Solving Rational and Fractional Functions - SAT Subject Test in Math I

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Question

Simplify $\frac{2}{3}+\frac{1}{4}+\frac{5}{6}$

Answer

Find the least common denominator (LCD) and convert each fraction to the LCD and then add. Simplify as necessary.

$\frac{2}{3}+\frac{1}{4}+\frac{5}{6}=\frac{8}{12}+\frac{3}{12}+\frac{10}{12}=\frac{21}{12}$

The result is an improper fraction because the numerator is larger than the denominator and can be simplified and converted to a mix numeral.

$\frac{21}{12}=\frac{7}{4}=1\frac{3}{4}$

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Question

Simplify:

$\left (8 x ^{3 } \right )^{-2}$

Answer

Raise a fraction to a negative power by raising its reciprocal to the power of the absolute value of the exponent. Then apply the properties of exponents as follows:

$\left (8 x ^{3 } \right )^{-2} = \frac{1}{\left (8 x ^{3 } \right )^{ 2}} = \frac{1}{8^{ 2} \left ( x ^{3 } \right )^{ 2}} =\frac{1}{64 x ^{3 \cdot 2 } } =\frac{1}{64 x ^{6 } }$

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Question

Simplify $\frac{1}{5}+\frac{3}{8}$

Answer

To simplify this problem we need to find the least common denominator between the two fractions. To do this we look at 5 and at 8. The least common number between these two is 40.

In order to rewrite each fraction in terms of a denominator of 40 we need to muliple as follows:

$\frac{1}{5}\frac{8}{8} + \frac{3}{8} \frac{5}{5}$

we are able to mulitply by 8/8 and 5/5 because those fractions are really just 1 written in a different format.

Now using order of opperations we get the following

$\frac{8}{40}+\frac{15}{40}$

Now we have a common denominator and can do our addition to get the simplfied number:

$\frac{23}{40}$

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Question

Find the simplified result:

$\frac{5}{6} + \frac{7}{8}$

Answer

Start by making both fractions into the same denominator. One option is $6 \times 8 = 48$

Then adjust the numerators by multiplying each fraction's numerator by the other fraction's denominator:

$\frac{5}{6}\cdot \frac{8}{8}+ \frac{7}{8}\cdot \frac{6}{6}$

$=\frac{40}{48}+\frac{42}{48}$

Then add the adjusted numerators:

$=\frac{82}{48}$

Then we simplify by dividing both numerator and denominator by 2:

$=\frac{2 \cdot 41}{2 \cdot 24}=\frac{41}{24}$

which gives us the final result.

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Question

Solve the following equation to find .

$\frac{3}{14} + \frac{2}{7} = x-\frac{1}{7}$

Answer

In order to be able to find , we must first find the least common denominator. In this case, it is :

$\frac{2}{7}= \frac{2\times 2}{7\times 2}=\frac{4}{14}$

$\frac{1}{7}=\frac{1\times 2}{7\times 2}=\frac{2}{14}$

The equation can now be written as:

$\frac{3}{14} + \frac{4}{14} = x-\frac{2}{14}$

Solving for , we get:

$\frac{3}{14}+\frac{4}{14} + \frac{2}{14}=x$

$x=\frac{9}{14}$

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Question

What's the least common denominator of and ?

Answer

When finding the least common denominator, the quickest way is to multiply the numbers out.

In this case and share a factor other than which is . We can divide those numbers by to get and leftover.

Now, they don't share a common factor so basically multiply them out with the shared factor. Answer is .

$2\cdot 4\rightarrow2(1\cdot 2)=4$

Another approach is to list out the factors of both number and find the factor that appears in both sets first.

We can see that appears in both sets before any other number thus, this is our answer.

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Question

Simplify:

$\frac{25st^2}{6s^3t^5}\div\frac{30s^{-9}}{45s^2t^7}$

Answer

To divid fractions, you need to multiply the first fraction by the reciprocal of the second.

$(\frac{25st^2}{6s^3t^5})\div(\frac{30s^{-9}}{45s^2t^7})=(\frac{25st^2}{6s^3t^5})\times(\frac{45s^2t^7}{30s^{-9}})$

Now, multiply the numerators and denominators together, then simplify.

$(\frac{25st^2}{6s^3t^5})\times(\frac{45s^2t^7}{30s^{-9}})=\frac{1125s^3t^9}{180s^{-6}t^5}=\frac{25s^9t^4}{4}$

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Question

Simplify:

$\frac{12m^2n}{16m^3n^4}\div\frac{8m^{-4}n}{18m^4n^2}$

Answer

To divide fractions, you need to multiply the first fraction by the reciprocal of the second.

$(\frac{12m^2n}{16m^3n^4})\div(\frac{8m^{-4}n}{18m^4n^2})=(\frac{12m^2n}{16m^3n^4})\times(\frac{18m^4n^2}{8m^{-4}n})$

Now, multiply the numerators and denominators, then simplify.

$(\frac{12m^2n}{16m^3n^4})\times(\frac{18m^4n^2}{8m^{-4}n})=\frac{216m^6n^3}{128m^{-1}n^5}=\frac{27m^7}{16n^2}$

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Question

Simplify the following:

$\frac{3}{x^2-1}+\frac{4}{x+1}$

Answer

To simplify the sum of the two fractions, we must find the common denominator.

Simplifying the denominator of the first fraction, we get

$\frac{3}{x^2-1}=\frac{3}{(x-1)(x+1)}$

because the denominator is a difference of two squares, which follows the form

Now, we can rewrite the sum as

$\frac{3}{(x+1)(x-1)}+\frac{4}{x+1}$

It is far easier to see the common denominator now:

$\frac{3}{(x-1)(x+1)}+\frac{4(x-1)}{(x-1)(x+1)}=\frac{3+4x-4}{(x-1)(x+1)}=\frac{4x-1}{x^2-1}$

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Question

Solve:

$\frac{3}{x+1}+\frac{4}{x^2-1}=0$

Answer

To solve this equation, we must find the least common denominator to add the fractions.

Keep in mind that the denominator of the second term is the difference of squares, which can be rewritten as

This is the least common denominator.

Now, we multiply both sides of the equationby the LCD on top and bottom (this is essentially 1):

$\frac{(x+1)(x-1)}{(x+1)(x-1)}\cdot (\frac{3}{x+1}+\frac{4}{x^2-1})=0$

After canceling terms, we get

$\frac{3(x-1)+4}{x^2-1}=0$

Now, we solve for x:

$x=-\frac{1}{3}$

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Question

Solve: $\frac{1}{\sqrt{3x+1}} = \frac{1}{3}$

Answer

Cross multiply to eliminate the fractions.

$(1)\sqrt{3x+1} = (1)(3)$

$\sqrt{3x+1} =3$

Square both sides.

$(\sqrt{3x+1})^2 =3^2$

Subtract 1 on both sides.

Divide by 3 on both sides.

$\frac{3x}{3}=\frac{8}{3}$

The answer is: $\frac{8}{3}$

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