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By Andreas Brandstädt, Van Bang Le, Jeremy P. Spinrad

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B. Sample response: This string is already easy because you can easily subtract 24 from 44 to get 20, so the answer is 520. You can do this mentally in one step, so the arrow string is already the easiest one. c. Sample response: ؊ 25 ؉ 54 29 ⎯ ⎯→ 4 ⎯⎯⎯→ 58 This string is easier because when you subtract 25 first, it leaves an easy number to work with. Answers to Check Your Work 51 Answers to Check Your Work d. Sample response: ؉2 ؉ 32 998 ⎯⎯→ 1,000 ⎯⎯⎯→ 1,032 This string is easier because when you add 2 to 998, you get lots of zeros that are easy to work with.

7 ؉ 73 e. 5 ⎯ ⎯⎯→ 35 ⎯⎯⎯→ 108 cm A stack of five chairs requires 108 cm of space. ؋7 ؉ 73 6⎯ ⎯⎯→ 42 ⎯⎯⎯→ 115 cm So six chairs will fit. ؋7 ؉ 73 7⎯ ⎯⎯→ 49 ⎯⎯⎯→ 122 cm Seven will not. Answers to Check Your Work 53 Answers to Check Your Work Section D Reverse Operations 1. a. 609 ഠ 6,581 km. Your answer should not contain decimals because the original measurement is rounded to the nearest mile. b. 6 ഠ 6,544 km c. 6,581 ؊ 6,544 ഠ 37 Discuss your answer with a classmate. Sample responses: • You do not know how 4,090 miles was measured.

Arithmetic trees are useful because they resolve any question about the order of the calculation. The problem is that they require a lot of room on your paper. Copy the first tree below. 15 6 4 ؉ ____ ؊ ____ Since the 6 ؉ 4 is simplified first, circle it on your copy. 15 6 4 ؉ ____ ؊ ____ The tree can then be simplified. 15 10 ؊ ____ Instead of the second arithmetic tree, you could write: 15 ؊ 6 ؉ 4 27. What does the circle represent? The whole circle is not necessary. People often write 15 ؊ (6 ؉ 4).

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