*If d represents the hundredths digit and e represents the thousandths digit in the decimal .4de, what is the value of this decimal rounded to the nearest tenth?*

(1) d – e is equal to a positive perfect square.

(2) sqrt (d) > e*e

(A) Statement (1) alone is sufficient, but statement(2) alone is not sufficient.

(B) Statement (2) alone is sufficient, but statement(1) alone is not sufficient.

(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

(D) Each statement ALONE is sufficient.

*(E) Statements (1) and (2) TOGETHER are NOTsufficient. *

Answer - E

From statement (1), we know that *d* *– e* must equal a positive perfect square. This means that *d* is greater than *e*. In addition, since any single digit minus any other single digit can yield a maximum of 9, *d* *– e* could only result in the perfect squares 9, 4, or 1.

However, this leaves numerous possibilities for the values of *d* and *e *respectively. For example, two possibilities are as follows:

d = 7, e = *3* (*d* *– e* = the perfect square 4)

*d* = 3, e = *2* (*d* *– e* = the perfect square 1)

In the first case, the decimal .*4de* would be .473, which, when rounded to the nearest tenth, is equal to .5. In the second case, the decimal would be .432, which, when rounded to the nearest tenth, is .4.

Thus, statement (1) is not sufficient on its own to answer the question.

Statement (2) tells us that sqrt d = *e*^{2}. Since *d* is a single digit, the maximum value for *d* is 9, which means the maximum square root of *d* is 3. This means that *e*^{2} must be less than 3. Thus the digit *e* can only be 0 or 1.

However, this leaves numerous possibilities for the values of *d* and *e *respectively. For example, two possibilities are as follows:

d = 9, *e* = 1

*d = 2*, *e* = 0

In the first case, the decimal .*4de* would be .491, which, when rounded to the nearest tenth, is equal to .5. In the second case, the decimal would be .420, which, when rounded to the nearest tenth, is .4.

Thus, statement (2) is not sufficient on its own to answer the question.

Taking both statements together, we know that *e* must be 0 or 1 and that *d **– e* is equal to 9, 4 or 1.

This leaves the following 4 possibilities:

*d* = 9*, **e* = 0

*d* = 5, *e* = 1

*d* = 4,* e* = 0

*d* = 1, *e* = 0

These possibilities yield the following four decimals: .490, .451, .440, and .410 respectively. The first two of these decimals yield .5 when rounded to the nearest tenth, while the second two decimals yield .4 when rounded to the nearest tenth.

Thus, both statements taken together are not sufficient to answer the question.

The correct answer is E: Statements (1) and (2) TOGETHER are NOT sufficient.