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:

$D=0.PQQQ\ldots$

$10^rD=P.QQQ\ldots$

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

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

$10^r\cdot10^{2s}D=$ $PQQ.QQQ\ldots$

Answer: D

Concepts:repeating decimal place value algebraic manipulation

Difficulty rating: 2060

Small Hint:

Shift the decimal point first past the $r$ nonrepeating digits and then past one block of $s$ repeating digits

Big Hint:

Subtract $10^rD$ from $10^{r+s}D$ and compare the result with choice D

Solution:

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.

1952 AMC 12 Problem 42

Answer: D

Difficulty rating: 2060

Small Hint:

Big Hint:

Solution:

Problem 42 in Other Years