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(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 NOT sufficient.

(1) 2/3 of the students in Section 1 are female.

(2) ½ of the students in Section 2 are male.

**Solution:** In isolation, neither of these statements is sufficient. When we combine them, we can find that in fact they aren’t sufficient in combination. We know the ratio of students in section one and those in section two, but we don’t’ know the number of students in each of the sections.

The correct answer is (E).

(1) One number is 5.

(2) The sum of the two numbers is 15.

**Solution:** The first statement is insufficient. Statement two states: 2x = y and x + y = 15 and it is therefore sufficient.

The correct answer is (B).

(1) [(r/(3s)] = (1/4)

(2) s = r + 4

**Solution:** r and s are both positive numbers. We can simplify the question to is r^2 , s^2 à r < s? Statement one and two are both sufficient in isolation.

The correct answer is (D).

(1) In 1985 Company R’s annual profit was $212,000; in 1989 Cmpany R’s annual profit was $242,000.

(2) Company R’s annual profit has increased by $7,500 each year since 1985.

**Solution:** Statement one is sufficient. Statement two is a classic trap. We have to remember that we’ve only been given the information about the base profit not in the question but in statement one. Statement two is not sufficient to answer the question.

Therefore the correct answer is (A).

(1) 6 ≤ X ≤ 81

(2) X is a multiple of 3

**Solution:** Statement one is insufficient. In combination, they are still insufficient.

The correct answer is (E).

(1) The perimeter of rectangle PQRS is 28 feet.

(2) Each diagonal of rectangle PQRS is 10 feet long.

**Solution:** We need to know the radius of the semicircle. Knowing the perimeter of the rectangle is not sufficient in isolation. Statement two is also insufficient in isolation, however, in combination they are sufficient to answer the question.

By the end linear equations rule, the correct answer is (C).

(1) x – y = 6

(2) y + z = 36

**Solution:** 4x = 5 y can be broken into different equations. 4x = 10z and 5y = 10z. if we’re told that x = 6 + y from the first statement then that is enough to answer the question. Similarly, statement two is sufficient on its own.

The correct answer is (D).