2004 AMC 10B Problem 17

Question

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

Small hint:

Write Jack’s age as

10x + y

and Bill’s as

10y + x

Big hint:

The condition

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

reduces to

8x = 19y + 5

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.