2004 AMC 10B Problem 17

Problem 17 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’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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