Fractions Chapter Two McGraw-Hill/Irwin Copyright © 2014 by The McGraw-Hill Companies, Inc. All rights reserved.

Learning unit objectives LU 2-1: Types of Fractions and Conversion Procedures Recognize the three types of fractions. Convert improper fractions to whole or mixed numbers and mixed numbers to improper fractions. Convert fractions to lowest and highest terms. LU 2-2: Fraction and Decimal Conversions Convert decimal fractions to decimals, proper fractions to decimals, mixed numbers to decimals, and pure and mixed decimals to decimal fractions. LU 2-3: Basic Math Functions with Fractions Add and subtract fractions. Multiply fractions. Divide fractions.

Types of Fractions Numerator Proper Fractions Denominator Proper fractions have a value less than 1; its numerator is smaller than its denominator. 1, 1, 1, 4, 18 4 2 12 7 55

Types of Fractions Denominator Numerator Improper Fractions 14, 7, 15, 22 14 6 14 19 Denominator Numerator Improper Fractions Improper fractions have a value equal to or greater than 1; its numerator is equal to or greater than its denominator.

Types of Fractions Mixed Numbers Mixed numbers are the sum of a whole number greater than zero and a proper fraction 1, 9, 7, 5, 9 6 10 8 6 11 5 5 8 33 139 2-5

Converting Improper Fractions to Whole or Mixed Numbers 2 Steps 1. Divide the numerator of the improper fraction by the denominator 15 = 1 2. a. If you have no remainder, the quotient is a whole number 16 1 5 5 3 R 1 5 16 15 1 = 3 2 b. If you have a remainder, the quotient is a mixed number. The remainder is placed over the old denominator as the proper fraction of the mixed number. 2-6

Converting Mixed Numbers to Improper Fractions 3 Steps 1. Multiply the denominator of the fraction by the whole number. (8 x 6) = 48 1 8 6 (8 x 6) = 48 48 + 1 = 49 2. Add the product from Step 1 to the numerator of the old fraction. 49 8 3. Place the total from Step 2 over the denominator of the old fraction to get the improper fraction. 2-7

Converting (Reducing) Fractions to Lowest Terms Find the largest whole number that will divide into both the numerator and denominator without leaving a remainder. 24 24 / 6 4 30 30 / 6 5 = = 2-8

Finding the Greatest Common Divisor 24 30 Step 1. Divide the numerator into the denominator. 1 24 30 24 6 4 6 24 24 Step 2. Divide the remainder in Step 1 into the divisor of Step 1. 24 / 6 4 30 / 6 5 = Step 3. Divide the remainder of Step 2 into the divisor of Step 2. Continue until the remainder is 0. 2-9

Converting (raising) Fractions to higher Terms Multiply the numerator and the denominator by the same whole number. 1 2 2 4 2 8 x = The fractions are equivalent in value. By converting, you divided it into more parts. 2-10

Raising Fractions to Higher Terms When Denominator is Known 4 = ? 7 28 2 Steps Divide the new denominator by the old denominator to get the common number that raises the fraction to higher terms. 4 7 28 28 4 x 4 = 16 16 28 2. Multiply the common number from Step 1 by the old numerator and place it as the new numerator over the new denominator. 2-11

Converting Proper Fractions to decimals Divide the numerator of the fraction by its denominator. 3 4 = .75 Round as necessary. 3 8 = .375 1 3 = .333 2-12

Converting mixed numbers to decimals Convert the fractional part of the mixed number to a decimal. 2 5 (Step 1) 2 5 8 = .40 Add the converted fractional part to the whole number. 8.00 +.40 (Step 2) 8.40 2-13

Converting pure and mixed numbers to Fractions 3 Steps 1. Place the digits to the right of the decimal point in the numerator of the fraction. Omit the decimal point. (Step 1) (Step 2) Places (Step 3) .3 3 3 1 3 10 1 2. Put a 1 in the denominator of the fraction. 3. Count the number of digits to the right of the decimal point. Add the same number of zeros to the denominator of the fraction. 2-14

Adding and subtracting Like Fractions Add or subtract the numerators and place the total over the denominator. 1 4 5 7 7 7 + = If the total of your numerators is the same as your original denominator, convert your answer to a whole number. If the total is larger than your original denominator, convert your answer to a mixed number. 4 1 3 7 7 7 - = 2-15

Adding or subtracting Unlike Fractions 4 Steps 1. Find the LCD. 1 1 1 1 3 8 9 12 + 2. Change each fraction to a like fraction with the LCD. 24 9 8 6 47 72 72 72 72 72 + = 3. Add or subtract the numerators and place the total over the LCD. 4. If necessary, reduce the answer to lowest terms. 2-16

Least Common Denominator (LCD) 3 5 7 21 The smallest nonzero whole number into which ALL denominators will divide evenly. + 7 42 21 What is the least common denominator? 2-17

Prime Numbers A prime number is a whole number greater than 1 that is only divisible by itself and 1. The number 1 is not a prime number. Examples 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43 2-18

Adding Mixed Numbers 7 7 20 20 3 12 5 20 1 5 4 20 24 4 20 20 4 20 4 4 3 Steps 1. Add the fractions. 2. Add the whole numbers. 3. Combine steps 1 & 2. Be sure you do not have an improper fraction in your final answer. If necessary, reduce the answer to lowest terms. 6 6 + 7 +7 Step 1 = 1 + 17 Step 2 18 1 5 Step 3 18 2-19

Subtracting Mixed Numbers When Borrowing Is Not Necessary: 3 Steps Step 1. Subtract fractions, making sure to find the LCD. 1 4 2 8 6 6 Step 2. Subtract whole numbers. Step 3. Reduce the fractions to lowest terms. - 3 3 8 8 1 8 2-20

Subtracting Mixed Numbers When Borrowing Is Necessary: 4 Steps 2 6 4 4 3 3 3 4 Step 1. Make sure the fractions have the LCD. 1 2 3 4 3 3 -1 2 Step 2. Borrow from the whole number. Step 3. Subtract whole numbers and fractions. -1 -1 Step 4. Reduce the fractions to lowest terms. 1 2-21

Multiplying Proper Fractions 2 Steps Step 1. Multiply the numerator and the denominator. 1 5 7 8 = x 5 56 Step 2. Reduce the answer to lowest terms. 2-22

Multiplying Mixed Numbers 1. Convert the mixed numbers to improper fractions. 1 1 7 3 7 1 3 2 3 2 2 2 2 3 = X 1 3. Reduce the answer to lowest terms. 2. Multiply the numerator and denominators. 2-23

Dividing Proper Fractions 1. Invert (turn upside down) the divisor (the second fraction). 2. Multiply the fractions. 1 2 1 3 3 8 3 8 2 16 = X . 3. Reduce the answer to lowest terms. 2-24

Dividing Mixed Numbers 1. Convert all mixed numbers to improper fractions. 3. Reduce the answer to lowest terms. 3 5 35 6 105 3 4 6 4 17 34 34 8 = 2 X 3 ÷ 2. Invert the divisor and multiply. 2-25