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ACT Math : Algebraic Fractions

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Example Question #1 : How To Evaluate A Fraction

Solve

Possible Answers:

no solution

–1

infinitely many solutions

Correct answer:

infinitely many solutions

Explanation:

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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Example Question #5 : How To Evaluate A Fraction

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

Possible Answers:

$\sec x$

$\tan x$

$\sin x\cos x$

$\cos x$

$\sin x$

Correct answer:

$\cos x$

Explanation:

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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Example Question #1 : How To Evaluate A Fraction

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

Possible Answers:

360

22 ¹/2

160

4 ¹/₂

Correct answer:

22 ¹/2

Explanation:

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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Example Question #1 : How To Evaluate A Fraction

If 5x=55 , x+y = 23 , and y-z=2 , find the value of $\frac{2x+y}{z}$ .

Possible Answers:

$\frac{9}{2}$

$\frac{13}{5}$

$\frac{17}{5}$

$\frac{19}{5}$

$\frac{16}{7}$

Correct answer:

$\frac{17}{5}$

Explanation:

In order to solve $\frac{2x+y}{z}$ , we must first find the values of , , and using the initial equations provided. Starting with 5x=55 :

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

Then:

x+y=23

11+y=23

y=12

Finally:

y-z=2

12-z=2

10=z

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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Example Question #6 : How To Evaluate A Fraction

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

Possible Answers:

$\frac{9}{7}$

$2 \frac{1}{3}$

$\frac{10}{3}$

$\frac{12}{7}$

Correct answer:

$\frac{12}{7}$

Explanation:

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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Example Question #7 : How To Evaluate A Fraction

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

Possible Answers:

$\frac{1}{12}$

$\frac{3}{55}$

$\frac{4}{27}$

$\frac{16}{81}$

$\frac{9}{10}$

Correct answer:

$\frac{16}{81}$

Explanation:

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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Example Question #251 : Algebra

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

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:

$\frac{1}{30}$

$\frac{1}{2}$

$\frac{1}{5}$

$\frac{2}{5}$

$\frac{1}{6}$

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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ACT Math : Algebraic Fractions

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

Example Question #1 : How To Evaluate A Fraction

Example Question #5 : How To Evaluate A Fraction

Example Question #1 : How To Evaluate A Fraction

Example Question #1 : How To Evaluate A Fraction

Example Question #6 : How To Evaluate A Fraction

Example Question #7 : How To Evaluate A Fraction

Example Question #251 : Algebra

Example Question #12 : How To Evaluate A Fraction

Example Question #11 : How To Evaluate A Fraction

Example Question #11 : How To Evaluate A Fraction

All ACT Math Resources

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