Data-Sufficiency Questions - GMAT Quantitative

includes all multiples of 2; includes all multiples of 3. comprises all multiples of either 2 or 3.

Knowing is a perfect square is neither necessary nor helpful, as, for example, , but (as 25 is neither a multiple of 2 nor a multiple of 3).

If you know that is a multiple of 99, then it must also be a multiple of any number that divides 99 evenly, one such number is 3. This means

The sum of the measures of the five interior angles of any pentagon is , so if Statement 1 is true, then the sum of the five numbers is 540, regardless of the individual values.

The mean of five data values multiplied by 5 is the sum of the values, so if Statement 2 is true, then their sum is .

Assume both statements. Then and . But this does not answer the question. For example,

If , then

making true.

But if , then

making false.

The relation comprises ten ordered pairs. If Statement 1 alone is known, then the domain comprises ten elements, each of which must appear in exactly one ordered pair. Therefore, no domain element is matched with more than one range element, and the relation is a function.

If Statement 2 alone is known, then the range comprises ten elements, each of which must appear in exactly one ordered pair. But nothing is known about the domain. If no domain element is repeated among the ordered pairs, the relation is a function; otherwise, the relation is not a function.

An arithmetic sequence is one in which the difference of each term in the sequence and the one preceding is constant.

Assume Statement 1 alone.

,

meaning that at least two such differences are unequal, and proving the sequence is not arithmetic.

Statement 2 alone only proves that two such differences are equal, but says nothing about any of the other (infinitely many) such differences. Therefore, it leaves the question unresolved.

Statement 1 tells you that and are of unlike sign, that and are of unlike sign, and that and are of unlike sign; that is, exactly one number from each pair is negative. Therefore, there are three negative numbers.

Statement 2 tells you that of , , and , there can be either exactly zero or two negative numbers; and that of , , and , there can be exactly one or three negative numbers. This means that the number of negative numbers among the six can be as few as one or as many as five.

From Statement 1 it follows that is a multiple of 5, but it does not answer the question by itself; 5 is prime, but the other multiples of 5 are composite. Similarly, from Statement 2 it follows that is a multiple of 7, but, 7 is prime and the other multiples of 7 are composite.

If we can assume both statements, then, since 5 and 7 are relatively prime, it follows that is a multiple of both 5 and 7; since now has at least four factors (1,5, 7, and 35), is composite.

If , then .

If , then

The two statements are equivalent. This is enough to allow the mean to be found:

The answer is that either statement alone is sufficient to answer the question.

If we know that , then the set is known to have one 24, five 26's, three 27's, two 28's, and one 29. The only way the set can have two modes is for and ; this makes 27 occur five times, just as frequently as 26.

If we know , however, the set is known to have one 24, four 26's, four 27's, two 28's, and one 29. There are two ways for the set to have two modes (26 and 27): for and , or for and .

The answer is that Statement 1 alone is sufficient to answer the question, but not Statement 2.

The median of a data set with an even number of elements is the arithmetic mean of the two elements that fall in the middle when the elements are arranged in ascending order. These two elements are both 6, so 6 is the median.

The mode of a data set is the element that occurs most frequently. Since 6 appears thre times, 7 appears two times, and all other elements appear once each, the mode is 6.

Therefore, the median and the mode are equal.

An arithmetic sequence is one in which the difference of each term in the sequence and the one preceding is constant.

The two statements together demonstrate that two such differences are equal to each other, and that two other such differences are equal to each other. No information, however, is given about any other differences, of which there are infinitely many. Therefore, the question of whether the sequence is arithmetic or not is unresolved.

Assume both statements. Then and . But this does not answer the question. For example,

If , then

making true.

But if , then

making false.

One must have information about both and to answer the question, so neither statement is sufficient by itself.

Now assume both statements to be true. Then, being positive and being negative, .

If Statement 1 alone is assumed, then, since there are only six domain elements and ten points in the relation, at least one of the domain elements must match with more than one range element. This forces the relation to not be a function.

If Statement 2 alone is assumed, then, since is a vertical line that passes through the graph twice, the relation fails the vertical line test and is therefore not a function.

Statement 1 tells you that and are of unlike sign, that and are of unlike sign, and that and are of unlike sign; that is, exactly one number from each pair is negative. Therefore, there are three negative numbers.

Statement 2 tells you that of , , and , there can be either exactly zero or two negative numbers; and that of , , and , there can be exactly one or three negative numbers. This means that the number of negative numbers among the six can be as few as one or as many as five.

From Statement 1 it follows that is a multiple of 5, but it does not answer the question by itself; 5 is prime, but the other multiples of 5 are composite. Similarly, from Statement 2 it follows that is a multiple of 7, but, 7 is prime and the other multiples of 7 are composite.

If we can assume both statements, then, since 5 and 7 are relatively prime, it follows that is a multiple of both 5 and 7; since now has at least four factors (1,5, 7, and 35), is composite.

If , then .

If , then

The two statements are equivalent. This is enough to allow the mean to be found:

The answer is that either statement alone is sufficient to answer the question.

If we know that , then the set is known to have one 24, five 26's, three 27's, two 28's, and one 29. The only way the set can have two modes is for and ; this makes 27 occur five times, just as frequently as 26.

If we know , however, the set is known to have one 24, four 26's, four 27's, two 28's, and one 29. There are two ways for the set to have two modes (26 and 27): for and , or for and .

The answer is that Statement 1 alone is sufficient to answer the question, but not Statement 2.

includes all multiples of 2; includes all multiples of 3. comprises all multiples of either 2 or 3.

Knowing is a perfect square is neither necessary nor helpful, as, for example, , but (as 25 is neither a multiple of 2 nor a multiple of 3).

If you know that is a multiple of 99, then it must also be a multiple of any number that divides 99 evenly, one such number is 3. This means