Bunuel wrote:
A decimal is called a “shrinking number” if its value is between 0 and 1 and each digit to the right of the decimal is not less than the digit to its immediate right. For instance, 0.86553221 is a shrinking number. If x is a shrinking number, which of the following must be true?
I. 9x/10 is a shrinking number.
II. 3.507/10.02 is a shrinking number.
III. a/40 is a shrinking number.
(A) I only
(B) II only
(C) I and II only
(D) III only
(E) I, II, and III
This question is a mess, as written. Statement II appears to be missing an
x, while Statement III has a mysterious
a where we would expect
x to be. Unless the
Manhattan Prep question-writers just tossed in a random variable to catch some people off guard, Statement III almost certainly looks different in the original question. Furthermore, looking at Statement II, we can disprove the purely arithmetic fraction right away. I multiplied both halves of the fraction by 1000 to move the decimal, and then I worked out the long division.
\(\frac{3507}{10020} = 0.35\)
Of course, 0.35 is NOT a shrinking number, per the definition provided in the problem. If
x were in there somewhere, we would be looking at a different proposition.
Finally, Statement I does NOT have to be true. We can consider nothing more than the first few digits of the given decimal, 0.865, to disprove it:
\(\frac{9 * 0.865}{10} = 7.785\)
After two minutes or so, I chose (B), having disproven anything with Statement I in it—answer choices (A), (C), and (E)—and Statement III seemed spurious.
Bunuel, would you please review the source material and correct the question accordingly? Many thanks.
- Andrew