A C E. Answers Investigation 3. Applications = 0.42 = = = = ,440 = = 42

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1 Answers Investigation Applications 1. a b , , ,440 d , e ,000 2,52 f a b If , then , and If ,544, then If 7, ,416, then If 2,405 5,17 12,787,85, then The correct choice is A. 8. The correct choice is F because both factors are largest given among the other products.. a ,800 b , d e ,228 f Failure to recognize that the 1 in 1 is actually, so the partial product should be 4,720, and the correct result should be 8, Again, failure to recognize that the 1 in 12 is actually 0, and the 2 in 12 is actually 20, so the partial products should be 6,540 and 2,700, and the correct result is 40, In this case it appears that the error is simply bringing down the decimal point instead of recognizing that the product will have two decimal places to make the correct result 8.68 (an estimate of should catch that). You multiply 1 by 1 to get In this case it appears that the calculation was done by multiplying each digit of one factor times the corresponding digit of the other factor (e.g instead of ). The correct product is 15, ; exact result is ; exact result is ; exact result is ; exact result is ; exact result is ; exact result is The whole-number product is 12,501, and the factors are approximately and 5. So the exact result must be close to 15. The only choice is rounds to $ a m 2 b. $288.8 (might be $288. if the seller rounds up) 2. a. The add a zero procedure only works for whole numbers and instead of actually adding what Shing means is to put a zero at the right end of the number. Decimal Ops 1 Investigation

2 Answers Investigation b. Even the put a zero at the end of the number rule doesn t work when a number has nonzero digits to the right of the decimal point. In that case, the rule is to move the decimal point one place to the right. This works because a decimal like.245 is equivalent to a fraction with a denominator that is some multiple of (in the case of.245, the equivalent common fraction is,245 1,000. When this number is multiplied by and the result simplified, we get, Multiplying by 0 is the same as multiplying by twice, by 1,000 is multiplying by three times, and by,000 is multiplying by four times. Thus the results are moving the decimal point 2 places to the right, places to the right, and 4 places to the right. 24. a Estimation of the quotient might give b Estimation of the quotient might give Estimation 0 0 of the quotient might give d Estimation of the quotient might give ,50 e Estimation of the quotient might give 2, f Estimation of the quotient might give a b. d ,400 25, , a b d e f a b. 1, ,400, d. 12,400.2,875 e f Only (a) and (c) give the same result because if you multiply dividend and divisor in (c) by 0, you get problem (a). That is, the fraction is equivalent to The fact family is: ; ; ; a. N a. b. N 0.06 N d. N 2.8 e. N 0.5 f. N 12 b , , Decimal Ops 2 Investigation

3 Answers Investigation a. This is correct. b. The correct answer is $ gallons $4 per gallon m m 6 m 4. a. Since there are 60 seconds in a minute, 2(60) b seconds 4 runners seconds per runner 44. a miles 45. a b d. 2.4 miles 2. hours 1.0 miles per hour 46. a. 2(21) ; or for each side b. 162 $0.15 $ yards 1. yards per bow 4., so she can make 4 bows and has nearly enough for one more bow. b miles 8.07 hours 17.4 miles per hour miles miles per hour Connections 48. a. 0 b. 00,000 d. 500,000 e. 50,200,000 f. 520,000,000 g. The exponent tells you how many zeros you should include at the end of the number. 4. a. 85 b d. 4,50 e. 27 f. 128 g. 12 h..8 or 8 i..8 or 8 j..8 or a b. 6 4 d. 4 e. 2 f..45 g. 4 h. 16 i. 8 5 j a. Carpet C is longest. b. Carpet B has the greatest area ( m 2 ). Carpet C costs $25.54; Carpet B costs $12.47; Carpet A costs $4.40, so Carpet A costs the most and Carpet B costs the least. Decimal Ops Investigation

4 Answers Investigation 52. a. Answers may vary. One possible length and width are / 7 ft and w 8 ft. b. Answers may vary. One possible length and width are / 7 ft and w 0.8 ft. Answers may vary. One possible length and width are / 0.7 ft and w 0.8 ft. 5. a b d e f Fraction Decimal a. The pattern suggests that for 1 n <, the repeating decimal for n ninths will be 0.nnn. For n, the decimal is 0. or or 1. b. would be 0. (so this must equal 1) would be would be and a. From left to right, the marks will be 1.8, 1.85, 1.0, 1.5, and 2.0. b. From left to right the marks will be 1.00, 1.025, 1.050, 1.075, and 1.. From left to right the marks will be 2., 2.5, 2.4, 2.45, and 2.5. d. From left to right the marks will be 1., 1.5, 2.00, 2.005, and e. One strategy would be to find the difference between the coordinates of the largest and smallest marked points and divide that difference into 4 equal pieces to see the step from each point to the next in cm ft m a. The area of the square is 2.25 m 2. b. The areas of the pieces are 0.25 m 2, 0.5m 2, 0.5m 2, and 1m 2. The sum of the areas of pieces in (b) is, of course, equal to the area of the whole square. What the second diagram shows is that (1.5)(1.5) ( )( ), a fact that is a consequence of the Distributive Property of Multiplication Over Addition. 61. Choice D is the only option that has divisor smaller than the dividend, thus giving a quotient greater than a b pounds 64. a. 150 b d. Assuming that the possible point values are 50, 0, and 200 (the only types mentioned so far in the problem), the likely result is correct on a 200 point question and incorrect on a 50 point question. Decimal Ops 4 Investigation

5 Answers Investigation Extensions 65. Belinda s estimate is closer to the exact value, a fact that one might expect because, by rounding 5.2 to only 5. Tanisha has made a large percentage round off. Belinda made a very small round off. 66. a. 0. b a b , , a. The decimal points could be placed in a variety of ways as long as there are a total of 4 digits to the right of the decimal points. One possibility is b. The decimal points could be placed in a variety of ways as long as there are a total of 4 digits to the right of the decimal points. One possibility is a. The number line can be marked in any number of different ways, as long as the increments between successive marks are equal and 2.1 fits in the middle. The most obvious, but not the only, way to mark the points from left to right would be 1., 2.0, 2.1, 2.2, and 2.. However, quite different sequences, such as 1.1, 1.6, 2.1, 2.6, and.1 are logically acceptable as well. One could also have 1.7, 1., 2.1, 2., and 2.5, et b. The mean of every sequence is 2.1 because it is a kind of balance point between equally-spaced numbers above and below. It is not possible to label the points in a way that gives a sum of, because there are five numbers and a sum of would imply a mean of 2, not the 2.1 that is required. 71. a. The greatest change (42.4 points) occurred between 168 and 172 when there must have been a change in the scoring system. 72. b. The next greatest change (285.5) occurred between 176 and 180. The least change (7.26) occurred between 16 and d , , miles per gallon The area of the rectangle is 4.2 cm 2. The given collection of small squares has 12 such presumably congruent squares, each with area n cm2. Any square with that area has each side of length 0.6 cm, so n 0.6 cm n 4.2, so n and n 1.5 cm (.84)n 4.2, so n and n 2.25 cm. Decimal Ops 5 Investigation