education, Inc. Upper Saddle River, NJ All Rights Reserved Reduce these examples to the lowest terms… FractionsSection 2-1 TIP: Check if the denominator can be divided by the numerator: = = =

Business Math, Ninth Edition Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel © 2012 Pearson Education, Inc. Upper Saddle River, NJ All Rights Reserved 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 numerator can be evenly divided. Example: The GCD of 15 and 20 is 5. Any number greater than 5 would result in a quotient with a remainder Find the greatest common divisor of two numbers FractionsSection 2-1 HOW TO:

Business Math, Ninth Edition Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel © 2012 Pearson Education, Inc. Upper Saddle River, NJ All Rights Reserved Divide the larger number by the smaller number: 42 divided by 28 = 1 R 14 Divide the divisor (28) by the remainder of the previous operation: (14) = 2 R 0. FractionsSection 2-1 Find the GCD Find the GCD of 42 and 28. When the R = 0, the divisor from that operation (14, in this case) is the GCD. 30, 3630, 12517, 51 GCD = 6 More Examples: GCD = 5 GCD = 17 HOW TO:

Business Math, Ninth Edition Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel © 2012 Pearson Education, Inc. Upper Saddle River, NJ All Rights Reserved 2 = FractionsSection Raise a fraction to higher terms Multiply “3” by “2” to get the equivalent numerator. Multiply “4” by “2” to get the equivalent denominator. is equal to Divide the two denominators. “4” goes into “8” two (2) times. 8 / 4 = 2 2 2

Business Math, Ninth Edition Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel © 2012 Pearson Education, Inc. Upper Saddle River, NJ All Rights Reserved Determine the equivalent fraction in higher terms: FractionsSection 2-1 Examples… = = =

Business Math, Ninth Edition Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel © 2012 Pearson Education, Inc. Upper Saddle River, NJ All Rights Reserved

Business Math, Ninth Edition Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel © 2012 Pearson Education, Inc. Upper Saddle River, NJ All Rights Reserved Learning Outcomes Add fractions with like (common) denominators. Find the least common denominator for two or more fractions. Add fractions and mixed numbers. Subtract fractions and mixed numbers.. 2-2

Business Math, Ninth Edition Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel © 2012 Pearson Education, Inc. Upper Saddle River, NJ All Rights Reserved Add the numerators. The denominator remains the same. If necessary, convert an improper fraction to a mixed number. Section = or Add Fractions with like (common) denominators Adding and Subtracting Fractions 2-2-1

Business Math, Ninth Edition Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel © 2012 Pearson Education, Inc. Upper Saddle River, NJ All Rights Reserved Adding fractions with different denominators. –First find the least common denominator (LCD). The LCD is the smallest number that can be divided evenly by each original denominator. Adding and Subtracting FractionsSection Find the least common denominator for two or more fractions

Business Math, Ninth Edition Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel © 2012 Pearson Education, Inc. Upper Saddle River, NJ All Rights Reserved Adding and Subtracting FractionsSection 2-2 Find the LCD for: and Convert to an equivalent fraction in eighths, Then add the two fractions together. Find the least common denominator for two or more fractions The common denominator can sometimes be found by inspection. –Mentally selecting a number that can be evenly divided by each denominator.

Business Math, Ninth Edition Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel © 2012 Pearson Education, Inc. Upper Saddle River, NJ All Rights Reserved Adding and Subtracting FractionsSection 2-2 Find the least common denominator for two or more fractions It is not as apparent which number might be the LCD, given different denominators. –Such as 12 and 30. You can use prime numbers to find the LCD. A prime number is a number greater than 1 that can be divided evenly by only itself and 1. TIP: The first ten prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29.

Business Math, Ninth Edition Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel © 2012 Pearson Education, Inc. Upper Saddle River, NJ All Rights Reserved Divide each one by the smallest prime number by which any of the numbers can be evenly divided. STEP 2 Adding and Subtracting FractionsSection 2-2 Find the least common denominator for two or more fractions Write the denominators in a row. STEP 1 HOW TO: MORE

Business Math, Ninth Edition Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel © 2012 Pearson Education, Inc. Upper Saddle River, NJ All Rights Reserved Adding and Subtracting FractionsSection 2-2 Find the least common denominator for two or more fractions Write a new row of numbers using: 1. The quotients from step Any numbers in the first row that cannot be evenly divided by the first prime number. STEP 3 HOW TO: MORE

Business Math, Ninth Edition Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel © 2012 Pearson Education, Inc. Upper Saddle River, NJ All Rights Reserved Adding and Subtracting FractionsSection 2-2 Find the least common denominator for two or more fractions Divide again by the smallest prime number by which any of the numbers can be evenly divided. STEP 4 STEP 5 Continue this process until you have a row of 1s. HOW TO: MORE

Business Math, Ninth Edition Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel © 2012 Pearson Education, Inc. Upper Saddle River, NJ All Rights Reserved Adding and Subtracting FractionsSection 2-2 Find the least common denominator for two or more fractions Multiply all the prime numbers used to divide the denominators. The product is the least common denominator. STEP 6 HOW TO:

Business Math, Ninth Edition Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel © 2012 Pearson Education, Inc. Upper Saddle River, NJ All Rights Reserved Denominators Prime numbers Add the fractions:, Adding and Subtracting FractionsSection 2-2 Find the least common denominator for two or more fractions HOW TO: First, find the LCD Multiply the prime numbers from the first column (2 x 2 x 2 x 5) to get the LCD. The LCD is 40 MORE

Business Math, Ninth Edition Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel © 2012 Pearson Education, Inc. Upper Saddle River, NJ All Rights Reserved Denominators Prime numbers, Adding and Subtracting FractionsSection 2-2 Find the least common denominator for two or more fractions HOW TO: The LCD is 40 Convert the fractions to the equivalent using 40 as the denominator. The LCD is 40 Add the fractions: becomes MORE

Business Math, Ninth Edition Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel © 2012 Pearson Education, Inc. Upper Saddle River, NJ All Rights Reserved If the numerator is greater than the denominator, it is an improper fraction and can be expressed as a mixed number. Inspect the fraction to determine if it is expressed in lowest terms., Adding and Subtracting FractionsSection 2-2 Find the least common denominator for two or more fractions HOW TO: Add the fractions: = + = +

Business Math, Ninth Edition Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel © 2012 Pearson Education, Inc. Upper Saddle River, NJ All Rights Reserved Adding and Subtracting FractionsSection 2-2 Adding fractions and mixed numbers Add the whole-number parts. Add the fraction parts and reduce to lowest terms. Change improper fractions to whole or mixed numbers. Add the whole-number parts.

Business Math, Ninth Edition Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel © 2012 Pearson Education, Inc. Upper Saddle River, NJ All Rights Reserved Adding and Subtracting FractionsSection 2-2 Subtracting mixed numbers – = Convert the fraction portion of each mixed number to equivalent fractions. Borrow “1” from the whole number to carry out the operation. – = Reduce to lowest terms, if necessary. STEP 1 STEP 2

Business Math, Ninth Edition Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel © 2012 Pearson Education, Inc. Upper Saddle River, NJ All Rights Reserved Adding and Subtracting FractionsSection 2-2 Subtracting fractions with like denominators Subtract the smaller numerator from the greater one—the denominator remains the same. Reduce to lowest terms, if necessary. – = = HOW TO:

Business Math, Ninth Edition Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel © 2012 Pearson Education, Inc. Upper Saddle River, NJ All Rights Reserved Subtracting fractions with different denominators Adding and Subtracting FractionsSection 2-2 HOW TO: 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. – = ? MORE

Business Math, Ninth Edition Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel © 2012 Pearson Education, Inc. Upper Saddle River, NJ All Rights Reserved Subtracting fractions with different denominators Adding and Subtracting FractionsSection 2-2 HOW TO: – = – = ? Find the LCD, which is 12. = Carry out the operation and reduce to lowest terms, if needed. Change to an equivalent fraction.

Business Math, Ninth Edition Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel © 2012 Pearson Education, Inc. Upper Saddle River, NJ All Rights Reserved Adding and Subtracting FractionsSection 2-2 – = – = – = Examples…

Business Math, Ninth Edition Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel © 2012 Pearson Education, Inc. Upper Saddle River, NJ All Rights Reserved Adding and Subtracting FractionsSection 2-2 Examples… How much will she have left? Maria has cups of flour, but only needs cups for her recipe.

Business Math, Ninth Edition Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel © 2012 Pearson Education, Inc. Upper Saddle River, NJ All Rights Reserved Adding and Subtracting FractionsSection 2-2 Examples… Julia needs yards of tape to finish a display. Bob brought her a yard piece from the supply room. How much will be left?

Business Math, Ninth Edition Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel © 2012 Pearson Education, Inc. Upper Saddle River, NJ All Rights Reserved

Business Math, Ninth Edition Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel © 2012 Pearson Education, Inc. Upper Saddle River, NJ All Rights Reserved Learning Outcomes Multiply fractions and mixed numbers. Divide fractions and mixed numbers. 2-3

Business Math, Ninth Edition Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel © 2012 Pearson Education, Inc. Upper Saddle River, NJ All Rights Reserved Multiplying and Dividing FractionsSection 2-3 Multiply fractions and mixed numbers To multiply two fractions: –Find the numerator of the product by multiplying the numerators of the fractions. –Find the denominator of the product by multiplying the denominators of the fractions. –Reduce to lowest terms, if needed. x Can this fraction be reduced further? 1 x 7 = 7 3 x 8 = 24 NO!

Business Math, Ninth Edition Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel © 2012 Pearson Education, Inc. Upper Saddle River, NJ All Rights Reserved x = ? 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. x Multiplying and Dividing FractionsSection 2-3 Multiply fractions and mixed numbers To keep things simple, if possible, reduce before multiplying. TIP:

Business Math, Ninth Edition Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel © 2012 Pearson Education, Inc. Upper Saddle River, NJ All Rights Reserved 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. Multiplying and Dividing FractionsSection 2-3 Multiply mixed numbers and whole numbers HOW TO:

Business Math, Ninth Edition Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel © 2012 Pearson Education, Inc. Upper Saddle River, NJ All Rights Reserved Multiplying and Dividing FractionsSection 2-3 An Example… x = ? == The “3” can be reduced to “1” and the “15” to “5” before multiplying. Product:Convert to a mixed number: xx 25 4 =

Business Math, Ninth Edition Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel © 2012 Pearson Education, Inc. Upper Saddle River, NJ All Rights Reserved No. When the multiplier is a proper fraction, the product is less than the original number. Are products always larger than their factors? 5 x = 3 This is also true when the multiplicand is a whole number, fraction or mixed number. x = Multiplying and Dividing FractionsSection 2-3 Products and factors

Business Math, Ninth Edition Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel © 2012 Pearson Education, Inc. Upper Saddle River, NJ All Rights Reserved The relationship between multiplying and dividing fractions involves a concept called reciprocals. –Two numbers are reciprocals if their product equals 1. Multiplying and Dividing FractionsSection 2-3 Divide fractions and mixed numbers is the reciprocal of What is the reciprocal of ? The reciprocal is 3

Business Math, Ninth Edition Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel © 2012 Pearson Education, Inc. Upper Saddle River, NJ All Rights Reserved Write the 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. HOW TO: Multiplying and Dividing FractionsSection 2-3 Divide fractions or mixed numbers

Business Math, Ninth Edition Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel © 2012 Pearson Education, Inc. Upper Saddle River, NJ All Rights Reserved ÷ = ? HOW TO: Multiplying and Dividing FractionsSection 2-3 Divide fractions or mixed numbers Change to its reciprocal, which is: Convert to an improper fraction: Change from division to multiplication. x =

Business Math, Ninth Edition Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel © 2012 Pearson Education, Inc. Upper Saddle River, NJ All Rights Reserved Madison Duke makes appliqués. A customer has ordered five appliqués. Multiplying and Dividing FractionsSection 2-3 Example… Does she need more fabric? Madison has yard of fabric, and each appliqué uses of a yard. ÷ becomes x 6 Simplify by dividing 4 and 6 by 2 x 3 The answer is She can only make 4 appliqués, and she needs more fabric.

Business Math, Ninth Edition Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel © 2012 Pearson Education, Inc. Upper Saddle River, NJ All Rights Reserved Multiplying and Dividing FractionsSection 2-3 Example… A home goods store is stacking decorative boxes on shelves. Six The shelf space is 45 inches. Each box is inches tall. How many boxes will fit on each shelf?

Business Math, Ninth Edition Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel © 2012 Pearson Education, Inc. Upper Saddle River, NJ All Rights Reserved

Business Math, Ninth Edition Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel © 2012 Pearson Education, Inc. Upper Saddle River, NJ All Rights Reserved EXERCISES SET A

Business Math, Ninth Edition Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel © 2012 Pearson Education, Inc. Upper Saddle River, NJ All Rights Reserved EXERCISES SET A

Business Math, Ninth Edition Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel © 2012 Pearson Education, Inc. Upper Saddle River, NJ All Rights Reserved EXERCISES SET A

Business Math, Ninth Edition Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel © 2012 Pearson Education, Inc. Upper Saddle River, NJ All Rights Reserved EXERCISES SET A

Business Math, Ninth Edition Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel © 2012 Pearson Education, Inc. Upper Saddle River, NJ All Rights Reserved EXERCISES SET A

Business Math, Ninth Edition Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel © 2012 Pearson Education, Inc. Upper Saddle River, NJ All Rights Reserved EXERCISES SET A

Business Math, Ninth Edition Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel © 2012 Pearson Education, Inc. Upper Saddle River, NJ All Rights Reserved EXERCISES SET A

Business Math, Ninth Edition Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel © 2012 Pearson Education, Inc. Upper Saddle River, NJ All Rights Reserved EXERCISES SET A

Business Math, Ninth Edition Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel © 2012 Pearson Education, Inc. Upper Saddle River, NJ All Rights Reserved EXERCISES SET A

Business Math, Ninth Edition Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel © 2012 Pearson Education, Inc. Upper Saddle River, NJ All Rights Reserved EXERCISES SET A

Business Math, Ninth Edition Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel © 2012 Pearson Education, Inc. Upper Saddle River, NJ All Rights Reserved

Business Math, Ninth Edition Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel © 2012 Pearson Education, Inc. Upper Saddle River, NJ All Rights Reserved PRACTICE TEST

Business Math, Ninth Edition Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel © 2012 Pearson Education, Inc. Upper Saddle River, NJ All Rights Reserved PRACTICE TEST

Business Math, Ninth Edition Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel © 2012 Pearson Education, Inc. Upper Saddle River, NJ All Rights Reserved PRACTICE TEST

Business Math, Ninth Edition Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel © 2012 Pearson Education, Inc. Upper Saddle River, NJ All Rights Reserved PRACTICE TEST

Business Math, Ninth Edition Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel © 2012 Pearson Education, Inc. Upper Saddle River, NJ All Rights Reserved PRACTICE TEST

Business Math, Ninth Edition Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel © 2012 Pearson Education, Inc. Upper Saddle River, NJ All Rights Reserved