ACT Math : Algebraic Fractions

Study concepts, example questions & explanations for ACT Math

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Example Questions

Example Question #12 : How To Evaluate A Fraction

Simplify:

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

Possible Answers:

\displaystyle \frac{27}{70}

\displaystyle \frac{81}{7}

\displaystyle \frac{135}{56}

\displaystyle \frac{47}{5}

\displaystyle \frac{98}{71}

Correct answer:

\displaystyle \frac{135}{56}

Explanation:

Begin by simplifying the numerator.

\displaystyle \frac{1}{4}+\frac{5}{7} has a common denominator of \displaystyle 28.  Therefore, we can rewrite it as:

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

Now, in our original problem this is really is:

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

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

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

Example Question #14 : How To Evaluate A Fraction

Simplify:

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

Possible Answers:

\displaystyle -\frac{551}{12}

\displaystyle \frac{8414}{41}

\displaystyle -\frac{4532}{1575}

\displaystyle -\frac{25956}{275}

\displaystyle \frac{8417}{14}

Correct answer:

\displaystyle -\frac{25956}{275}

Explanation:

Begin by simplifying the numerator and the denominator.

Numerator

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

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

Denominator

\displaystyle \frac{1}{9}-\frac{2}{7} has a common denominator of \displaystyle 63.  Therefore, we have:

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

Now, reconstructing our fraction, we have:

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

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

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

Example Question #15 : How To Evaluate A Fraction

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

 

Possible Answers:

\displaystyle \frac{1}{2}

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

 

\displaystyle 2x^2-32

\displaystyle 2

None of the other answer choices are correct.

Correct answer:

\displaystyle 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

Example Question #16 : How To Evaluate A Fraction

Solve for \displaystyle x:

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

Possible Answers:

\displaystyle -\frac{81}{13}

\displaystyle -\frac{5}{7}

\displaystyle -\frac{280}{57}

\displaystyle \frac{100}{21}

\displaystyle \frac{43}{2}

Correct answer:

\displaystyle -\frac{280}{57}

Explanation:

Begin by isolating the variables:

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

Now, the common denominator of the variable terms is \displaystyle 8. The common denominator of the constant values is \displaystyle 3. Thus, you can rewrite your equation:

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

Simplify:

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

Cross-multiply:

\displaystyle 19x * 3 = -35*8

Simplify:

\displaystyle 57x = -280

Finally, solve for \displaystyle x:

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

Example Question #31 : Algebraic Fractions

Evaluate Actmath_18_159_q9when x=11. Round to the nearest tenth.

 

Possible Answers:

1.8

0.2

0.3

1.9

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.

Example Question #32 : Algebraic Fractions

Find the inverse equation of:

\displaystyle 3y-2x=20

 

Possible Answers:

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

\displaystyle \frac{2x+20}{3}=y

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

\displaystyle \frac{1}{2}x+10=y

Correct answer:

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

Explanation:

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

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

 

 

Example Question #5 : Algebraic Fractions

Find the inverse equation of  \displaystyle 5x-4y=18.

Possible Answers:

\displaystyle y=\frac{x+18}{4}

\displaystyle x=\frac{4y+18}{5}

\displaystyle x=y

\displaystyle y=\frac{4x+18}{5}

\displaystyle y=\frac{4x+5}{18}

Correct answer:

\displaystyle y=\frac{4x+18}{5}

Explanation:

\displaystyle 5x-4y=18

1. Switch the \displaystyle x and \displaystyle y variables in the above equation.

\displaystyle 5y-4x=18

 

2. Solve for \displaystyle y:

\displaystyle 5y-4x=18

\displaystyle 5y-4x+4x=18+4x

\displaystyle 5y=4x+18

\displaystyle \frac{5y}{5}=\frac{4x+18}{5}

\displaystyle y=\frac{4x+18}{5}

 

Example Question #33 : Algebraic Fractions

When \displaystyle x=2,  \displaystyle y=18.

When \displaystyle x=9\displaystyle y=4.

If \displaystyle x varies inversely with \displaystyle y, what is the value of \displaystyle y when \displaystyle x=12?

Possible Answers:

\displaystyle y=x

\displaystyle y=3

\displaystyle y=6

\displaystyle y=1

\displaystyle y=1.5

Correct answer:

\displaystyle y=3

Explanation:

If \displaystyle y varies inversely with \displaystyle x\displaystyle y=\frac{K}{x}.

 

1. Using any of the two \displaystyle x,y combinations given, solve for \displaystyle K:

Using \displaystyle (2,18):

\displaystyle 18=\frac{K}{2}

\displaystyle K=36

 

2. Use your new equation \displaystyle y=\frac{36}{x} and solve when \displaystyle x=12:

\displaystyle y=\frac{36}{12}=3

 

Example Question #34 : Algebraic Fractions

x

y

\displaystyle 5

\displaystyle 4.8

\displaystyle 6.4

\displaystyle 3.75

\displaystyle 3

\displaystyle n

\displaystyle 20

\displaystyle 1.2

If \displaystyle y varies inversely with \displaystyle x, what is the value of \displaystyle n?

Possible Answers:

\displaystyle 12

\displaystyle 4

\displaystyle 17

\displaystyle 8

\displaystyle 6

Correct answer:

\displaystyle 8

Explanation:

An inverse variation is a function in the form: \displaystyle xy = k or \displaystyle y = \frac{k}{x}, where \displaystyle k is not equal to 0. 

Substitute each \displaystyle \left ( x,y \right ) in \displaystyle xy = k.

\displaystyle 5(4.8) = 24

\displaystyle 6.4(3.75) = 24

\displaystyle 20(1.2) = 24

Therefore, the constant of variation, \displaystyle k, must equal 24. If \displaystyle y varies inversely as \displaystyle x\displaystyle 3n must equal 24. Solve for \displaystyle n.

\displaystyle 3n = 24

\displaystyle n = 8

Example Question #1 : How To Find Inverse Variation

Two numbers \displaystyle x and \displaystyle y vary inversely, and \displaystyle x=3 when \displaystyle y=8. If this is true, what is the value of \displaystyle x when \displaystyle y =6?

Possible Answers:

\displaystyle 10

\displaystyle 12

\displaystyle 3

\displaystyle 4

\displaystyle 6

Correct answer:

\displaystyle 4

Explanation:

If \displaystyle x=3 when \displaystyle y=8, and the variation is direct, then \displaystyle xy = 24. Using this, we know that if \displaystyle y=6\displaystyle x= \frac{24}{6} = 4.

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