7.1. Decimals 7.2. Operations with Decimals

Transcription

1 CHAPTER 7 Decimals, Ratio, Proportion, and Percent Addition 7.1. Decimals 7.2. Operations with Decimals (1) Using fractions: = , = , = 17, = (2) Decimal approach align the decimal points, add the numbers in columns as if they were whole numbers, and insert a decimal in the answer immediately beneath the decial points of the numbers being added. Subtraction (1) Using fractions: = , or = 13, = 10, =

2 2 7. DECIMALS, RATIO, PROPORTION, AND PERCENT (2) Decimal approach as with addition or =) Multiplication (1) Estimate: 7 11 = 77 (2) Using fractions: = , 293 = = 10 0 = Note that the location of the decimal matches the estimate.

3 7.2. OPERATIONS WITH DECIMALS 3 (3) Decimal approach multiply as though without decimal points, and then insert a decimal point in the answer so that the number of digits to the right of the decimal in the answer equals the sum of the number of digits to the right of the decimal points in the numbers being multiplied = Again, the placement of the decimal point makes sense in view of the estimate Estimate: = = 4 The placement of the decimal point corresponds with the estimate.

4 4 7. DECIMALS, RATIO, PROPORTION, AND PERCENT Division: (1) Estimate: (2) Using fractions: 6.03 = 6 3 = 6 3 = = = = = = 250 (3) Decimal approach replace the original problem by an equivalent problem where the divisor is a whole number (1) Estimate: (2) Using fractions: 6.03 = 6 3 = 6 3 = = = = = = 250

5 7.2. OPERATIONS WITH DECIMALS 5 (3) Decimal approach replace the original problem by an equivalent problem where the divisor is a whole number

6 6 7. DECIMALS, RATIO, PROPORTION, AND PERCENT Repeating Decimals (1) Fractions in simplified form with only 2 s and 5 s as prime factors in the denominator convert to terminating decimals.

7 7.2. OPERATIONS WITH DECIMALS 7 (2) Fractions in simplified form with factors other than 2 and 5 in the denominator convert to repeating decimals =.4166 =.416 with 6 indicating the 6 repeats indefinitely.

8 8 7. DECIMALS, RATIO, PROPORTION, AND PERCENT = The 27 is called the repetend. Decimals with a repetend are 11 called repeating decimnals. The number of digits in the repetend is the period of the decimal. Terminating decimals are decimals with a repetend of 0, e.g., 0.3 = 0.30.

9 7.2. OPERATIONS WITH DECIMALS 9 Every fraction can be written as a repeating decimal. Ts see why this is so, consider 5. In dividing by 7, there are 7 possible remainders, 0 through 6. Thus 7 a remainder must repeat by the 7th division: = Theorem (Fractions with Repeating, Nonterminating Decimal Representations). Let a be a fraction in simplest form. b Then a has a repeating b decimal representation that does not terminate if and only if b has a prime factor other than 2 or 5.

10 10 7. DECIMALS, RATIO, PROPORTION, AND PERCENT Changing a repeating decimal into a fraction has a period of 3, so we use 10 3 = 0. Let n = Then 0n = n = n = n = n = Change.439 to a fraction..439 has a period of 1, so we use 10 1 = 10. Let n =.439. Then 10n = n = n = So.439 =.44 = n = 3.96 n = = = 44 {z} Notice n =.44 = We have two decimal numerals for the same number. When 9 repeats, you cvan drop the repetend and increase the preivious digit by 1 to get a terminating decimal. Theorem. Every fraction has a repeating decimal representation, and every repeating decimal has a fraction representation.

11 7.3. RATIO AND PROPORTION Ratio and Proportion On a given farm, the ratio of cattle to hogs is 7 : 4. (This is read 7 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: a) 7 11 a) 4 11 of the livestock are cattle. of the livestock are hogs. 4)There are 7 4 5) There are 4 7 as many cattle as hogs. as many hogs as cattle. 6) 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, with b 6= 0. Note. 1) Ratios allow us to compare the relative sizes of 2 quantities. 2) The ratio a : b can also be represented by the fraction a b.

12 12 7. DECIMALS, RATIO, PROPORTION, AND PERCENT 3) Ratios can involve any real numbers: 3.5 : 1 or 3.5 1, 7 2 : 3 4 or 7/2 3/4, p 2 : or p 2 4) Ratios can be used to express 3 typres of comparisons: a) part-to-part A cattle to hog ratio of 7 : 4. b) part-to-whole A hog to livestock ratio of 4 : 11. c) whole-to-part Livestock to cattle ratio of 11 : 7. 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). there would then be 60 corresponding groups of 4 hogs each, or 60 4 = 240 hogs. Definition (Equality of Ratios). Let a b and c d be any two ratios. Then a b = c d Note. if and only if ad = bc. 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 the product of the means. 2) Just as with fractions, if n 6= 0, an bn = a or an : bn = a : b. b

13 Definition. A proportion is a statement that 2 ratios are equal RATIO AND PROPORTION 13 Write a fraction in simplest form that is equivalent to the ratio 39 : : 91 = = = 3 7 Are the ratios 7 : 12 and 36 : 60 equal?. Extremes: 7 60 = 420 Means: = 432 The ratios are not equal. Solve for the unknown in the proportion B 8 = B = =) 18B = =) 18B = 16+2 =) 18B = 18 =) B = 1 Solve for the unknown in the proportion 3x 4 = 12 6 x. 18x = 4(12 x) =) 18x = 48 4x =) 22x = 48 =) x = = Solve the follwing proportions mentally: 1) 26 miles for 6 hours is equal to for 24 hours. 104

14 14 7. DECIMALS, RATIO, PROPORTION, AND PERCENT 2) 750 people for each 12 square miles is equal to people for each 16 square miles. 0 If one inch on a map represents 35 miles and two cities are 0 miles apart, how many inches apart would the be on the map? Use a table: We have 1 35 = x 0 scale actual inches 1 x miles 35 0 (notice how the unit align). 35x = 0 x = 0 35 = = A softball pitcher has given up 18 earned runs in 39 innings. How many earned runs does she give up per seven-inning game (ERA) season game earned runs 18 x innings = x 7 39x = 126 x = =

15 7.4. PERCENT Percent Percent means per hundred and % is used to represent percent. 60 percent = 60% = 60 =.60 In general, Conversions: 530 percent = 530% = 53 = 5.30 n% = n (1) Percents to fractions use the definition 37% = 37 (definition). (2) Percents to decimals go percent to fraction to decimal 67% = 67 =.67 Shortcut drop % sign and move the dcimal two places to the left. 54% =.54 5% = % = 3.72 (3) Decimals to percents reverse the shortcut of step (2) (move the decimal two places to the right and add the % sign.

16 16 7. DECIMALS, RATIO, PROPORTION, AND PERCENT.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 and 5 s as factors) can be expressed as a fraction with a denominator of. 5 8 = = 62.5 =.625 = 62.5% 3 (long division).429 = 42.9% 7 Common Equivalents Percent Fraction 5% % % % % % % % 3 4

17 Find mentally: 196 is 200% of. 25% of 244=. 40 is % of x = 196 =) x = PERCENT = = = 5 4 = = % + 25% = 125% is 50% of. 1 x = 731 =) x = = % of 300 is. Find 15% of % = % % = = = Find 300% of % = 10% + 5% = = = = 240 Find % of = 70

18 18 7. DECIMALS, RATIO, PROPORTION, AND PERCENT Estimate mentally: 21% of % of 431. (10 + 1)% = Solving Percent Problems (1) Grid approach. 1 5 of 35 = of 430 = = 47 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 squares are marked. Each square represents 14, 000 = 140 dollars (1% of $14,000). Thus 30 squares represent 30% of $14,000 or 30 $140 = $4200.

19 7.4. PERCENT 19 (2) Proportion approach since percents can be written as a ratio. A volleyball team wins 105 games, which is 70% of the games played. How many games were played? 70 = 105 x percent actual wins games x =) 70x = 10, 500 =) x = 150 games played If Frank saves $28 of his $240 weekly salary, what percent does he save? actual percent saved 28 x salary = x Frank saves % =) 240x = 2800 =) x = 240 = 35 3 (3) Equation approach (x is unknown; p, n, and a are fixed numbers). Translation of Problem Equation p (a) p% of n is x n = x p (b) p% of x is a x = a x (c) x% of n is a n = a

20 20 7. DECIMALS, RATIO, PROPORTION, AND PERCENT Sue is paid $ a week plus a 6% comission on sales. Find her weekly earnings if the sales for the week are $ Translation (a): x = = Salary = $ $34.50 = $ A department store marked down all summer clothing 25%. The following week, remaining items were marked down 15% o the sale price. When John bought 2 tank tops, he presented a coupon that gave him an additional 20% o. What percent of the original price did John save? solution. x = percent saved, Translation (c): P = original price x P = P price John paid 80 = P (2nd markdown) 80 h 85 = P = P 80 x P = P.51P =.49P x =.49 x = 49% i (1st markdown) h i P