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SAT Math : Algebraic Fractions

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Example Question #1 : How To Evaluate A Fraction

Evaluate the following equation when x=5 and round your answer to the nearest hundredth.

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

Possible Answers:

-1.94

4.33

5.28

5.00

2.32

Correct answer:

2.32

Explanation:

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

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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Example Question #2 : How To Evaluate A Fraction

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

Possible Answers:

$\frac{1}{4}$

$\frac{1}{64}$

$\frac{1}{16}$

$\frac{1}{32}$

$\frac{1}{2}$

Correct answer:

$\frac{1}{4}$

Explanation:

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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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 #1 : How To Evaluate A Fraction

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

Possible Answers:

0.3

0.2

1.9

1.8

Correct answer:

1.8

Explanation:

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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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 #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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SAT Math : Algebraic Fractions

Study concepts, example questions & explanations for SAT Math

All SAT Math Resources

Example Questions

Example Question #1 : How To Evaluate A Fraction

Example Question #2 : How To Evaluate A Fraction

Example Question #1 : How To Evaluate A Fraction

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 #21 : Algebraic Fractions

All SAT Math Resources

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