DSQ: Understanding fractions - GMAT Quantitative

From the initial information, we can determine that of the balls are green.

The first statement alone tells you that half of the blue marbles and half of the red marbles are large, so half of the green marbles must be large as well. But this alone does not tell you how many marbles there are total.

The second statement alone tells you how many large marbles are red or green, but you do not have any way of figuring out how many small marbles or total marbles are.

But if you put the statements together, you know the following:

Half of the red marbles and half of the green marbles comprise the thirty-six large marbles:

There are 120 marbles.

The answer is that both statements together are sufficent to answer the question, but neither statement alone is sufficient.

The two statements together provide insufficient information.

and are two examples of fractions that satisfy both conditions; note that the first is represented in decimal form by a terminating decimal, and the second, by a repeating decimal.

Both and satisfy the conditions of Statement 1; the first fraction is in lowest terms, the second is not.

Both and satisfy the conditions of Statement 2; the first fraction is in lowest terms, the second is not.

Neither statement alone is enough to answer the question. Both together, however, are enough to prove the fraction is not in lowest terms; a fraction which satisfies both statements has both its numerator and denominator divisible by 2.

The two statements together are insufficient. and each satisfy the conditions of both statements, but the first fraction is in lowest terms and the second is not.

From the question, we know that only one of the fractions in the answer choices is larger than . The correct answer must therefore be the largest of the answer choices. This observation is important, because it means we can eliminate some of the answer choices without performing any calculations. For example, is smaller than , so eliminate from the possibilities. is smaller than both and , so eliminate it as well. Similarly, is smaller than , so eliminate too.

This leaves us with only two answer choices: and

First consider . We can rewrite this as , which is clearly smaller than . Eliminate .

By process of elimination, we've thus shown that is the correct answer choice.

Statement 1 alone does not prove is or is not an integer.

For example, if , then

.

If , then

.

is an integer in only one scenario, but Statement 1 is true in both.

Now assume Statement 2 alone, and let . Then is an integer by Statement 2, and

Suppose is an integer. Then , the difference of integers, is itself an integer. But we know that this is equal to , which is not an integer, so, by contradiction, is not an integer.

From the initial information, we can determine that of the balls are green.

The first statement alone tells you that half of the blue marbles and half of the red marbles are large, so half of the green marbles must be large as well. But this alone does not tell you how many marbles there are total.

The second statement alone tells you how many large marbles are red or green, but you do not have any way of figuring out how many small marbles or total marbles are.

But if you put the statements together, you know the following:

Half of the red marbles and half of the green marbles comprise the thirty-six large marbles:

There are 120 marbles.

The answer is that both statements together are sufficent to answer the question, but neither statement alone is sufficient.

The two statements together provide insufficient information.

and are two examples of fractions that satisfy both conditions; note that the first is represented in decimal form by a terminating decimal, and the second, by a repeating decimal.

Both and satisfy the conditions of Statement 1; the first fraction is in lowest terms, the second is not.

Both and satisfy the conditions of Statement 2; the first fraction is in lowest terms, the second is not.

Neither statement alone is enough to answer the question. Both together, however, are enough to prove the fraction is not in lowest terms; a fraction which satisfies both statements has both its numerator and denominator divisible by 2.

The two statements together are insufficient. and each satisfy the conditions of both statements, but the first fraction is in lowest terms and the second is not.

From the question, we know that only one of the fractions in the answer choices is larger than . The correct answer must therefore be the largest of the answer choices. This observation is important, because it means we can eliminate some of the answer choices without performing any calculations. For example, is smaller than , so eliminate from the possibilities. is smaller than both and , so eliminate it as well. Similarly, is smaller than , so eliminate too.

This leaves us with only two answer choices: and

First consider . We can rewrite this as , which is clearly smaller than . Eliminate .

By process of elimination, we've thus shown that is the correct answer choice.

Statement 1 alone does not prove is or is not an integer.

For example, if , then

.

If , then

.

is an integer in only one scenario, but Statement 1 is true in both.

Now assume Statement 2 alone, and let . Then is an integer by Statement 2, and

Suppose is an integer. Then , the difference of integers, is itself an integer. But we know that this is equal to , which is not an integer, so, by contradiction, is not an integer.

From the initial information, we can determine that of the balls are green.

The first statement alone tells you that half of the blue marbles and half of the red marbles are large, so half of the green marbles must be large as well. But this alone does not tell you how many marbles there are total.

The second statement alone tells you how many large marbles are red or green, but you do not have any way of figuring out how many small marbles or total marbles are.

But if you put the statements together, you know the following:

Half of the red marbles and half of the green marbles comprise the thirty-six large marbles:

There are 120 marbles.

The answer is that both statements together are sufficent to answer the question, but neither statement alone is sufficient.

The two statements together provide insufficient information.

and are two examples of fractions that satisfy both conditions; note that the first is represented in decimal form by a terminating decimal, and the second, by a repeating decimal.

Both and satisfy the conditions of Statement 1; the first fraction is in lowest terms, the second is not.

Both and satisfy the conditions of Statement 2; the first fraction is in lowest terms, the second is not.

Neither statement alone is enough to answer the question. Both together, however, are enough to prove the fraction is not in lowest terms; a fraction which satisfies both statements has both its numerator and denominator divisible by 2.

The two statements together are insufficient. and each satisfy the conditions of both statements, but the first fraction is in lowest terms and the second is not.

From the question, we know that only one of the fractions in the answer choices is larger than . The correct answer must therefore be the largest of the answer choices. This observation is important, because it means we can eliminate some of the answer choices without performing any calculations. For example, is smaller than , so eliminate from the possibilities. is smaller than both and , so eliminate it as well. Similarly, is smaller than , so eliminate too.

This leaves us with only two answer choices: and

First consider . We can rewrite this as , which is clearly smaller than . Eliminate .

By process of elimination, we've thus shown that is the correct answer choice.

Statement 1 alone does not prove is or is not an integer.

For example, if , then

.

If , then

.

is an integer in only one scenario, but Statement 1 is true in both.

Now assume Statement 2 alone, and let . Then is an integer by Statement 2, and

Suppose is an integer. Then , the difference of integers, is itself an integer. But we know that this is equal to , which is not an integer, so, by contradiction, is not an integer.

From the initial information, we can determine that of the balls are green.

The first statement alone tells you that half of the blue marbles and half of the red marbles are large, so half of the green marbles must be large as well. But this alone does not tell you how many marbles there are total.

The second statement alone tells you how many large marbles are red or green, but you do not have any way of figuring out how many small marbles or total marbles are.

But if you put the statements together, you know the following:

Half of the red marbles and half of the green marbles comprise the thirty-six large marbles:

There are 120 marbles.

The answer is that both statements together are sufficent to answer the question, but neither statement alone is sufficient.

The two statements together provide insufficient information.

and are two examples of fractions that satisfy both conditions; note that the first is represented in decimal form by a terminating decimal, and the second, by a repeating decimal.