Operations - GRE Quantitative Reasoning

Card 0 of 136

Question

Which of the following is true?

Quantity A: $\frac{12}{7}-\frac{3}{5}$

Quantity B: $\frac{11}{6}-\frac{7}{8}$

Answer

First, consider each quantity separately.

Quantity A

$\frac{12}{7}-\frac{3}{5}$

These two fractions do not have a common factor. Their common denominator is . Thus, we multiply the fractions as follows to give them a common denominator:

$\frac{12}{7}\frac{5}{5}-\frac{3}{5}\frac{7}{7}$

This is the same as:

$\frac{60}{35}-\frac{21}{35}=\frac{39}{35}$

Quantity B

$\frac{11}{6}-\frac{7}{8}$

The common denominator of these two values is . Therefore, you multiply the fractions as follows to give them a common denominator:

$\frac{11}{6}\frac{4}{4}-\frac{7}{8}\frac{3}{3}$

This is the same as:

$\frac{44}{24}-\frac{21}{24}=\frac{23}{24}$

Since Quantity A is larger than and Quantity B is a positive fraction less than , we know that Quantity A is larger without even using a calculator.

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Question

Which of the following is true?

Quantity A: $\frac{12}{7}-\frac{3}{5}$

Quantity B: $\frac{11}{6}-\frac{7}{8}$

Answer

First, consider each quantity separately.

Quantity A

$\frac{12}{7}-\frac{3}{5}$

These two fractions do not have a common factor. Their common denominator is . Thus, we multiply the fractions as follows to give them a common denominator:

$\frac{12}{7}\frac{5}{5}-\frac{3}{5}\frac{7}{7}$

This is the same as:

$\frac{60}{35}-\frac{21}{35}=\frac{39}{35}$

Quantity B

$\frac{11}{6}-\frac{7}{8}$

The common denominator of these two values is . Therefore, you multiply the fractions as follows to give them a common denominator:

$\frac{11}{6}\frac{4}{4}-\frac{7}{8}\frac{3}{3}$

This is the same as:

$\frac{44}{24}-\frac{21}{24}=\frac{23}{24}$

Since Quantity A is larger than and Quantity B is a positive fraction less than , we know that Quantity A is larger without even using a calculator.

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Question

What is the result of adding of $\frac{2}{7}$ to $\frac{1}{4}$ ?

Answer

Let us first get our value for the percentage of the first fraction. 20% of 2/7 is found by multiplying 2/7 by 2/10 (or, simplified, 1/5): (2/7) * (1/5) = (2/35)

Our addition is therefore (2/35) + (1/4). There are no common factors, so the least common denominator will be 35 * 4 or 140. Multiply the numerator and denominator of 2/35 by 4/4 and the numerator of 1/4 by 35/35.

This yields:

(8/140) + (35/140) = 43/140, which cannot be reduced.

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Question

Reduce to simplest form: $\frac{1}{4}+(\frac{4}{3}\times \frac{3}{8})-(\frac{1}{4}\div \frac{3}{8})$

Answer

Simplify expressions inside parentheses first: $\dpi{100} \small \left (\frac{4}{3} \times \frac{3}{8} \right ) = \frac{12}{24} = \frac{1}{2}$ and $\dpi{100} \small \left (\frac{1}{4} \div \frac{3}{8} \right ) = \left (\frac{1}{4} \times \frac{8}{3} \right ) = \frac{8}{12} = \frac{2}{3}$

Now we have: $\frac{1}{4} + \frac{1}{2} - \frac{2}{3}$

Add them by finding the common denominator (LCM of 4, 2, and 3 = 12) and then multiplying the top and bottom of each fraction by whichever factors are missing from this common denominator:

$\dpi{100} \small \frac{1\times 3}{4\times 3} + \frac{1\times 6}{2\times 6} - \frac{2\times 4}{3\times 4} =\frac{3}{12} + \frac{6}{12} - \frac{8}{12} = \frac{1}{12}$

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Question

Quantity A: $\frac{15}{x}+\frac{12}{y}$

Quantity B: $\frac{15y+12x}{xy}$

Which of the following is true?

Answer

Start by looking at Quantity A. The common denominator for this expression is . To calculate this, you perform the following multiplications:

$\frac{15}{x}\frac{y}{y}+\frac{12}{y}\frac{x}{x}$

This is the same as:

$\frac{15y}{xy}+\frac{12x}{xy}$ , or $\frac{15y+12x}{xy}$

This is the same as Quantity B. They are equal!

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Question

What is the result of adding of $\frac{2}{7}$ to $\frac{1}{4}$ ?

Answer

Let us first get our value for the percentage of the first fraction. 20% of 2/7 is found by multiplying 2/7 by 2/10 (or, simplified, 1/5): (2/7) * (1/5) = (2/35)

Our addition is therefore (2/35) + (1/4). There are no common factors, so the least common denominator will be 35 * 4 or 140. Multiply the numerator and denominator of 2/35 by 4/4 and the numerator of 1/4 by 35/35.

This yields:

(8/140) + (35/140) = 43/140, which cannot be reduced.

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Question

Reduce to simplest form: $\frac{1}{4}+(\frac{4}{3}\times \frac{3}{8})-(\frac{1}{4}\div \frac{3}{8})$

Answer

Simplify expressions inside parentheses first: $\dpi{100} \small \left (\frac{4}{3} \times \frac{3}{8} \right ) = \frac{12}{24} = \frac{1}{2}$ and $\dpi{100} \small \left (\frac{1}{4} \div \frac{3}{8} \right ) = \left (\frac{1}{4} \times \frac{8}{3} \right ) = \frac{8}{12} = \frac{2}{3}$

Now we have: $\frac{1}{4} + \frac{1}{2} - \frac{2}{3}$

Add them by finding the common denominator (LCM of 4, 2, and 3 = 12) and then multiplying the top and bottom of each fraction by whichever factors are missing from this common denominator:

$\dpi{100} \small \frac{1\times 3}{4\times 3} + \frac{1\times 6}{2\times 6} - \frac{2\times 4}{3\times 4} =\frac{3}{12} + \frac{6}{12} - \frac{8}{12} = \frac{1}{12}$

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Question

Quantity A: $\frac{15}{x}+\frac{12}{y}$

Quantity B: $\frac{15y+12x}{xy}$

Which of the following is true?

Answer

Start by looking at Quantity A. The common denominator for this expression is . To calculate this, you perform the following multiplications:

$\frac{15}{x}\frac{y}{y}+\frac{12}{y}\frac{x}{x}$

This is the same as:

$\frac{15y}{xy}+\frac{12x}{xy}$ , or $\frac{15y+12x}{xy}$

This is the same as Quantity B. They are equal!

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Question

What is the result of adding of $\frac{2}{7}$ to $\frac{1}{4}$ ?

Answer

Let us first get our value for the percentage of the first fraction. 20% of 2/7 is found by multiplying 2/7 by 2/10 (or, simplified, 1/5): (2/7) * (1/5) = (2/35)

Our addition is therefore (2/35) + (1/4). There are no common factors, so the least common denominator will be 35 * 4 or 140. Multiply the numerator and denominator of 2/35 by 4/4 and the numerator of 1/4 by 35/35.

This yields:

(8/140) + (35/140) = 43/140, which cannot be reduced.

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Question

Reduce to simplest form: $\frac{1}{4}+(\frac{4}{3}\times \frac{3}{8})-(\frac{1}{4}\div \frac{3}{8})$

Answer

Simplify expressions inside parentheses first: $\dpi{100} \small \left (\frac{4}{3} \times \frac{3}{8} \right ) = \frac{12}{24} = \frac{1}{2}$ and $\dpi{100} \small \left (\frac{1}{4} \div \frac{3}{8} \right ) = \left (\frac{1}{4} \times \frac{8}{3} \right ) = \frac{8}{12} = \frac{2}{3}$

Now we have: $\frac{1}{4} + \frac{1}{2} - \frac{2}{3}$

Add them by finding the common denominator (LCM of 4, 2, and 3 = 12) and then multiplying the top and bottom of each fraction by whichever factors are missing from this common denominator:

$\dpi{100} \small \frac{1\times 3}{4\times 3} + \frac{1\times 6}{2\times 6} - \frac{2\times 4}{3\times 4} =\frac{3}{12} + \frac{6}{12} - \frac{8}{12} = \frac{1}{12}$

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Question

Quantity A: $\frac{15}{x}+\frac{12}{y}$

Quantity B: $\frac{15y+12x}{xy}$

Which of the following is true?

Answer

Start by looking at Quantity A. The common denominator for this expression is . To calculate this, you perform the following multiplications:

$\frac{15}{x}\frac{y}{y}+\frac{12}{y}\frac{x}{x}$

This is the same as:

$\frac{15y}{xy}+\frac{12x}{xy}$ , or $\frac{15y+12x}{xy}$

This is the same as Quantity B. They are equal!

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Question

Car A traveled 120 miles with 5 gallons of fuel.

Car B can travel 25 miles per gallon of fuel.

Quantity A: The fuel efficiency of car A

Quantity B: The fuel efficiency of car B

Answer

Let's make the two quantities look the same.

Quantity A: 120 miles / 5 gallons = 24 miles / gallon

Quantity B: 25 miles / gallon

Quantity B is greater.

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Question

Quantity A:

The -value of the equation $y = \frac{3}{4}x-2$ when $y = \frac{-1}{3}$

Quantity B:

$\frac{3}{4}$

Answer

In order to solve quantitative comparison problems, you must first deduce whether or not the problem is actually solvable. Since this consists of finding the solution to an -coordinate on a line where nothing too complicated occurs, it will be possible.

Thus, your next step is to solve the problem.

Since $y = \frac{3}{4}x-2$ and $y = \frac{-1}{3}$ , you can plug in the -value and solve for :

$y = \frac{3}{4}x-2$

Plug in y:

$\frac{-1}{3} = \frac{3}{4}x-2$

Add 2 to both sides:

$\frac{5}{3} = \frac{3}{4}x$

Divide by 3/4. To divide, first take the reciprocal of 3/4 (aka, flip it) to get 4/3, then multiply that by 5/3:

$\frac{20}{9} = x$

Make the improper fraction a mixed number:

$2\tfrac{2}{9}$

Now that you have what x equals, you can compare it to Quantity B.

Since $2\tfrac{2}{9}$ is bigger than 2, the answer is that Quantity A is greater

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Question

What is equivalent to $\frac{\frac{1}{5}}{\frac{4}{15}}$ ?

Answer

Remember that when you divide by a fraction, you multiply by the reciprocal of that fraction. Therefore, this division really is:

$\frac{1}{5}\cdot \frac{15}{4}$

At this point, it is merely a matter of simplification and finishing the multiplication:

$\frac{1}{5}\cdot\frac{15}{4}=\frac{15}{20}=\frac{3}{4}$

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Question

Which of the following is equivalent to $\frac{7}{\frac{4}{5}}$ ?

Answer

To begin with, most students find it easy to remember that...

$\frac{7}{\frac{4}{5}}=\frac{\frac{7}{1}}{\frac{4}{5}}$

From this, you can apply the rule of division of fractions. That is, multiply by the reciprocal:

Therefore,

$\frac{\frac{7}{1}}{\frac{4}{5}} = \frac{7}{1} \cdot \frac{5}{4}=\frac{35}{4}$

Since nothing needs to be reduced, this is your answer.

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Question

Car A traveled 120 miles with 5 gallons of fuel.

Car B can travel 25 miles per gallon of fuel.

Quantity A: The fuel efficiency of car A

Quantity B: The fuel efficiency of car B

Answer

Let's make the two quantities look the same.

Quantity A: 120 miles / 5 gallons = 24 miles / gallon

Quantity B: 25 miles / gallon

Quantity B is greater.

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Question

Quantity A:

The -value of the equation $y = \frac{3}{4}x-2$ when $y = \frac{-1}{3}$

Quantity B:

$\frac{3}{4}$

Answer

In order to solve quantitative comparison problems, you must first deduce whether or not the problem is actually solvable. Since this consists of finding the solution to an -coordinate on a line where nothing too complicated occurs, it will be possible.

Thus, your next step is to solve the problem.

Since $y = \frac{3}{4}x-2$ and $y = \frac{-1}{3}$ , you can plug in the -value and solve for :

$y = \frac{3}{4}x-2$

Plug in y:

$\frac{-1}{3} = \frac{3}{4}x-2$

Add 2 to both sides:

$\frac{5}{3} = \frac{3}{4}x$

Divide by 3/4. To divide, first take the reciprocal of 3/4 (aka, flip it) to get 4/3, then multiply that by 5/3:

$\frac{20}{9} = x$

Make the improper fraction a mixed number:

$2\tfrac{2}{9}$

Now that you have what x equals, you can compare it to Quantity B.

Since $2\tfrac{2}{9}$ is bigger than 2, the answer is that Quantity A is greater

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Question

What is equivalent to $\frac{\frac{1}{5}}{\frac{4}{15}}$ ?

Answer

Remember that when you divide by a fraction, you multiply by the reciprocal of that fraction. Therefore, this division really is:

$\frac{1}{5}\cdot \frac{15}{4}$

At this point, it is merely a matter of simplification and finishing the multiplication:

$\frac{1}{5}\cdot\frac{15}{4}=\frac{15}{20}=\frac{3}{4}$

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Question

Which of the following is equivalent to $\frac{7}{\frac{4}{5}}$ ?

Answer

To begin with, most students find it easy to remember that...

$\frac{7}{\frac{4}{5}}=\frac{\frac{7}{1}}{\frac{4}{5}}$

From this, you can apply the rule of division of fractions. That is, multiply by the reciprocal:

Therefore,

$\frac{\frac{7}{1}}{\frac{4}{5}} = \frac{7}{1} \cdot \frac{5}{4}=\frac{35}{4}$

Since nothing needs to be reduced, this is your answer.

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Question

Car A traveled 120 miles with 5 gallons of fuel.

Car B can travel 25 miles per gallon of fuel.

Quantity A: The fuel efficiency of car A

Quantity B: The fuel efficiency of car B

Answer

Let's make the two quantities look the same.

Quantity A: 120 miles / 5 gallons = 24 miles / gallon

Quantity B: 25 miles / gallon

Quantity B is greater.

Compare your answer with the correct one above

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