Statement (1) specifies that (p + q)² = 4pq or p² + q² + 2pq = 4pq.
So, p² + q² − 2pq = 0.
Hence, (p − q)² = 0, which implies p = q.
Thus, statement (1) ALONE is sufficient.
Statement (2) specifies that (p + q)(p − q) − 150(p − q) = 0.
Hence, (p − q)(p + q − 150) = 0, which implies p = q OR p + q = 150.
Thus, statement (2) ALONE is not sufficient.

Statement (1) specifies the percentage of votes Challenger B got, but nothing about the votes Challenger C got. Thus, statement (1) ALONE is not sufficient.
Statement (2) indicates that Challenger C got 60% (i.e., 60000 / 100000 x 100%) of the votes cast, which implies that no other candidate could have secured more than 40% of the votes.
Thus, statement (2) ALONE is sufficient.

Statement (1) specifies the price of a pound of tomatoes before the increase, but nothing about the actual increase. Thus, statement (1) ALONE is not sufficient.
Statement (2) specifies the increase in the price, but nothing about the original price.
Thus, statement (2) ALONE is not sufficient.
When considered together, the percentage increase in the price is (20/50) × 100% = 40%.
Thus, BOTH statements TOGETHER are sufficient.

Let S = number of laborers who speak only Spanish, F = number of laborers who speak only French, B = number of laborers who speak both languages and T = total number of laborers. Then,
S + F + B = T.
∴ Fraction of laborers who speak only Spanish = (S/T) = 1 − (F/T) − (B/T).
Statement (1) specifies (B/T), but not (F/T).
Thus, statement (1) ALONE is not sufficient.
Statement (2) specifies only F, but neither B nor T.
Thus, statement (2) ALONE is not sufficient.
When both statements are considered, (B/T) and F are known. However, without knowing T, the required fraction cannot be calculated.
Thus, statements (1) and (2) TOGETHER are NOT sufficient.
This type of problem may be represented conveniently as two intersecting sets on a Venn diagram.

Statement (1) provides no useful information because the product of two or more consecutive integers is always even. Thus, statement (1) ALONE is not sufficient.
Statement (2) specifies that the sum of the integers is zero.
Let the seven consecutive integers be n, n + 1, ... , n + 6.
Then, 7n + 21 = 0 or n = −21/7 = −3.
The seven consecutive integers are −3, −2, −1, 0, 1, 2 and 3 (whose largest number is odd).
Thus, statement (2) ALONE is sufficient.
Alternative Approach :
The set of 7 consecutive integers can consist of 3 odd integers and 4 even integers, or 4 odd integers and 3 even integers.
If the set consists of 3 odd integers (whose sum is odd) and 4 even integers (whose sum is even), then the sum of the 7 integers cannot go to 0.
So, the set of 7 consecutive integers must consist of 4 odd integers (whose sum is even) and 3 even integers (whose sum is also even).
∴ The set has the form OEOEOEO where E represents an even integer and O represents an odd integer.
This implies that the largest number is odd. Thus, statement (2) ALONE is sufficient.

Let p, q and r be the amounts that P, Q and R received, respectively. Then, p + q + r = 50000.
Statement (1) gives p = (q + r) / 4. It is not possible to conclude who got the least amount from this equation, because there are many possibilities for p, q and r. For example, if p = 10000, then q and r can have values of 15000 and 25000, or5000 and 35000.Thus, statement (1) ALONE is not sufficient.
Statement (2) gives r = p + q. This implies that R got the highest amount. However, it is not possible to conclude who (P or Q) got the least amount.Thus, statement (2) ALONE is not sufficient.
However, if both statements are considered together, then the three equations may be solved for the three unknowns to obtain the following unique solution: p = 10000, q = 15000, and r = 25000. Therefore, P got the least amount.Thus, BOTH statements TOGETHER are sufficient.

A parallelogram is a quadrilateral whose opposite sides are parallel. Note that the opposite sides and opposite angles of a parallelogram are equal.
Statement (1) specifies that 2 adjacent sides of the quadrilateral are equal.
However, nothing is specified about the other 2 sides, which need not be equal.
So, ABCD need not necessarily be a parallelogram.
Thus, statement (1) ALONE is not sufficient.
Statement (2) implies that 3 sides of the quadrilateral are equal.
However, nothing is specified about the fourth side, which need not be equal (so, the angles between the sides may vary).
Thus, statement (2) ALONE is not sufficient.
When considered together, the two statements tell us that the 4 sides of the quadrilateral are equal making it a parallelogram (in fact, a rhombus which is a parallelogram with 4 equal sides).
Thus, BOTH statements TOGETHER are sufficient.

Statement (1) does not guarantee that the new group can be divided into teams of 10 each.
For example, the group could have had 16 scouts initially, in which case the new group (of 21 scouts) cannot be divided into teams of 10 scouts each without any remainder.
On the other hand, the group could have had 15 scouts initially, in which case the new group (of 20 scouts) can be divided into teams of 10 scouts each without any remainder.
Thus, statement (1) ALONE is not sufficient.
Statement (2) implies that the number of scouts in the group initially is 5 less than a multiple of 10.
So, if 5 scouts are now added to the group, the new group can be divided into teams of 10 scouts each.
Thus, statement (2) ALONE is sufficient.

For (x + y) to be an even number, both x and y must be even or both must be odd.
Statement (1) specifies that xy is even; then either x or y must be even, but both need not necessarily be even.
For example, 2 x 3 = 6 (even number), then 2 + 3 = 5 (odd number).
However, 2 x 2 = 4 (even number), then 2 + 2 = 4 (even number).
∴ If the product is even, then the sum can be either odd or even.
Thus, statement (1) ALONE is not sufficient.
Statement (2) specifies that y / x is odd; then both x and y must be odd or both must be even.
For example, 15 / 3 = 5 (odd number), then 15 + 3 = 18 (even number).
Also, 14 / 2 = 7 (odd number), then 14 + 2 = 16 (even number).
∴ If the quotient is odd, then the sum is necessarily even as proven below.
Since (y / x) is odd, y / x = 2m + 1, where m is an integer.
Multiplying both sides by x, one obtains y = 2mx + x .
Adding x to both sides, x + y = 2mx + 2x.
Since the right-hand side is divisible by 2, it is clear that (x + y) is even.
Thus, statement (2) ALONE is sufficient.

Statement (1) specifies that z < −2.
Adding 4 to both sides of the inequality, one obtains 4 + z < 2.
Clearly, (4 + z) cannot be equal to 2.
Thus, statement (1) ALONE is sufficient.
Statement (2) specifies that z < 0.
Adding 4 to both sides of the inequality, we get 4 + z < 4.
However, the value of (4 + z) may or may not be 2.
Thus, statement (2) ALONE is not sufficient.
Alternatively, solving 4 + z = 2 gives z = −2. So, the question may be re-worded as 'Is z = −2?'
Statement (1) ALONE is sufficient, because it specifies z ≠ −2 (since z < −2).
Statement (2) ALONE is not sufficient, because z may be any negative integer (not necessarily −2).

Statement (1) specifies the value of (x + 13y).
So, the required average is (7x + 91y) / 2 = 7 (x + 13y) / 2 = 7 x 100 / 2 = 350.
Thus, statement (1) ALONE is sufficient.
Statement (2) specifies the value of (x + y), from which it is not possible to evaluate (7x + 91y).
Thus, statement (2) ALONE is not sufficient.

Statement (1) specifies that the product of the two digits is 7.
Since 7 is a prime number, the two digits are 1 and 7.
The required two-digit number could be either 17 or 71.
Thus, statement (1) ALONE is not sufficient.
Statement (2) specifies that the larger of the two digits is in the units place. There are many two-digit numbers that satisfy this criterion.
Thus, statement (2) ALONE is not sufficient.
When both statements are taken together, the required number is 17 (since the larger of the two digits is in the units place) and not 71.
Thus, BOTH statements TOGETHER are sufficient.

The integers whose cubes are less than 100 are 1, 2, 3 and 4.
1³ = 1; 2³ = 8; 3³ = 27; 4³ = 64. So, p is 1, 8, 27 or 64.
Statement (1) indicates that p is either 8 or 64 (since 8 and 64 are even, whereas 1 and 27 are odd).
Thus, statement (1) ALONE is not sufficient.
Statement (2) indicates that p is either 1 or 64 (since 1² = 1 and 8² = 64).
Thus, statement (2) ALONE is not sufficient.
However, when both statements are considered together, p = 64 (since it is less than 100, the cube of 4, the square of 8, and even).
Thus, BOTH statements TOGETHER are sufficient.

Statement (1) indicates that the money could have been spent in the following three ways:
(a) Five items worth $40.
(b) One item worth $80 and three items worth $40.
(c) Two items worth $80 and one item worth $40.
Thus, statement (1) ALONE is not sufficient.
Statement (2) simply states that more than one item worth $80 was purchased.
Thus, statement (2) ALONE is not sufficient.
When both statements are combined, it may be concluded that two items worth $80 and one item worth $40 were purchased.Thus, BOTH
statements TOGETHER are sufficient.

Since the line passes through the origin (0, 0) and the point (a, b), its slope is b/a.
Statement (1) specifies a = 3b, which gives the the slope b/a = 1/3.
Thus, statement (1) ALONE is sufficient.
Statement (2) specifies only a. Without knowing b, the slope cannot be determined.
Thus, statement (2) ALONE is not sufficient.