Algebraic Fractions - PSAT Math

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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

Find the inverse equation of:

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Answer

To solve for an inverse, we switch x and y and solve for y. Doing so yields:

$\frac{3x-20}{2}=y$

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Question

Evaluate when x=11. Round to the nearest tenth.

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Answer

Wherever there is an x, plug in 11 and perform the given operations. The numerator will be equal to 83 and the denominator will be equal to 46. 83 divided by 46 is equal to 1.804… and since the second decimal place is not greater than or equal to 5, the first decimal place stays the same when rounding so the final answer is 1.8.

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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

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

Simplify the given fraction:

$\frac{120}{6000}$

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Answer

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

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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

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

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

A school's tornado shelter has enough food to last 20 children for 6 days. If 24 children ended up taking shelter together, for how many fewer days will the food last?

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Answer

Because the number of days goes down as the number of children goes up, this problem type is inverse variation. We can solve this problem by the following steps:

206=24x

120=24x

x=120/24

x=5

In this equation, x represents the total number of days that can be weathered by 24 students. This is down from the 6 days that 20 students could take shelter together. So the difference is 1 day less.

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