MANHATTAN PREP EXPLANATIONIf x2 < x, then what must be true of x? If you don’t know this one already, pull out a flashcard and prepare to Know the Code. (If you know this already, you can skip to the discussion of statement 1, below.)
The given says that x-squared is actually smaller than x. Most numbers get bigger when they’re squared; what numbers get smaller? Try some real numbers to figure it out.
Any negative values for x will get bigger when squared, because the square will be positive. The numbers 0 and 1 stay the same when squared. Any numbers greater than 1 get bigger when squared. Only numbers between 0 and 1 get smaller when squared.
Know the Code: x2 < x really means 0 < x < 1.
Next, “terminating decimal” means a decimal that ends at some point; that is, it does not keep going forever. 0.538 is an example of a terminating decimal. The value for pi is not a terminating decimal; nor is the decimal value of 2/3. Both decimals go on forever.
(1) INSUFFICIENT: If x is multiplied by 10, then the tenths place of x turns into the units digit (an integer), but the other decimal places are still decimal places. For instance, if x = 0.315, then 10x = 3.15. What used to be the hundredths digit (1) has become the tens digit. Using just this example, the answer to the question is yes, x does have a nonzero hundredths digit.
Can you also find a “no” example that makes statement 1 true? What if x = 0.305? In this case, 10x = 3.05 (not an integer), so this number fits statement 1. This time, though, x does NOT have a nonzero hundredths digit. The hundredths digit is 0, so the answer to the question is no.
Since the hundredths digit can be either zero or nonzero, the statement is insufficient.
(2) INSUFFICIENT: If 100x is an integer, then only the first two decimal places of x can potentially have nonzero digits. For instance, if x = 0.38, then100x = 38, which makes statement 2 true. Therefore 0.38 is a valid possible value for x. On the other hand, if x = 0.381, then 100x = 38.1, which makes statement 2 false. Therefore, 0.381 is not an acceptable possible value for x.
The valid example above, x = 0.38, returns a “yes” answer: yes, x does have a nonzero hundredths digit. Can you think of a number that makes statement 2 true but returns a “no” answer instead? For example, if x = 0.3 (which can also be written 0.30), then 100x = 30, an integer. In this case, no, x does NOT have a nonzero hundredths digit; the hundredths digit is zero.
Since the hundredths digit can be either zero or nonzero, the statement is insufficient.
(1) AND (2) SUFFICIENT: From statement (1), because 10x is NOT an integer, x must have a nonzero digit somewhere to the right of the tenths place—in the hundredths place or further to the right (because there still needs to be a decimal after x is multiplied by 10).
From statement (2), because 100x IS an integer, every digit after the hundredths place has to be zero (because there can be no more decimals after x is multiplied by 100).
If there must be a nonzero digit somewhere to the right of the tenths place, but all of the digits to the right of the hundredths place must be zero, then x must have a nonzero digit in the hundredths place.
The correct answer is (C).