2004 AMC 12B Problem 15

Below is the professionally curated solution for Problem 15 of the 2004 AMC 12B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2004 AMC 12B solutions, or check the answer key.

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Concepts:agesplace valuelinear equation

Difficulty rating: 1510

15.

The two digits in Jack's age are the same as the digits in Bill's age, but in reverse order. In five years Jack will be twice as old as Bill will be then. What is the difference in their current ages?

99

1818

2727

3636

4545

Solution:

Let Jack be 10x+y10x + y and Bill be 10y+x.10y + x. Then 10x+y+5=2(10y+x+5),10x + y + 5 = 2(10y + x + 5), so 8x=19y+5.8x = 19y + 5. Testing digits, only y=1,x=3y = 1, x = 3 works, so Jack is 3131 and Bill is 13.13. The difference is 3113=18.31 - 13 = 18.

Thus, the correct answer is B.

Problem 15 in Other Years