Operations - GRE Quantitative Reasoning

Card 0 of 136

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

There are 340 students at Saint Louis High School in the graduating senior class. Of these students, 9/10 are going to college. Of those going to college, 2/5 are going to Saint Louis University. How many students are going to Saint Louis University?

Answer

122 students are going to Saint Louis University. To answer this question, the following equation can be used: 340(9/10)(2/5) . This is then rounded down to 122 students attending Saint Louis University.

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Question

At a certain company, one quarter of the employees take the bus to work and one third drive. Of the remaining employees, half walk, one third ride a bike, and the rest take the subway.

Out of the total number of employees, what fraction ride a bike to work?

Answer

First we want to find the fraction of employees that neither take the bus nor drive, so we’ll add the fractions that do take the bus or drive and subtract that result from the total.

Bus: $\frac{1}{4}$

Drive: $\frac{1}{3}$

Remaining: $1-(\frac{1}{4}+\frac{1}{3})=1-(\frac{7}{12})=\frac{5}{12}$

Now we need the fraction representing one third of these remaining employees (the fraction that ride a bike). Since "of " means multiply, we'll multiply.

$\frac{1}{3}*\frac{5}{12}=\frac{5}{36}$

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Question

If $\frac{4}{7}$ of a number is , what is $\frac{6}{7}$ of that number?

Answer

The least common multiple of 4 and 6 is 12.

So we know if $\frac{4}{7}$ of the number is then

$3\cdot \frac{4}{7}=\frac{12}{7}$ of the number is

$3\cdot 22=66$ .

So then it follows that

$\frac{1}{2}\cdot \frac{12}{7}=\frac{12}{14}=\frac{6}{7}$ of the number is

$\frac{1}{2}\cdot 66=33$ .

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Question

If and , what is the value of $\frac{x}{y}$ ?

Answer

$11x=3\Rightarrow x=\frac{3}{11}$

$3y=11\Rightarrow y=\frac{11}{3}$

$\frac{x}{y}= \frac{\frac{3}{11}}{\frac{11}{3}}=\frac{3}{11}\cdot \frac{3}{11}=\frac{9}{121}$

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Question

Simplify:

$\frac{44}{18}\frac{3}{11}\frac{5}{8}$

Answer

Multiplying fractions is very easy. All you do is multiply all the numerators by each other and all the denominators by each other. You do not have to do anything that has to do with fancy common denominators like you do for adding and subtracting. For a question like this, it is often easiest just to cancel factors before you start your final multiplication. First, note:

$\frac{44}{18}\frac{3}{11}\frac{5}{8}=\frac{4435}{18118}$

Now, cancel the from the :

$\frac{435}{188}$

Next, the in the numerator cancels with the in the denominator:

$\frac{35}{182}$

Finally, the in the numerator cancels with the in the denominator:

$\frac{5}{62}=\frac{5}{12}$

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Question

Simplify:

$\frac{15}{2}\frac{12}{45}\frac{3}{4}$

Answer

Multiplying fractions is very easy. All you do is multiply all the numerators by each other and all the denominators by each other. You do not have to do anything that has to do with fancy common denominators like you do for adding and subtracting. For a question like this, it is often easiest just to cancel factors before you start your final multiplication. First, note:

$\frac{15}{2}\frac{12}{45}\frac{3}{4}=\frac{15123}{2454}$

Now, cancel the in the denominator with the in the numerator:

$\frac{1533}{245}$

Next, the in the numerator cancels with the in the denominator:

$\frac{15}{25}$

Finally, cancel the in the denominator with the in the numerator:

$\frac{3}{2}$

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Question

Solve for :

$\frac{1}{3}(4x+\frac{12}{5})=2$

Answer

Begin by distributing the group on the left side of the equation. Remember that it is easy to multiply fractions. You only need to multiply the denominators and numerators. There are no "fancy" steps in between.

Therefore,

$\frac{1}{3}(4x+\frac{12}{5})=2$

is the same as:

$(\frac{1}{3}4x+\frac{1}{3}\frac{12}{5})=2$

You can cancel part of the second fraction out, so you get:

$\frac{4x}{3}+\frac{4}{5}=2$

Now, subtract $\frac{4}{5}$ from both sides:

$\frac{4x}{3}=2-\frac{4}{5}$

Simplifying the right side of the equation, you get...

$\frac{4x}{3}=\frac{10}{5}-\frac{4}{5}$

$\frac{4x}{3}=\frac{6}{5}$

Now, multiply both sides by $\frac{3}{4}$ :

$\frac{3}{4}\frac{4x}{3}=\frac{6}{5}\frac{3}{4}$

Simplify:

$x=\frac{18}{20} = \frac{9}{10}$

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Question

Solve for :

$\frac{x}{4}=\frac{3x}{8}-12$

Answer

Begin by isolating the factors:

$\frac{x}{4}-\frac{3x}{8}=-12$

Now, the common denominator of these two fractions is . Therefore, multiply $\frac{x}{4}$ by $\frac{2}{2}$ :

$\frac{2}{2}*\frac{x}{4}-\frac{3x}{8}=-12$

$\frac{2x}{8}-\frac{3x}{8}=-12$

Now, you can subtract the left values:

$-\frac{x}{8}=-12$

Now, multiply both sides by :

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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

Simplify:

$\frac{15}{12}-\frac{2}{9}$

Answer

Just like adding fractions, when you subtract fractions, you need to find a common denominator. For and , the least common denominator is . In order to do your subtraction, you need to multiply appropriately to give your fractions this denominator:

$\frac{15}{12}-\frac{2}{9}=\frac{15}{12}\frac{3}{3}-\frac{2}{9}\frac{4}{4}$

Which is the same as...

$\frac{45}{36}-\frac{8}{36}$

Now, you can subtract the numerators and retain the denominator:

$\frac{37}{36}$

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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

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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