2014 AMC 12A Problem 6

Attempt Problem 6 of the 2014 AMC 12A 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 2014 AMC 12A solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

The difference between a two-digit number and the number obtained by reversing its digits is

5

times the sum of the digits of either number. What is the sum of the two-digit number and its reverse?

44

55

77

99

110

Write the number as

10a+b

and its reverse as

10b+a

The difference is

9(a-b)

, and setting it equal to

5(a+b)

gives

2a=7b

Let the larger number be

10a+b.

Then

\begin{aligned} &(10a+b)-(10b+a)=9(a-b)\\ &\quad{}=5(a+b), \end{aligned}

which simplifies to

2a=7b.

The only nonzero digits satisfying this are

a=7

and

b=2,

so the number is

72

and its reverse is

27.

Their sum is

72+27=99.

Thus, the correct answer is D.

Problem 6 in other years