2004 AMC 10B Problem 17

Below is the professionally curated solution for Problem 17 of the 2004 AMC 10B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2004 AMC 10B solutions, or check the answer key.

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Concepts:ages digits Diophantine Equation

Difficulty rating: 1410

17.

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?

$9$

$18$

$27$

$36$

$45$

Solution:

Let Jack's age be $10x + y$ and Bill's be $10y + x.$ In five years $10x + y + 5 = 2(10y + x + 5),$ which simplifies to $8x = 19y + 5.$

Since $x$ and $y$ are digits, the only solution is $y = 1,$ $x = 3.$

So Jack is $31$ and Bill is $13,$ a difference of $18.$

Thus, the correct answer is B.

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