Decimals, Ratio, Proportion, And Percent

CHAPTER 7Decimals, Ratio, Proportion, and Percent7.1. Decimals7.2. Operations with DecimalsAdditionExample. 3.71 13.809(1) Using fractions:3.71 13.809 371 13, 809 3710 13, 809 17, 519 17.5191001000100010001000(2) Decimal approach – align the decimal points, add the numbers in columns asif they were whole numbers, and insert a decimal in the answer immediatelybeneath the decial points of the numbers being added.3.713.710 13.809 13.809or17.51917.519SubtractionExample. 13.8093.71(1) Using fractions:13.8093.71 13, 8091000371 13, 809 100100013710 10, 099 10.09910001000

27. DECIMALS, RATIO, PROPORTION, AND PERCENT(2) Decimal approach – as with addition.13.8093.71or10.099Example. 14.313.8093.71017.5197.96114.37.961 )14.3007.9616.339MultiplicationExample. 7.3 11.41(1) Estimate: 7 11 77(2) Using fractions:73 1141 73 · 1141 83, 293 83.2931010010 · 1001000Note that the location of the decimal matches the estimate.7.3 11.41

7.2. OPERATIONS WITH DECIMALS3(3) Decimal approach – multiply as though without decimal points, and theninsert a decimal point in the answer so that the number of digits to theright of the decimal in the answer equals the sum of the number of digitsto the right of the decimal points in the numbers being multiplied.7.3 11.41 11.41 7.3Again, the placement of the decimal point makes sense in view of the estimate.Example. 421.2 .0076Estimate:400 .01 400 1 4100The placement of the decimal point corresponds with the estimate.

47. DECIMALS, RATIO, PROPORTION, AND PERCENTDivision:Example. 6.5 0.026(1) Estimate:6 .03 6 3100 600 6 20010033(2) Using fractions:6.5 0.026 65266500266500 25010 1000 1000 100026(3) Decimal approach – replace the original problem by an equivalent problemwhere the divisor is a whole numberExample. 6.5 0.026(1) Estimate:6 .03 6 3100 600 6 20010033(2) Using fractions:6.5 0.026 65266500266500 25010 1000 1000 100026

7.2. OPERATIONS WITH DECIMALS5(3) Decimal approach – replace the original problem by an equivalent problemwhere the divisor is a whole numberExample. 1470.3838 26.57

67. DECIMALS, RATIO, PROPORTION, AND PERCENTRepeating Decimals(1) Fractions in simplified form with only 2’s and 5’s as prime factors in thedenominator convert to terminating decimals.Example.Example.

7.2. OPERATIONS WITH DECIMALS7(2) Fractions in simplified form with factors other than 2 and 5 in the denominator convert to repeating decimals.5Example.125 .4166 · · · .416 with 6 indicating the 6 repeats indefinitely.12

87. DECIMALS, RATIO, PROPORTION, AND PERCENTExample.3113 0.27. The “27” is called the repetend. Decimals with a repetend are11called repeating decimnals. The number of digits in the repetend is the periodof the decimal.Terminating decimals are decimals with a repetend of 0, e.g., 0.3 0.30.

7.2. OPERATIONS WITH DECIMALS9Every fraction can be written as a repeating decimal. Ts see why this is so,5consider . In dividing by 7, there are 7 possible remainders, 0 through 6. Thus7a remainder must repeat by the 7th division:5Example.75 0.7142857Theorem (Fractions with Repeating, Nonterminating Decimal Represenaatations). Letbe a fraction in simplest form. Thenhas a repeatingbbdecimal representation that does not terminate if and only if b has a primefactor other than 2 or 5.

107. DECIMALS, RATIO, PROPORTION, AND PERCENTExample. Changing a repeating decimal into a fraction.18.634 has a period of 3, so we use 103 1000.Let n 18.634. Then 1000n 18634.634.1000n 18634.634634 · · ·n 18.634634 · · ·999n 1861618616n 999Example. Change .439 to a fraction.439 has a period of 1, so we use 101 10.Let n .439. Then 10n .439.10n 4.39999 · · ·n .43999 · · ·9n 3.963.96 396n 9900So .439 .44 .440.4411 10025 {z}Notice n .44We have two decimal numerals for the same number. When 9 repeats, you cvandrop the repetend and increase the preivious digit by 1 to get a terminatingdecimal.Theorem. Every fraction has a repeating decimal representation, andevery repeating decimal has a fraction representation.

7.3. RATIO AND PROPORTION117.3. Ratio and ProportionExample. On a given farm, the ratio of cattle to hogs is 7 : 4. (This is read7 to 4.).What this means:1) For every 7 cattle, there are 4 hogs.2) For every 4 hogs, there are 7 cattle.3) Assuming there are no other types of livestock on the farm:7a)of the livestock are cattle.114a)of the livestock are hogs.1174)There are as many cattle as hogs.445) There are as many hogs as cattle.76) Again assuming no other types of livestock:a) 7 of 11 livestock are cattle.a) 4 of 11 livestock are hogs.Definition. A ratio is an ordered pair of numbers, written a : b, withb 6 0.Note.1) Ratios allow us to compare the relative sizes of 2 quantities.a2) The ratio a : b can also be represented by the fraction .b

127. DECIMALS, RATIO, PROPORTION, AND PERCENT3) Ratios can involve any real numbers:Example.3.5 : 1 or3.5,17 37/2: or,2 43/4p2 : orp2 4) Ratios can be used to express 3 typres of comparisons:a) part-to-partA cattle to hog ratio of 7 : 4.b) part-to-wholeA hog to livestock ratio of 4 : 11.c) whole-to-partLivestock to cattle ratio of 11 : 7.Example. Suppose our farm has 420 cattle. How many hogs are there?Solution. The cattle can be broken up into 60 groups of 7 (420 7). therewould then be 60 corresponding groups of 4 hogs each, or 60 · 4 240 hogs. Definition (Equality of Ratios).aca cLet and be any two ratios. Then if and only if ad bc.bdb dNote.1) a and d are called the extremes and b and c are called the means“a : b {z }c : d if and only if ad bc.” means{z }extremes“Two ratios are equal if and only if the product of the extremes equals theproduct of the means.”an a2) Just as with fractions, if n 6 0, or an : bn a : b.bnb

7.3. RATIO AND PROPORTION13Definition.A proportion is a statement that 2 ratios are equal.Example.Write a fraction in simplest form that is equivalent to the ratio 39 : 91.39 13 · 3 339 : 91 91 13 · 7 7Example.Are the ratios 7 : 12 and 36 : 60 equal?.Extremes: 7 · 60 420The ratios are not equal.Means: 12 · 36 432Example.B 2 14Solve for the unknown in the proportion .818 1 118B 8·2 ) 18B 8 2 ) 18B 16 2 ) 18B 18 ) B 144Example.3x 12 xSolve for the unknown in the proportion .4618x 4(12x) ) 18x 484x ) 22x 48 ) x Example.Solve the follwing proportions mentally:1) 26 miles for 6 hours is equal tofor 24 hours.10448 24 22 11

147. DECIMALS, RATIO, PROPORTION, AND PERCENT2) 750 people for each 12 square miles is equal tosquare miles.people for each 161000Example.If one inch on a map represents 35 miles and two cities are 1000 miles apart,how many inches apart would the be on the map?Use a table:scale actualinches 1xmiles 35 10001x (notice how the unit align).35 100035x 10001000 2004x 28 28.573577Example.We haveA softball pitcher has given up 18 earned runs in 39 innings. How many earnedruns does she give up per seven-inning game (ERA)season gameearned runs 18xinnings39718 x 39 739x 126126 42x 3.233913

7.4. PERCENT157.4. PercentPercent means per hundred and % is used to represent percent.6060 percent 60% .6010053530 percent 530% 5.30100In general,nn% (definition).100Conversions:(1) Percents to fractions – use the definitionExample.37% 37100(2) Percents to decimals – go percent to fraction to decimalExample.67% 67 .67100Shortcut – drop % sign and move the dcimal two places to the left.Example.54% .545% .05372% 3.72(3) Decimals to percents – reverse the shortcut of step (2) (move the decimaltwo places to the right and add the % sign.

167. DECIMALS, RATIO, PROPORTION, AND PERCENTExample.73 73%2.17 217%.235 23.5%(4) Fractions to percents – go fraction to decimal to percent.Note. fractions with terminating decimals (denominator only has 2’s and5’s as factors) can be expressed as a fraction with a denominator of 100.Example.562562.5 .625 62.5%8 10001003 (long division) .429 42.9%7Common EquivalentsPercent Fraction15%2010%11020%1525%1433 13 %1350%1266 23 %2375%34

7.4. PERCENTExample. Find mentally:196 is 200% of.2x 196 ) x 25% of 244 40 is.1 196 9821 244 614% of 32.40 51 1 100% 25% 125%32 44731 is 50% of.1x 731 ) x 2 731 14622166 23 % of 300 is.222166 % 100% 66 % 1 333 21 300 300 200 5003Find 15% of 40.Find 300% of 120.Find 33 13 % of 210.1115% 10% 5% 10 20 1 1 40 4 2 610 202 120 2401 210 70317

187. DECIMALS, RATIO, PROPORTION, AND PERCENTExample. Estimate mentally:21% of 34.11.2% of 431.1of 35 751 (10 1)% of 430 43 4 4710 100Solving Percent Problems 1(1) Grid approach.Example. A car was purchased for 14,000 with a 30% down payment.How much was the down payment?Let the grid below represent the total cost of 14,000. Since the down payment is 30%, 30 of 100 squares are marked.Each square represents14, 000 140 dollars (1% of 14,000).100Thus 30 squares represent 30% of 14,000 or30 140 4200.

7.4. PERCENT19(2) Proportion approach – since percents can be written as a ratio.Example. A volleyball team wins 105 games, which is 70% of the gamesplayed. How many games were played?percent actualwins70105games 100x70105 ) 70x 10, 500 ) x 150 games played100xExample. If Frank saves 28 of his 240 weekly salary, what percent doeshe save?actual percentsaved 28xsalary 24010028x2800 35 ) 240x 2800 ) x 240 1002403Frank saves 11 23 %.(3) Equation approach (x is unknown; p, n, and a are fixed numbers).Translation of Problem Equationp (a) p% of n is xn x100 p (b) p% of x is ax a100 x (c) x% of n is an a100

207. DECIMALS, RATIO, PROPORTION, AND PERCENTExample. Sue is paid 315.00 a week plus a 6% comission on sales. Findher weekly earnings if the sales for the week are 575.00.6Translation (a): x · 575 34.5.100Salary 315.00 34.50 349.50.Example. A department store marked down all summer clothing 25%. Thefollowing week, remaining items were marked down 15% o the sale price. WhenJohn bought 2 tank tops, he presented a coupon that gave him an additional20% o . What percent of the original price did John save?solution.x percent saved,P original priceTranslation (c):xP P100 P P PxP P100x .49100x 49%price John paid80· (2nd markdown)100i80 h 85·· (1st markdown)100 10080 h 85 75 i··P100 100 100.51P .49P

10 7. DECIMALS, RATIO, PROPORTION, AND PERCENT Example. Changing a repeating decimal into a fraction. 18.634