Simplifying Fractions - GRE Quantitative Reasoning

Card 0 of 18

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

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

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

For how many integers, , between 26 and 36 is it true that

$\frac{3}{x}_{,}\frac{4}{x}_{,}$ and $\frac{5}{x}$ are all in lowest terms?

Answer

If is even, then $\frac{4}{x}$ is not in lowest terms, since both and 4 are divisible by 2. Therefore, the only possibilities are 27,29,31,33, and 35. But $\frac{3}{27}= \frac{1}{9}$ , $\frac{3}{33}= \frac{1}{11}$ , and $\frac{5}{35}= \frac{1}{7}$ , so only two integers satisfy the given condition: 29 and 31.

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Question

Mrs. Lawrence's class has students, of which are girls. If you were to choose a student at random, what's the probability that the student chosen would be a boy?

Answer

In order to find out the probability of choosing a boy, you must first find out how many boys there are. Since there are girls out of students, students should be boys.

Therefore, the probability of choosing a boy is,

$\frac{21}{35}$ .

However, this is not one of the answer choices; therefore, you must reduce the fraction.

In order to reduce a fraction, you have to find their GCM, or greatest common multiple. This is the biggest number that will go into both the numerator and denominator . The largest number is . Divide both the top and bottom by , and you will get the answer:

$\frac{3}{5}$

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Question

Simplify the fraction:

$\frac{0.0003(0.4)(0.0002)}{0.00000006(0.01)}$

Answer

To simplify the fraction

$\frac{0.0003(0.4)(0.0002)}{0.00000006(0.01)}$

It may be helpful to write it in terms of scientific notation:

$\frac{3\cdot 10^{-4}(4\cdot 10^{-1})(2\cdot 10^{-4})}{6\cdot 10^{-8}(1\cdot 10^{-2})}$

$\frac{24\cdot 10^{-4-1-4}}{6\cdot 10^{-8-2}}$

$\frac{24\cdot 10^{-9}}{6\cdot 10^{-10}}$

$4\cdot10^{-9+10}$

$4\cdot10$

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Question

Simply the fraction:

$\frac{4a^3+12a^2b+12ab^2+4b^3}{3a^2+6ab+3b^2}$

Answer

To simplify the fraction

$\frac{4a^3+12a^2b+12ab^2+4b^3}{3a^2+6ab+3b^2}$

Begin by factoring out common terms in the numerator and denominator:

$\frac{4(a^3+3a^2b+3ab^2+b^3)}{3(a^2+2ab+b^2)}$

Now notice that the coefficients for the polynomials in the numerator and denominator match the form of Pascal's Triangle:

The fraction can be rewritten as:

$\frac{4(a+b)^3}{3(a+b)^2}$

And finally reduced to

$\frac{4}{3}(a+b)$

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