GRE Math : Operations

Study concepts, example questions & explanations for GRE Math

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Example Questions

Example Question #1 : How To Add Fractions

What is the result of adding \(\displaystyle 20\%\) of \(\displaystyle \frac{2}{7}\) to \(\displaystyle \frac{1}{4}\)?

Possible Answers:

\(\displaystyle \frac{47}{140}\)

\(\displaystyle \frac{43}{140}\)

\(\displaystyle \frac{3}{39}\)

\(\displaystyle \frac{23}{11}\)

\(\displaystyle \frac{3}{28}\)

Correct answer:

\(\displaystyle \frac{43}{140}\)

Explanation:

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.

Example Question #1 : Operations

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

Possible Answers:

\frac{1}{12}\(\displaystyle \frac{1}{12}\)

\frac{1}{4}\(\displaystyle \frac{1}{4}\)

\frac{3}{4}\(\displaystyle \frac{3}{4}\)

\frac{3}{8}\(\displaystyle \frac{3}{8}\)

\frac{1}{3}\(\displaystyle \frac{1}{3}\)

Correct answer:

\frac{1}{12}\(\displaystyle \frac{1}{12}\)

Explanation:

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

 

Now we have: \frac{1}{4} + \frac{1}{2} - \frac{2}{3}\(\displaystyle \frac{1}{4} + \frac{1}{2} - \frac{2}{3}\)

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

Example Question #2 : Operations

Quantity A: \(\displaystyle \frac{15}{x}+\frac{12}{y}\)

Quantity B: \(\displaystyle \frac{15y+12x}{xy}\)

Which of the following is true?

Possible Answers:

The relationship between the two quantities cannot be determined.

Quantity B is larger.

Quantity A is larger.

The two quantities are equal.

Correct answer:

The two quantities are equal.

Explanation:

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

\(\displaystyle \frac{15}{x}*\frac{y}{y}+\frac{12}{y}*\frac{x}{x}\)

This is the same as:

\(\displaystyle \frac{15y}{xy}+\frac{12x}{xy}\), or \(\displaystyle \frac{15y+12x}{xy}\)

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

Example Question #41 : Fractions

Solve for \(\displaystyle x\):

\(\displaystyle \frac{x}{4}=\frac{3x}{8}-12\)

Possible Answers:

\(\displaystyle 14\)

\(\displaystyle -12\)

\(\displaystyle 96\)

\(\displaystyle -82\)

\(\displaystyle 56\)

Correct answer:

\(\displaystyle 96\)

Explanation:

Begin by isolating the \(\displaystyle x\) factors:

\(\displaystyle \frac{x}{4}-\frac{3x}{8}=-12\)

Now, the common denominator of these two fractions is \(\displaystyle 8\).  Therefore, multiply \(\displaystyle \frac{x}{4}\) by \(\displaystyle \frac{2}{2}\):

\(\displaystyle \frac{2}{2}*\frac{x}{4}-\frac{3x}{8}=-12\)

\(\displaystyle \frac{2x}{8}-\frac{3x}{8}=-12\)

Now, you can subtract the left values:

\(\displaystyle -\frac{x}{8}=-12\)

Now, multiply both sides by \(\displaystyle -8\):

\(\displaystyle x=96\)

Example Question #1041 : Gre Quantitative Reasoning

Simplify:

\(\displaystyle \frac{15}{12}-\frac{2}{9}\)

Possible Answers:

\(\displaystyle \frac{5}{9}\)

\(\displaystyle \frac{13}{3}\)

\(\displaystyle \frac{13}{21}\)

\(\displaystyle \frac{37}{36}\)

\(\displaystyle \frac{21}{22}\)

Correct answer:

\(\displaystyle \frac{37}{36}\)

Explanation:

Just like adding fractions, when you subtract fractions, you need to find a common denominator. For \(\displaystyle 12\) and \(\displaystyle 9\), the least common denominator is \(\displaystyle 36\). In order to do your subtraction, you need to multiply appropriately to give your fractions this denominator:

\(\displaystyle \frac{15}{12}-\frac{2}{9}=\frac{15}{12}*\frac{3}{3}-\frac{2}{9}*\frac{4}{4}\)

Which is the same as...

\(\displaystyle \frac{45}{36}-\frac{8}{36}\)

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

\(\displaystyle \frac{37}{36}\)

Example Question #1 : How To Subtract Fractions

Which of the following is true?

Quantity A: \(\displaystyle \frac{12}{7}-\frac{3}{5}\)

Quantity B: \(\displaystyle \frac{11}{6}-\frac{7}{8}\)

Possible Answers:

The two quantities are equal.

The relationship between the quantities cannot be determined.

Quantity B is larger.

Quantity A is larger.

Correct answer:

Quantity A is larger.

Explanation:

First, consider each quantity separately.

 

Quantity A

 \(\displaystyle \frac{12}{7}-\frac{3}{5}\)

These two fractions do not have a common factor. Their common denominator is \(\displaystyle 35\). Thus, we multiply the fractions as follows to give them a common denominator:

\(\displaystyle \frac{12}{7}*\frac{5}{5}-\frac{3}{5}*\frac{7}{7}\)

This is the same as:

\(\displaystyle \frac{60}{35}-\frac{21}{35}=\frac{39}{35}\)

 

Quantity B

 \(\displaystyle \frac{11}{6}-\frac{7}{8}\)

The common denominator of these two values is \(\displaystyle 24\).  Therefore, you multiply the fractions as follows to give them a common denominator:

\(\displaystyle \frac{11}{6}*\frac{4}{4}-\frac{7}{8}*\frac{3}{3}\)

This is the same as:

\(\displaystyle \frac{44}{24}-\frac{21}{24}=\frac{23}{24}\)

Since Quantity A is larger than \(\displaystyle 1\) and Quantity B is a positive fraction less than \(\displaystyle 1\), we know that Quantity A is larger without even using a calculator.

Example Question #451 : Arithmetic

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?

Possible Answers:

122

103

136

The answer cannot be determined from the given information.

306

Correct answer:

122

Explanation:

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. 

Example Question #1 : How To Multiply Fractions

If \(\displaystyle \frac{4}{7}\) of a number is \(\displaystyle 22\), what is \(\displaystyle \frac{6}{7}\) of that number?

Possible Answers:

\(\displaystyle 44\)

\(\displaystyle 33\)

\(\displaystyle 12\)

\(\displaystyle 6\)

\(\displaystyle 11\)

Correct answer:

\(\displaystyle 33\)

Explanation:

The least common multiple of 4 and 6 is 12.

So we know if \(\displaystyle \frac{4}{7}\) of the number is \(\displaystyle 22\) then 

\(\displaystyle 3\cdot \frac{4}{7}=\frac{12}{7}\) of the number is 

\(\displaystyle 3\cdot 22=66\).

So then it follows that

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

\(\displaystyle \frac{1}{2}\cdot 66=33\).

Example Question #2 : How To Multiply Fractions

If \(\displaystyle 11x=3\) and \(\displaystyle 3y=11\), what is the value of \(\displaystyle \frac{x}{y}\)?

Possible Answers:

\(\displaystyle \frac{11}{3}\)

\(\displaystyle \frac{3}{121}\)

\(\displaystyle \frac{121}{9}\)

\(\displaystyle 1\)

\(\displaystyle \frac{9}{121}\)

Correct answer:

\(\displaystyle \frac{9}{121}\)

Explanation:

\(\displaystyle 11x=3\Rightarrow x=\frac{3}{11}\)

\(\displaystyle 3y=11\Rightarrow y=\frac{11}{3}\)

\(\displaystyle \frac{x}{y}= \frac{\frac{3}{11}}{\frac{11}{3}}=\frac{3}{11}\cdot \frac{3}{11}=\frac{9}{121}\)

Example Question #3 : How To Multiply Fractions

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?

Possible Answers:

\(\displaystyle \small \frac{5}{36}\)

\(\displaystyle \small \frac{2}{21}\)

\(\displaystyle \small \frac{8}{15}\)

\(\displaystyle \small \frac{6}{15}\)

\(\displaystyle \small \frac{1}{12}\)

Correct answer:

\(\displaystyle \small \frac{5}{36}\)

Explanation:

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: \(\displaystyle \frac{1}{4}\)

Drive: \(\displaystyle \frac{1}{3}\)

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

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.

\(\displaystyle \frac{1}{3}*\frac{5}{12}=\frac{5}{36}\)

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