Chapter 6. Math Review. Copyright 2012, 2009, 2003 by Saunders, an imprint of Elsevier Inc. All rights reserved. 1

Lesson Objectives Identifying proper, improper, and equivalent fractions. Changing improper fractions to mixed numbers. Reducing fractions to lowest terms. Finding the lowest common denominator for fractions. Adding, subtracting, multiplying, and dividing fractions and mixed numbers. Rounding decimals to whole numbers, tenths, hundredths, and thousandths. Copyright 2012, 2009, 2003 by Saunders, an imprint of Elsevier Inc. All rights reserved. 2

Lesson Objectives (cont d.) Converting fractions and mixed numbers to decimals. Adding, subtracting, multiplying, and dividing decimals. Changing percents to decimals. Changing decimals to percents. Multiplying and dividing percents. Using fractions to figure percentages. Using proportions to figure percents. Copyright 2012, 2009, 2003 by Saunders, an imprint of Elsevier Inc. All rights reserved. 3

Fractions Proper fractions numerator is always a lower number than denominator Example 1/2, 4/5, 3/8 Improper fractions numerator is equal to or greater than the denominator Example 11/6, 5/5, 3/2 Equivalent fractions two or more fractions have the same portion of a whole Example 5/10, 1/2, 3/6 Copyright 2012, 2009, 2003 by Saunders, an imprint of Elsevier Inc. All rights reserved. 4

Simplifying Fractions to Lowest Term To simplify a fraction, divide the numerator and denominator by the largest number that will divide into both 5/10 5 will divide into 5 and 10 5/10 5/5 = 1/2 5/10 simplified is 1/2 Copyright 2012, 2009, 2003 by Saunders, an imprint of Elsevier Inc. All rights reserved. 5

Mixed Numbers Improper fractions may be changed to a mixed number Mixed number is a whole number plus a fraction Example 19/11 can be divided by 11 19/11 11/11 = 1 8/11 a mixed number Example 12/8 can be divided by 8 12/8 8/8 = 1-4/8 Reduce 1-4/8 by dividing 4/8 by 4/4 1-4/8 = 1-1/2 Copyright 2012, 2009, 2003 by Saunders, an imprint of Elsevier Inc. All rights reserved. 6

Adding Fractions and Mixed Numbers with the Same Denominator Add the numerators and carry forth the denominator Example 5/6 + 2/6 = 5 + 2/6 7/6 Now change to mixed number by dividing by 6/6 7/6 6/6 = 1-1/6 If numbers to be added are mixed numbers, change to fraction Copyright 2012, 2009, 2003 by Saunders, an imprint of Elsevier Inc. All rights reserved. 7

Adding Fractions and Mixed Numbers with Different Denominators Make both denominators the same Find the lowest common denominator (LCD) - lowest number by which each denominator can be divided evenly Replace the denominator with that number Find the equal fraction for each number by dividing the original denominator into LCD and multiply numerator by the number obtained Then add the numerators and simplify as appropriate Copyright 2012, 2009, 2003 by Saunders, an imprint of Elsevier Inc. All rights reserved. 8

Adding Fractions and Mixed Numbers with the Same Denominator Example - 5/6 + 7/10 Both denominators may be divided into 30 5/6 30 6 = 5 Multiply 5 x numerator (5) = 25 so 5/6 = 25/30 7/10 30 10 =3 Multiply 3 x numerator (7) = 21 so 7/10 = 21/30 25/30 + 21/30 = 46/30 Simplify to 1-16/30 or 1-8/15 Copyright 2012, 2009, 2003 by Saunders, an imprint of Elsevier Inc. All rights reserved. 9

Subtracting Fractions Find LCD and calculate equal fractions Subtract the calculated numerators and carry forward the denominators Simplify as appropriate Example 7/9 1/6 = 7/9 = 14/18 1/6 = 3/18 14/18-3/18 = 11/18 7/9 1/6 = 11/18 (This cannot be simplified) If mixed number, change to improper fraction first Copyright 2012, 2009, 2003 by Saunders, an imprint of Elsevier Inc. All rights reserved. 10

Multiplying Fractions Multiply the numerators Then multiply the denominators Simplify fractions as appropriate Example 2/5 x 3/4 = 2 x 3 = 6 5 x 4 = 20 6/20 = 3/10 after being simplified Copyright 2012, 2009, 2003 by Saunders, an imprint of Elsevier Inc. All rights reserved. 11

Multiplying Mixed Numbers Shortcut for Multiplying Mixed Numbers When changing mixed number to improper fraction, multiply the whole number and denominator then add numerator to obtain the mixed number Cancel any numerators and denominators that can be divided equally by the same number Copyright 2012, 2009, 2003 by Saunders, an imprint of Elsevier Inc. All rights reserved. 12

Multiplying Mixed Numbers Change mixed number to improper fraction Cancel any numerators by denominators if possible Multiply the numerator and denominator line. Simplify as appropriate 5 1/3 x 3/4 = 16/3 x 3/4 =16/4 16/4 = 4 Copyright 2012, 2009, 2003 by Saunders, an imprint of Elsevier Inc. All rights reserved. 13

Dividing Fractions To divide fractions, invert the second fraction (or the reciprocal) and multiply the two fractions Example 5/6 11/12 = 5/6 x 12/11 5/6 1 x 12 2 /11 = 10/11 5/6 11/12 = 10/11 To divide a mixed number change mixed number to improper fraction Copyright 2012, 2009, 2003 by Saunders, an imprint of Elsevier Inc. All rights reserved. 14

Review of Fractions To change improper fraction to mixed number, divide numerator by denominator. To change mixed number to improper fraction, multiply the denominator by whole number and add numerator to product (the numerator of the mixed number; denominator of fraction is retained). To reduce a fraction to lowest term, divide both numerator and denominator by greatest common number. To add or subtract fractions with a common denominator, add or subtract numerators and write the sum or difference as the numerator and common denominator remains the same. Simplify fraction as appropriate. Copyright 2012, 2009, 2003 by Saunders, an imprint of Elsevier Inc. All rights reserved. 15

Review of Fractions For fractions with uncommon denominators, find LCD and perform the addition or subtraction as indicated. Reduce fractions or change to mixed number as appropriate. To multiply fractions, multiply the numerators and the denominators; reduce answer to a fraction or change to a mixed number as appropriate. To multiply mixed number, change mixed number to improper fraction and multiply as above. To divide fractions, invert divisor fraction and multiply fractions. Simplify as appropriate. To divide mixed number, change to improper fraction, invert divisor fraction and multiply fraction. Simplify as appropriate. Copyright 2012, 2009, 2003 by Saunders, an imprint of Elsevier Inc. All rights reserved. 16

Decimals Decimals are fractional parts of a number. Decimals appear to the right of a whole number. If no digit to left of decimal point, always add 0 to the left. Decimal may be changed to fraction or mixed number: Numbers to left of decimal remains whole number Numbers to right of decimal becomes numerator for fraction Denominator becomes power of 10 with zero for each place in decimal Example 12.467 becomes 12 467/1000 Copyright 2012, 2009, 2003 by Saunders, an imprint of Elsevier Inc. All rights reserved. 17

Rounding Decimals Decimals can be rounded to nearest whole or multiples of 10 Rounding shortens decimal by dropping one or more digits behind decimal point Digit 5 determines rounding 5 or above round up Below 5 round down Example: 5.456 can be rounded to 5.46, 5.5, or 5 depending on the desired position for rounding Copyright 2012, 2009, 2003 by Saunders, an imprint of Elsevier Inc. All rights reserved. 18

Converting Fractions to Decimals To convert proper fraction to decimal, divide numerator by denominator Example 9/10 = 9 10 = 0.9 (notice the 0 in front of decimal) To convert improper fraction to decimal, divide numerator by denominator. Will have a whole number plus a decimal. Example 14/8 14 8 = 1.5 Copyright 2012, 2009, 2003 by Saunders, an imprint of Elsevier Inc. All rights reserved. 19

Adding and Subtracting Decimals Set up list of numbers with decimal point of each number written directly under the decimal point above Do math calculation and place decimal point in assigned position Place decimal point behind any whole number Copyright 2012, 2009, 2003 by Saunders, an imprint of Elsevier Inc. All rights reserved. 20

Multiplying Decimals Align numbers without regard to decimal points and complete multiplication Insert decimal point by counting number of decimal places in each line. Place decimal point in the product, being sure to count from right to left. Determine the product of 92.3 x 4.66 Copyright 2012, 2009, 2003 by Saunders, an imprint of Elsevier Inc. All rights reserved. 21

Dividing Decimals Change divisor with decimal to whole number by moving decimal point to right in divisor; move decimal that number of digits in the dividend as moved in divisor; add zeros to right in dividend for number of places moved in divisor if insufficient numbers are not present. Determine the quotient of 48.2 0.68 to the hundredth. Copyright 2012, 2009, 2003 by Saunders, an imprint of Elsevier Inc. All rights reserved. 22

Review of Decimals To read decimal, number to left of decimal is read as whole number; decimal point is read as and, with numbers to right of decimal read as number of spaces to right (including decimal point as one space) with th added to number (0.5 would be tenth). To round decimal to certain place, begin at furthermost right digit and move to left. If number is 5 to 9, round up; if number is 4 to 0, round down. To add or subtract decimals, align decimal points and calculate as with other mathematical calculations. If no decimal point, add decimal point behind whole number to assist with alignment. Be sure decimal point is correctly placed in answer. Copyright 2012, 2009, 2003 by Saunders, an imprint of Elsevier Inc. All rights reserved. 23

Review of Decimals To multiply decimals, no alignment of decimal points is used. Multiply the numbers; count the number of decimal places and insert decimal point that many spaces in answer, moving right to left in answer. To divide decimals, make divisor whole number and place decimal point in number of places moved in the dividend. Add zeros as needed to supply adequate number of digits. Divide, placing decimal in the correct placement in the quotient. Copyright 2012, 2009, 2003 by Saunders, an imprint of Elsevier Inc. All rights reserved. 24

Review of Decimals To change fraction to decimal, divide numerator of fraction by denominator. If improper fraction, a whole number plus a fraction will occur. To change decimal to fraction, place the decimal number as the numerator with the denominator being one place for number of places in decimal with the decimal point being one place. Reduce fraction as appropriate. Copyright 2012, 2009, 2003 by Saunders, an imprint of Elsevier Inc. All rights reserved. 25

Percents Percent mean hundredths or parts per 100 May be seen as a fraction, a whole number, or a mixed number When changing percent expressed as fraction to percent expressed as decimal, the result is still a percent that must be changed to decimal number for further calculations Copyright 2012, 2009, 2003 by Saunders, an imprint of Elsevier Inc. All rights reserved. 26

Changing Percents to Decimals Drop % sign and divide by 100 (Word percent means part of 100) This causes decimal point to move 2 places to left Percents that are a fraction must be changed to decimal percents before dividing by 100 Copyright 2012, 2009, 2003 by Saunders, an imprint of Elsevier Inc. All rights reserved. 27

Multiplying Percents When asked to find percent of number, of means to multiply a number by percent Change percent to decimal and multiply number by decimal Input correct number of decimal places, moving right to left Always drop zeros that appear after decimal as trailing zeros Find 3% of 42 Copyright 2012, 2009, 2003 by Saunders, an imprint of Elsevier Inc. All rights reserved. 29

Dividing Percents 15 is what percent of 45? To answer the question of what Set up problem as fraction Simplify fraction if possible Change decimal to percent Step 1: Write as a fraction: 15/45 Step 2: Simplify the fraction: 15/45 = 1/3 Step 3: Divide: 3 1.00 = 0.33 Step 4: Change step 3 answer to a percent: 0.33 = 33% Copyright 2012, 2009, 2003 by Saunders, an imprint of Elsevier Inc. All rights reserved. 31

Using Fractions to Figure Percentages Unknown number is x = is identified by is % is found over 100 What is 20% of 200? Copyright 2012, 2009, 2003 by Saunders, an imprint of Elsevier Inc. All rights reserved. 32

Using Fractions to Figure Percentages Formula for fractions a = c b d First fraction shows numbers given in the problem Second fraction show number as percentage of 100 The unknown is listed as x a the amount or part of whole being compared to base b base that follows word of c percent of 100 (number or unknown followed by %) d always 100 when solving for percents Copyright 2012, 2009, 2003 by Saunders, an imprint of Elsevier Inc. All rights reserved. 33

Using an Equation to Figure Percentage What (x) is (=) 20% of (X) 200? What is x is represents = of translates to times (X) Copyright 2012, 2009, 2003 by Saunders, an imprint of Elsevier Inc. All rights reserved. 37

Ratio and Proportion Ratio shows relationship between two numbers Proportion shows relationship between two ratios Proportional method used to find relationships between ratios, including finding unknowns and dosage calculations Two colons (::) shows relationship between two ratios One colon (:) shows relationship between two numbers Copyright 2012, 2009, 2003 by Saunders, an imprint of Elsevier Inc. All rights reserved. 38

Solving for x in Proportion Product of means must equal product of extremes Solve for x in this proportion: 1: x :: 2 : 8 Solving for unknown easy when ratio/proportion used Copyright 2012, 2009, 2003 by Saunders, an imprint of Elsevier Inc. All rights reserved. 39

Using Ratios and Proportions to Similar for solving for x Formula a : b :: c : d Figure Percents a amount or part of whole being compared to the base b means or whole number in the problem or the standard used for comparison c always a percent of 100 d will always be 100 or total amount of item Copyright 2012, 2009, 2003 by Saunders, an imprint of Elsevier Inc. All rights reserved. 40

Using Ratios and Proportions to Figure Percent 15 is what % of 45? 15 and 45 are numbers being compared What is % So the formula will be 15 : 45 :: x% : 100 45 x = 1500 (15 x 100) X = 33.3% Copyright 2012, 2009, 2003 by Saunders, an imprint of Elsevier Inc. All rights reserved. 41

Using Ratios and Proportions to Figure Percent 21 is 60% of what number? 21 : x :: 60 : 100 Only one number of the first ratio is given so x becomes part of first ratio 60 x = 2100 x = 35 Copyright 2012, 2009, 2003 by Saunders, an imprint of Elsevier Inc. All rights reserved. 42

Using Ratios and Proportions to Figure Percent (cont d.) What number is 60% of 35? x : 35 :: 60 : 100 100 x = 2100 x = 21 Copyright 2012, 2009, 2003 by Saunders, an imprint of Elsevier Inc. All rights reserved. 43

Review of Ratio/Proportion If two fractions are equivalent, cross-products will be equal. If cross-products in proportional equation are equal, fractions are equivalent or proportion is true. If same number is added or subtracted from each member of an equality, sum or differences will remain equal. To change a percent to a decimal, multiply by 0.01 or move the decimal point two places to left. Copyright 2012, 2009, 2003 by Saunders, an imprint of Elsevier Inc. All rights reserved. 44

Review of Ratio/Proportion To solve for unknown proportion, multiply the means and the extremes, setting the equation with the x on the left and the quotient of two numbers on right. To solve percent proportion, the what translates to the unknown; of translates to times or x ; is translates to equal or = ; and % may be either 1/100 or 0.01 for multiplication of the percentage. Always read percentage proportions and insert the translation into the formula and then solve for unknown. Copyright 2012, 2009, 2003 by Saunders, an imprint of Elsevier Inc. All rights reserved. 45