Problem:
Ashley, Betty, Carlos, Dick, and Elgin went shopping. Each had a whole number of dollars to spend, and together they had . The absolute difference between the amounts Ashley and Betty had to spend was . The absolute difference between the amounts Betty and Carlos had was , between Carlos and Dick was , between Dick and Elgin was , and between Elgin and Ashley was . How much did Elgin have?
Answer Choices:
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Solution:
Let , and denote the amounts Ashley, Betty, Carlos, Dick, and Elgin had for shopping, respectively. Then , and . The sum of the left sides is zero, so the sum of the right sides must also be zero. In other words, we must choose some subset of which has the same element-sum as its complement. Since , the sum of the members of is . Hence is either the set or its complement . Thus either and are the only positive differences or , and are. In the former case, expressing , and in terms of , we get , which yields . In the latter case, the same strategy yields , which leads to non-integer values. Hence .