How to evaluate a fraction - GRE Quantitative Reasoning

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Question

Simplify:

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

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Answer

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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Question

Solve

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Answer

The common denominator of the left side is x(x–1). Multiplying the top and bottom of 1/x by (x–1) yields

Since this statement is true, there are infinitely many solutions.

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Question

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

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Answer

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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Question

If 3x = 12, y/4 = 10, and 4z = 9, what is the value of (10xyz)/xy?

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Answer

Solve for the variables, the plug into formula.

x = 12/3 = 4

y = 10 * 4 = 40

z= 9/4 = 2 1/4

10xyz = 3600

Xy = 160

3600/160 = 22 1/2

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Question

Mary walked to school at an average speed of 2 miles per hour and jogged back along the same route at an average speed of 6 miles per hour. If her total traveling time was 1 hour, what was the total number of miles in the round trip?

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Answer

Since Mary traveled 3 times as quickly coming from school as she did going to school (6 miles per hour compared to 2 miles per hour), we know that Mary spent only a third of the time coming from school as she did going. If x represents the number of hours it took to get to school, then x/3 represents the number of hours it took her to return.

Knowing that the total trip took 1 hour, we have:

x + x/3 = 1

3x/3 + 1x/3 = 1

4_x_/3 = 1

x = 3/4

So we know it took Mary 3/4 of an hour to travel to school (and the remaining 1/4 of an hour to get back).

Remembering that distance = rate * time, the distance Mary traveled on her way to school was (2 miles per hour) * (3/4 of an hour) = 3/2 miles. Furthermore, since she took the same route coming back, she must have traveled 3/2 of a mile to return as well.

Therefore, the the total number of miles in Mary's round trip is 3/2 miles + 3/2 miles = 6/2 miles = 3 miles.

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Question

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?

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Answer

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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Question

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?

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Answer

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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Question

If $w=\frac{1}{8}$ then which of the following is equal to $w^\frac{2}{3}$ ?

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Answer

To raise $\frac{1}{8}$ to the exponent $\frac{2}{3}$ , square $\frac{1}{8}$ and then take the cube root.

$w^\frac{2}{3}=\sqrt[3]{w^2}$

$(\frac{1}{8})^\frac{2}{3}=\sqrt[3]{(\frac{1}{8})^2}=\sqrt[3]{\frac{1}{64}}$

$\sqrt[3]{\frac{1}{64}}=\frac{1}{4}$

$(\frac{1}{8})^{\frac{2}{3}}=\frac{1}{4}$

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Question

For this question, the following trigonometric identities apply:

$\csc x=\frac{1}{\sin x}$ ,

$\sec x=\frac{1}{\cos x}$

$\cot x=\frac{1}{\tan x}$

Simplify: $\sin ^{2}x\csc x\cot x$

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Answer

To begin a problem like this, you must first convert everything to $\sin x$ and $\cos x$ alone. This way, you can begin to cancel and combine to its most simplified form.

Since $\csc x=\frac{1}{\sin x}$ and $\cot x=\frac{\cos x}{\sin x}$ , we insert those identities into the equation as follows.

$\sin ^{2}x\cdot \frac{1}{\sin x}\cdot \frac{\cos x}{\sin x}$

From here we combine the numerator and denominators of each fraction together to easily see what we can combine and cancel.

$\frac{\sin^{2}x\cdot \cos x}{\sin x\cdot \sin x}$

Since there is a $\sin ^{2}x$ in the numerator and the denominator, we can cancel them as they divide to equal 1. All we have left is $\cos x$ , the answer.

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Question

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

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Answer

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:

$2\frac{x}{10}+2\frac{y}{6}=\frac{x}{5}+\frac{y}{3}=\frac{3x}{35}+\frac{5y}{53}=\frac{3x+5y}{15}$

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Question

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

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Answer

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

This simplifies to 7.

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Question

If $x =\frac{1}{2}$ and $y=\frac{2}{3}$ , then which of the following is equal to $\frac{2}{x+y}$ ?

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Answer

In order to solve $\frac{2}{x+y}$ , first substitute the values of and provided in the problem:

$\frac{2}{\frac{1}{2}+\frac{2}{3}}$

Find the Least Common Multiple (LCM) of the fractional terms in the denominator and find the equivalent fractions with the same common denominator:

$\frac{2}{\frac{1}{2}+\frac{2}{3}}$

$=\frac{2}{\frac{3}{6}+\frac{4}{6}}$

$=\frac{2}{\frac{7}{6}}$

Finally, in order to divide by a fraction, we must multiply by the reciprocal of the fraction:

$=2\times \frac{6}{7}$

$=\frac{12}{7}$

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Question

If , , and , find the value of $\frac{2x+y}{z}$ .

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Answer

In order to solve $\frac{2x+y}{z}$ , we must first find the values of , , and using the initial equations provided. Starting with :

$x=\frac{55}{5}=11$

Then:

Finally:

With the values of , , and in hand, we can solve the final equation:

$\frac{2x+y}{z}$

$=\frac{2(11)+12}{10}$

$=\frac{22+12}{10}$

$=\frac{34}{10}$

$=\frac{17}{5}$

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Question

Find the value of if $w=\bigg(\frac{1}{3}+y\bigg)^{2}$ and $y=\frac{1}{9}$ .

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Answer

In order to solve for , first substitute $y=\frac{1}{9}$ into the equation for :

$w=\bigg(\frac{1}{3}+y\bigg)^{2}$

$w=\bigg(\frac{1}{3}+\frac{1}{9}\bigg)^{2}$

Then, find the Least Common Multiple (LCM) of the two fractions and generate equivalent fractions with the same denominator:

$w=\bigg(\frac{3}{9}+\frac{1}{9}\bigg)^{2}$

Finally, simplify the equation:

$w=\bigg(\frac{4}{9}\bigg)^{2}$

$w=\frac{16}{81}$

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Question

Evaluate the following equation when and round your answer to the nearest hundredth.

$\frac{x^{3}-2x^{2}-3x+5}{5x+3}$

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Answer

$\frac{x^{3}-2x^{2}-3x+5}{5x+3}$

1. Plug in wherever there is an in the above equation.

$\frac{5^{3}-(2)(5^{2})-(3)(5)+5}{(5)(5)+3}$

2. Perform the above operations.

$\frac{5^{3}-(2)(25)-(3)(5)+5}{(5)(5)+3}$

$\frac{125-50-15+5}{25+3}$

$\frac{65}{28}=2.32$

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Question

Simplify:

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

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Answer

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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Question

Simplify:

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

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Answer

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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Question

Solve for :

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

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Answer

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:

Simplify:

Finally, solve for :

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

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Question

Simplify the expression

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

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Answer

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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Question

Solve

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Answer

The common denominator of the left side is x(x–1). Multiplying the top and bottom of 1/x by (x–1) yields

Since this statement is true, there are infinitely many solutions.

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