1) Abracadabra has four different coins with values as shown below. Suppose you had just one of each of the four different coins. How many different amounts can be made using one or more of the four different coins?
2) Tom multiplied a number by 2 1/2 and got 50 as an answer. However, he should have divided the number by 2 1/2 to get the correct answer. What is the correct answer? 3) The sum of the ages of Al and Bill is 25; the sum of the ages of Al and Carl is 20; the sum of the ages of Bill and Carl is 31. Who is the oldest of the three and how old is he? Solutions 1) List the possible amounts: 1 coin: produces 1, 2, 4, 8 2 coins: produces 1 + 2, 1 + 4, 1 + 8, 2 + 4, 2 + 8, 4 + 8 3 coins: produces 1 + 2 + 4, 1 + 2 + 8, 1 + 4 + 8, 2 + 4 + 8 4 coins: produces 1 + 2 + 4 + 8 There are 15 different amounts that can be made. Notice that the amounts are the natural numbers from 1 to 15 inclusive. 2) What number multiplied by 2 1/2 gives 50 as an answer? This question is equivalent to finding what number is equal to 50 divided by 2 1/2? That number is 20. To get the correct answer, divide 20 by 2 1/2 producing 8. 3) Method 1 Algebra: Let A, B, and C represent the respective ages of Al, Bill, and Carl. Given (1) A + B = 25 Given (2) A + C = 20 Given (3) B + C = 31 Subtract (2) from (1) (4) B  C = 5 Add (3) and (4) (5) 2B = 36 Divide both sides of (5) by 2 (6) B = 18 From (1) and (2), B is older than C. From (1) and (3), C is older than A. Therefore, Bill is the oldest and he is 18 years old. Method 2 Add conditions (1), (2), and (3) given in Method 1. The sum should be 2A + 2B + 2C = 76. Divide both sides by 2. The result should be (*) A + B + C = 38. Condition (2) states that A + C = 20. Replace A + C in (*) by 20. The result is B + 20 = 38. Clearly B = 18. Therefore Bill is the oldest and he is 18 years old.
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