1952 AMC 12 Problem 42

Question

Let

D

represent a repeating decimal. If

P

denotes the

r

figures of

D

which do not repeat themselves, and

Q

denotes the

s

figures which do repeat themselves, then the incorrect expression is:

D=0.PQQQ\ldots

10^rD=P.QQQ\ldots

10^{r+s}D=PQ.QQQ\ldots

10^r(10^s-1)D=Q(P-1)

10^r\cdot10^{2s}D=

PQQ.QQQ\ldots

Shift the decimal point first past the

r

nonrepeating digits and then past one block of

s

repeating digits

Subtract

10^rD

from

10^{r+s}D

and compare the result with choice D

After shifting past the nonrepeating block,

10^rD=P.QQQ\ldots,

and after shifting one repeating block farther,

10^{r+s}D=PQ.QQQ\ldots.

Subtracting yields the standard relation

\begin{gathered} 10^r(10^s-1)D \\ =P(10^s-1)+Q. \end{gathered}

This is not

Q(P-1),

as stated in choice D.

Thus, the incorrect expression, and hence the correct answer, is D.

Problem 42 in other years