1997 AHSME Problems/Problem 22
Problem
Ashley, Betty, Carlos, Dick, and Elgin went shopping. Each had a whole number of dollars to spend, and together they had 
dollars. The absolute difference between the amounts Ashley and Betty had to spend was 
dollars. The absolute difference between the amounts Betty and Carlos had was 
dollars, between Carlos and Dick was 
dollars, between Dick and Elgin was 
dollars, and between Elgin and Ashley was 
dollars. How many dollars did Elgin have?
Solution
Working backwards, if 
, then 
. Since 
is a positive integer, 
.
Since , we know that 
. But if 
, which is the smallest possible "plus" value, then 
, which is too much money.
Hence, 
. But since 
must be a positive integer, that leaves only two possibilities: 
or 
, which correspond with 
and 
.
Concentrating only on 
, we have leading to 
, which leads to 
, which leads to 
. Thus far we have given out 
dollars. This means that Dick must have 
dollars. However, the difference between Carlos and Dick is not dollars.
Thus, the right answer must be 
. Verifying, if , then 
, 
, which leads to 
. Thus far, we have given out 
dollars, leaving 
dollars for Dick. Dick does indeed have dollars more than Carlos, and dollars more than Elgin.
See also
| 1997 AHSME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 21 |
Followed by Problem 23 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 | ||
| All AHSME Problems and Solutions | ||
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