GRE Math : Algebraic Fractions

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Example Question #21 : Algebraic Fractions

$\frac{7^{12}-7^{10}}{7^{11}-7^{9}}=$

Possible Answers:

343

Correct answer:

Explanation:

Factor out 7 from the numerator: $\frac{7(7^{11}-7^{9})}{7^{11}-7^{9}}$

This simplifies to 7.

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Example Question #12 : How To Evaluate A Fraction

If pizzas cost dollars and sodas cost dollars, what is the cost of pizzas and sodas in terms of and ?

Possible Answers:

$5x+\frac{3y}{15}$

3x+5y

15x+5y

$\frac{3x+5y}{15}$

15xy

Correct answer:

$\frac{3x+5y}{15}$

Explanation:

If 10 pizzas cost x dollars, then each pizza costs x/10. Similarly, each soda costs y/6. We can add the pizzas and sodas together by finding a common denominator:

$2*\frac{x}{10}+2*\frac{y}{6}=\frac{x}{5}+\frac{y}{3}=\frac{3x}{3*5}+\frac{5y}{5*3}=\frac{3x+5y}{15}$

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Example Question #11 : How To Evaluate A Fraction

Gre9

According the pie chart, the degree measure of the sector representing the number of workers spending 5 to 9 years in the same role is how much greater in the construction industry chart than in the financial industry chart?

Possible Answers:

$25^{\circ}$

$18^{\circ}$

$70^{\circ}$

$35^{\circ}$

$7^{\circ}$

Correct answer:

$25^{\circ}$

Explanation:

Since the values in the pie charts are currently in terms of percentages (/100), we must convert them to degrees (/360, since within a circle) to solve the question. The "5 to 9 years" portion for the financial and construction industries are 18 and 25 percent, respectively. As such, we can cross-multiply both:

18/100 = x/360

x = 65 degrees

25/100 = y/360

y = 90 degrees

Subtract: 90 – 65 = 25 degrees

Alternatively, we could first subtract the percentages (25 – 18 = 7), then convert the 7% to degree form via the same method of cross-multiplication.

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Example Question #11 : How To Evaluate A Fraction

6 contestants have an equal chance of winning a game. One contestant is disqualified, so now the 5 remaining contestants again have an equal chance of winning. How much more likely is a contestant to win after the disqualification?

Possible Answers:

$\frac{1}{30}$

$\frac{2}{5}$

$\frac{1}{6}$

$\frac{1}{2}$

$\frac{1}{5}$

Correct answer:

$\frac{1}{30}$

Explanation:

When there are 6 people playing, each contestant has a 1/6 chance of winning. After the disqualification, the remaining contestants have a 1/5 chance of winning.

1/5 – 1/6 = 6/30 – 5/30 = 1/30.

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Example Question #13 : How To Evaluate A Fraction

Simplify:

$\frac{\frac{1}{4}+\frac{5}{7}}{\frac{2}{5}}$

Possible Answers:

$\frac{27}{70}$

$\frac{135}{56}$

$\frac{81}{7}$

$\frac{47}{5}$

$\frac{98}{71}$

Correct answer:

$\frac{135}{56}$

Explanation:

Begin by simplifying the numerator.

$\frac{1}{4}+\frac{5}{7}$ has a common denominator of . Therefore, we can rewrite it as:

$\frac{7}{28}+\frac{20}{28}=\frac{27}{28}$

Now, in our original problem this is really is:

$\frac{\frac{27}{28}}{\frac{2}{5}}$

When you divide by a fraction, you really multiply by the reciprocal:

$\frac{\frac{27}{28}}{\frac{2}{5}}=\frac{27}{28} \cdot\frac{5}{2}=\frac{135}{56}$

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Example Question #14 : How To Evaluate A Fraction

Simplify:

$\frac{\frac{81}{5}+\frac{7}{25}}{\frac{1}{9}-\frac{2}{7}}$

Possible Answers:

$-\frac{551}{12}$

$\frac{8414}{41}$

$-\frac{25956}{275}$

$-\frac{4532}{1575}$

$\frac{8417}{14}$

Correct answer:

$-\frac{25956}{275}$

Explanation:

Begin by simplifying the numerator and the denominator.

Numerator

$\frac{81}{5}+\frac{7}{25}$ has a common denominator of . Therefore, we have:

$\frac{405}{25}+\frac{7}{25} = \frac{412}{25}$

Denominator

$\frac{1}{9}-\frac{2}{7}$ has a common denominator of . Therefore, we have:

$\frac{7}{63}-\frac{18}{63}=-\frac{11}{63}$

Now, reconstructing our fraction, we have:

$\frac{\frac{412}{25}}{-\frac{11}{63}}$

To make this division work, you multiply the numerator by the reciprocal of the denominator:

$\frac{\frac{412}{25}}{-\frac{11}{63}} = \frac{412}{25} \cdot -\frac{63}{11} =-\frac{25956}{275}$

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Example Question #11 : How To Evaluate A Fraction

Simplify: $(\frac{x-4}{0.5})\div (\frac{1}{x+4})$

Possible Answers:

$\frac{3}{2x^2}-16$

2x^2-32

$\frac{1}{2}$

None of the other answer choices are correct.

Correct answer:

2x^2-32

Explanation:

Recall that dividing is equivalent multiplying by the reciprocal. Therefore, ((x - 4) / (1 / 2)) / (1 / (x + 4)) = ((x - 4) * 2) * (x + 4) / 1.

Let's simplify this further:

(2x – 8) * (x + 4) = 2x² – 8x + 8x – 32 = 2x² – 32

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Example Question #16 : How To Evaluate A Fraction

Solve for :

$\frac{3x}{8}+12=\frac{1}{3}-2x$

Possible Answers:

$\frac{43}{2}$

$-\frac{5}{7}$

$-\frac{280}{57}$

$\frac{100}{21}$

$-\frac{81}{13}$

Correct answer:

$-\frac{280}{57}$

Explanation:

Begin by isolating the variables:

$\frac{3x}{8}+2x=\frac{1}{3}-12$

Now, the common denominator of the variable terms is . The common denominator of the constant values is . Thus, you can rewrite your equation:

$\frac{3x}{8}+\frac{16x}{8}=\frac{1}{3}-\frac{36}{3}$

Simplify:

$\frac{19x}{8}=-\frac{35}{3}$

Cross-multiply:

19x * 3 = -35*8

Simplify:

57x = -280

Finally, solve for :

$x = -\frac{280}{57}$

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Example Question #11 : How To Evaluate A Fraction

Simplify the expression

$\frac{x^3y+3x^2y^2}{x^2-9y^2}$

Possible Answers:

$\frac{x}{(x-3y)}$

$\frac{x^2y}{(x-3y)}$

$\frac{x^2y}{(x+3y)}$

$\frac{x^2}{(x-3y)}$

$\frac{xy}{(x+3y)}$

Correct answer:

$\frac{x^2y}{(x-3y)}$

Explanation:

Begin by pulling out like factors in the numerator:

$\frac{x^3y+3x^2y^2}{x^2-9y^2}$

$\frac{x^2y(x+3y)}{x^2-9y^2}$

Now rewrite the denominator, since it is a difference of squares:

$\frac{x^2y(x+3y)}{(x+3y)(x-3y)}$

Cancelling like terms in the numerator and denominator leaves:

$\frac{x^2y}{(x-3y)}$

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Example Question #11 : How To Evaluate A Fraction

Reduce the following fraction

$\frac{a^2b^3c+ab^2c^2}{5a^2b}$

Possible Answers:

$\frac{ac(ab+c)}{5b}$

$\frac{5ab(ab+c)}{c}$

$\frac{ab(ab+c)}{5c}$

$\frac{bc(ab+c)}{5a}$

Correct answer:

$\frac{bc(ab+c)}{5a}$

Explanation:

To reduce this fraction we need to factor the numerator and find like terms in the denominator to cancel out.

The fraction

$\frac{a^2b^3c+ab^2c^2}{5a^2b}$

can be rewritten as

$\frac{ab^2c(ab+c)}{5a^2b}$

by factoring.

From here cancel like terms in the numerator and denominator:

$\frac{bc(ab+c)}{5a}$

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GRE Math : Algebraic Fractions

Study concepts, example questions & explanations for GRE Math

All GRE Math Resources

Example Questions

Example Question #21 : Algebraic Fractions

Example Question #12 : How To Evaluate A Fraction

Example Question #11 : How To Evaluate A Fraction

Example Question #11 : How To Evaluate A Fraction

Example Question #13 : How To Evaluate A Fraction

Example Question #14 : How To Evaluate A Fraction

Example Question #11 : How To Evaluate A Fraction

Example Question #16 : How To Evaluate A Fraction

Example Question #11 : How To Evaluate A Fraction

Example Question #11 : How To Evaluate A Fraction

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