The average of 9 GMAT scores is 600. Is the average of the top 3 scores greater than 670?

(1) None of the 9 scores are the same.
(2) The average of the bottom 3 scores is 460.

Statement 1 by itself is insufficient to answer the question with a definite yes or no answer, observe:
{560, 570, 580, 590, 600, 610, 620, 630, 640}
and
{490, 500, 510, 590, 600, 610, 690, 700, 710}
are two sets of 9 GMAT scores with an average of 600 and no duplicate scores. In the first set the average of the top 3 is 630, and in the second set the average of the top 3 is 700.


Statement 2 by itself is insufficient to answer the question with a definite yes or no answer. Let x be the average of the top 6 scores. With statement 2, we have [3(460) + 6(x)]/9 = 600, or x = 670. We could have score sets of
{460, 460, 460, 670, 670, 670, 670, 670, 670}
or
{460, 460, 460, 600, 600, 600, 740, 740, 740}
With the first set, the answer to the question is no because the average of the top 3 scores is equal to 670, not greater. With the second set the answer to the question is yes.

What if we combine the information from the two statements? We know that the average of the top 6 is 670, and we know that none of the scores are the same. Then the average of the top 3 has to be greater than 670.

Since we need both statement to supply a definite answer to the question, the answer is C.

p, q, and r are integers and p + q + r is an odd number. Which of the following must always be true?

(i) p² + q² + r² is odd
(ii) p - q + r is odd
(iii) pq + qr + rp is odd

(A) i and ii
(B) i and iii
(C) ii and iii
(D) i only
(E) iii only

Let's evaluate either (i) or (iii) first since these two appear most frequently. Since the expression in (i) is simpler, it should be first.

If p + q + r is odd, then it is either the sum of three odd numbers, or one odd and two even numbers. When you square a number, it stays either odd or ever, so the expression p² + q² + r² is also the sum of either three odd numbers, or one odd and two even numbers. Thus p² + q² + r² is odd. This means we can cross out choices C and E.

The next statement to evaluate is (ii), since it is the simplest. Notice that p - q + r = (p + q + r) - 2q. The expression (p + q + r) - 2q is an odd number minus and even number, hence an odd number. Therefore (ii) is true as well.

So the correct answer is A, and we didn't even have to check (iii). As an exercise for extra practice, you should verify that if p + q + r is odd, then pq + qr + rp can be even sometimes.

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Quadrilateral ABCD has side lengths as shown above. What is the length of BC?

(A) 3sqrt(7)
(B) 2sqrt(7)
(C) sqrt(23)
(D) 3sqrt(2)
(E) sqrt(15)

Notice in the figure above, ∆ABD and ∆BCD are right triangles that share the same hypotenuse BD. We can figure out the length of BD by using the side lengths of ∆ABD.
So BD has length 8. Now we can apply the Pythagorean Theorem one more time to find BC. 8² - 7² = 64 -49 = 15 = (BC)², so the length of BC is sqrt(15), answer choice E.

If y is an integer and y³ - 4y = 48, what is y² + 4?

(A) 29
(B) 24
(C) 20
(D) 16
(E) 13

Let's start by factoring the left side of y³ - 4y = 48. We have y³ - 4y = y(y² - 4) = y(y-2)(y+2). If we rearrange terms a little, it is (y-2)y(y+2) = 48

Notice that y-2, y, and y+2 must be either three consecutive even integers, or three consecutive odd integers. Since the product, 48, is even, these must be three consecutive even integers. So we need to find three consecutive even integers whose product is 48. If you experiment a little, you should come up with 2∙4∙6 = 48, and therefore, y = 4.

4² + 4 = 20, so the correct answer is C.

Let Z_n denote the nth term in a sequence Z, and suppose Z_n is defined by equation Z_n = .5(Z_n-1)² - 6. If Z₁ = 2, what is the 5th term of the sequence?

(A) -2
(B) 2
(C) 6.5
(D) 355
(E) 728

Let's apply the formula to generate successive terms after the first. The formula tells you that in order to get the next term, you take the previous term, square it and multiply by .5, and then subtract 6.

Z₁ = 2
Z₂ = .5(2)² - 6 = -4
Z₃ = .5(-4)² - 6 = 2
Z₄ = .5(2)² - 6 = -4
Z₅ = .5(-4)² - 6 = 2

The fifth term of the sequence is Z₅, so the correct answer is B.

x > y, x ≠ -y

(Col A) (x²y + xy²)/(x² + y²)
(Col B) (x³y + xy³)/(x³ + y³)

This problem is best solved by plugging numbers in for x and y. For the first set, let's pick x with a larger absolute value than y, eg x = 2 and y = -1:
Col A becomes (-4 + 2)/(4 + 1) = -2/5
Col B becomes (-8 - 2)/(8 - 1) = -10/7
Col A is larger with this set of numbers.

For the second set, let's pick y with a larger absolute value than x, eg y = -2 and x = 1:
Col A becomes (-2 + 4)/(1 + 4) = 2/5
Col B becomes (-2 - 8)/(1 - 8) = 10/7
Col B is larger with this set of numbers.

The correct answer is D

Six pipes are stacked as shown in the figure above. Each pipe has a radius of 1. What is the height of the stack h?

(A) 6
(B) 2 + 2sqrt(3)
(C) 3sqrt(3)
(D) 1 + 3sqrt(3)
(E) 4 + sqrt(3)

The key to finding the exact height of the stack is break up the vertical distance into disjoint segments and add up the lengths carefully so that you do not over- or under-count anything. Hopefully the figure below will give an idea of how to do this.

The sum of the vertical segments is 1 + 2sqrt(3) + 1 = 2 + 2sqrt(3), so B is correct.

Here is a data sufficiency question. It is a question on slopes of lines and tests basic concepts about lines in geometry and coordinate geometry.

Question

Are lines p (with slope m) and q (with slope n) perpendicular to each other?
1. m + 2 = n
2. m + n = 0

Correct Answer: Choice C. BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.

Explanatory Answer

If two lines are perpendicular, then the product of the slopes of the two lines will be equal to -1.

In this case, if the product m * n = -1, then the two lines will be perpendicular to each other. If the product is not equal to -1, then they are not perpendicular. We need to assess that conclusively.

Statement 1 m + 2 = n
m could be -1 and n could be 1, in which case the product is -1. Alternatively, m could be 4 and n could be 6 in which case the product is not -1.

As we are not able to conclude using the information in statement 1, it is not sufficient. Choices A and D can be eliminated. We are left with choices B, C or E.

Statement 2 m + n = 0.
m could be -1 and n could be 1 or vice versa. In that case, m * n = -1.
m could be any other number and n could be -m. In that case m * n will not be equal to -1. Hence, statement 2 is also not sufficient. We can eliminate choice B. We are left with choices C or E.

Combining the two statements, we know that m = -n from statement 2. Substituting that in statement 1, we get m + 2 = -m or 2m = -2 or m = -1. Hence, n = 1. Hence, the product m * n = -1.

As the information provided in the two statements is sufficient to answer the question, choice C is the correct answer.