2004 AMC 10B Problem 17

Attempt Problem 17 of the 2004 AMC 10B below, then check your answer against the professionally curated solution 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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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?

$9$

$18$

$27$

$36$

$45$

Answer: B

Concepts:ages digits Diophantine Equation

Difficulty rating: 1410

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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