Fractions - GRE Quantitative Reasoning

Card 0 of 1968

Question

Simplify:

(2_x_ + 4)/(x + 2)

Answer

(2_x_ + 4)/(x + 2)

To simplify you must first factor the top polynomial to 2(x + 2). You may then eliminate the identical (x + 2) from the top and bottom leaving 2.

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Question

Quantitative Comparison: Compare Quantity A and Quantity B, using additional information centered above the two quantities if such information is given.

10 < n < 15

Quantity A Quantity B

7/13 4/n

Answer

To determine which quantity is greater, we must first determine the range of potential values for Quantity B. Let's call this quantity m. This is most efficiently done by dividing 4 by the highest and lowest possible values for n.

4/10 = 0.4

4/15 = 0.267

So the possible values for m are 0.267 < m < 0.4

Now let's find the value for 7/13, to make comparison easier.

7/13 = 0.538

Given this, no matter what the value of n is, 7/13 will still be a higher proportion, so Quantity B is greater.

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Question

Simplify:

$\frac{x^{2}y-5x^{2}y^{2}}{x^{2}y}$

Answer

With this problem the first thing to do is cancel out variables. The x2 can all be divided by each other because they are present in each system. The equation will now look like this:

$\frac{y-5y^{2}}{y}$

Now we can see that the equation can all be divided by y, leaving the answer to be:

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Question

Simplify:

$\frac{x^2-y^2}{x+y}$

Answer

_x_2 – _y_2 can be also expressed as (x + y)(x – y).

Therefore, the fraction now can be re-written as (x + y)(x – y)/(x + y).

This simplifies to (x – y).

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Question

Simplify the following expression:

$\frac{4x^6}{8x^3}$

Answer

Following this equation, you divide 4 by 8 to get 1/2. When a variable is raised to an exponent, and you are dividing, you subtract the exponents, so 6 – 3 = 3.

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Question

Simplify the given fraction:

$\frac{125}{2000}$

Answer

125 goes into 2000 evenly 16 times. 1/16 is the fraction in its simplest form.

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Question

Simplify the given fraction:

$\frac{120}{6000}$

Answer

120 goes into 6000 evenly 50 times, so we get 1/50 as our simplified fraction.

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Question

A train travels at a constant rate of meters per second. How many kilometers does it travel in minutes? $(1\ km=1000\ m)$

Answer

Set up the conversions as fractions and solve:

$\dpi{100} \small \frac{20m}{1sec}\times \frac{60sec^}{1min}\times \frac{1km}{1000m}\times \frac{10min}{1}$

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Question

Simplify. $\frac{4x^{4}z^{3}}{2xz^{2}}$

Answer

To simplify exponents which are being divided, subtract the exponents on the bottom from exponents on the top. Remember that only exponents with the same bases can be simplified

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Question

Simplify the following expression:

$\frac{2x^{4}-32}{2x^{2}-8}$

Answer

Factor both the numerator and the denominator:

$\frac{2(x^{2}-4)(x^{2}+4)}{2(x^{2}-4)}$

After reducing the fraction, all that remains is:

$(x^{2}+4)$

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Question

Simplify:

$\frac{x\sqrt{2}-2\sqrt{2}}{\sqrt{2x}+2}$

Answer

Notice that the $\sqrt{2}$ term appears frequently. Let's try to factor that out:

$\frac{x\sqrt{2}-2\sqrt{2}}{\sqrt{2x}+2} = \frac{\sqrt{2}(x-2)}{\sqrt{2}(\sqrt{x} +\sqrt{2})} = \frac{x-2}{\sqrt{x} +\sqrt{2}}$

Now multiply both the numerator and denominator by the conjugate of the denominator:

$\frac{x-2}{\sqrt{x} +\sqrt{2}} * \frac{\sqrt{x} -\sqrt{2}}{\sqrt{x} -\sqrt{2}} =\frac{(x-2)*(\sqrt{x} -\sqrt{2})}{x-2} =\sqrt{x} -\sqrt{2}$

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Question

Reduce the fraction:

$\frac{12}{96}$

Answer

The numerator and denominator are both divisible by 12. Thus, we divide both by 12 to get our final answer.

If we instead divide by another common factor, we may need to complete the process again to make sure that we have completely reduced the fraction.

In mathematical words we get the following:

$\frac{12}{96}=\frac{1 \cdot 12}{8 \cdot 12}=\frac{1}{8}$

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Question

Which quantity is greater?

Quantity A

$\frac{6}{33}+\frac{17}{24}$

Quantity B

$\frac{17}{32}+\frac{7}{25}$

Answer

This can be solved using 2 methods.

The most time-efficient solution would recognize that $\frac{17}{24}$ is the largest value and nearly equals the sum the other fraction by itself.

The more time consuming method would be to convert each fraction to decimal form and calculate the sum of each quantity.

Quantity A:

Quantity B:

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Question

Simplify.

$\frac{(x^3+8x^2-69x-252)}{(x^2+5x-84)}$

Answer

When we factor the numerator and denominator, we get:

$\frac{(x-7)(x+12)(x+3)}{(x-7)(x+12)}$ .

After cancelling , we are left with .

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Question

Which of the following fractions is between $\frac{1}{2}$ and $\frac{3}{4}$ ?

Answer

With common denominators, the range is from

$\frac{6}{12} to \frac{9}{12}$

or

$\frac{8}{16} to \frac{12}{16}$ .

The only fraction that falls in either of these ranges is $\frac{9}{16}$ .

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Question

Column A: $\frac{2}{5}%$

Column B:

Answer

2/5% = 0.40% = 0.004. Therefore, Column B is greater.

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Question

For every 1000 cookies baked, 34 are oatmeal raisin.

Quantity A: Percent of cookies baked that are oatmeal raisin

Quantity B: 3.4%

Answer

Simplify Quantity A by dividing the number of oatmeal raisin cookies by the total number of cookies to find the percentage of oatmeal raisin cookies. Since a percentage is defined as being out of 100, either multiply the resulting decimal by 100 or reduce the fraction until the denominator is 100. You will find that the two quantities are equal.

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Question

50 students took an exam. There were 4 A's, 9 B's, 15 C's, 8 D's, and the rest of the students failed. What percent of the students failed?

Answer

students failed.

$\frac{14}{50}=\frac{28}{100}=0.28$

which equals 28%.

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Question

36 is x% of 133. What is x

Answer

36 is x% of 133

that means that 36 = (x%)(133)

x% = 36/133 X 100 = 27%

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Question

is percent of

is percent of

Quantity A:

Quantity B:

Answer

To do this problem, translate each expression into mathematical terms:

is percent of :

$a=\frac{15}{100}(20)=\frac{300}{100}=3$

is percent of :

$7=\frac{b}{100}(140)=b\frac{140}{100}$

$b=\frac{100}{140}(7)=\frac{100}{20}=5$

Quantity B is greater.

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