2004 AMC 12B Problem 15

Problem 15 of 25IntermediateAlgebra Number Theory

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?

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Solution

Let Jack be

10x + y

and Bill be

10y + x.

Then

10x + y + 5 = 2(10y + x + 5),

8x = 19y + 5.

Testing digits, only

y = 1, x = 3

works, so Jack is

31

and Bill is

13.

The difference is

31 - 13 = 18.

Thus, the correct answer is B.

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