AMC12 2004 B

AMC12 2004 B · Q15

AMC12 2004 B · Q15. It mainly tests Linear equations, Remainders & modular arithmetic.

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?

Jack的年龄的两个数字与Bill的年龄的两个数字相同，但顺序相反。五年后，Jack的年龄将是Bill当时年龄的两倍。他们目前年龄的差是多少？

(A) 9 9

(B) 18 18

(D) 36 36

(E) 45 45

Answer

Correct choice: (B)

正确答案：（B）

Solution

If Jack's current age is $\overline{ab}=10a+b$, then Bill's current age is $\overline{ba}=10b+a$. In five years, Jack's age will be $10a+b+5$ and Bill's age will be $10b+a+5$. We are given that $10a+b+5=2(10b+a+5)$. Thus $8a=19b+5 \Rightarrow a=\dfrac{19b+5}{8}$. For $b=1$ we get $a=3$. For $b=2$ and $b=3$ the value $\frac{19b+5}8$ is not an integer, and for $b\geq 4$, $a$ is more than $9$. Thus the only solution is $(a,b)=(3,1)$, and the difference in ages is $31-13=\boxed{\mathrm{(B)\ }18}$.

若Jack当前年龄为$\overline{ab}=10a+b$，则Bill当前年龄为$\overline{ba}=10b+a$。五年后，Jack的年龄为$10a+b+5$，Bill的年龄为$10b+a+5$。题设给出$10a+b+5=2(10b+a+5)$。因此$8a=19b+5 \Rightarrow a=\dfrac{19b+5}{8}$。当$b=1$时，得$a=3$。当$b=2$和$b=3$时，$\frac{19b+5}8$不是整数；当$b\geq 4$时，$a$大于$9$。因此唯一解为$(a,b)=(3,1)$，年龄差为$31-13=\boxed{\mathrm{(B)\ }18}$。

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