Fractions - Algebra 2

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Question

Divide the fractions: $\frac{\frac{2}{3}}{5}\div \frac{\frac{1}{2}}{3}$

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Answer

We will need to solve each complex fraction first.

Rewrite the complex fractions using a division sign, change the sign to a multiplication sign, and then take the reciprocal of the second term.

$\frac{\frac{2}{3}}{5} = \frac{2}{3}\div 5 =\frac{2}{3}\times \frac{1}{5} = \frac{2}{15}$

$\frac{\frac{1}{2}}{3} = \frac{1}{2}\div 3 = \frac{1}{2}\times \frac{1}{3} = \frac{1}{6}$

Divide the two fractions.

$\frac{2}{15}\div \frac{1}{6} = \frac{2}{15}\times \frac{6}{1} = \frac{12}{15}$

Reduce this fraction.

The answer is: $\frac{4}{5}$

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Question

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

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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 the expression.

$\frac{x-3y}{6x^{3}y}-\frac{x+3y}{6x^{3}y}$

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Answer

Since the denominator is the same in both expressions, we can perform the subtraction in the numerator. The result will have the same denominator of 6x3y.

$\frac{x-3y-(x+3y)}{6x^{3}y}$

$\frac{-6y}{6x^{3}y}$

Simplify the expression:

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

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Question

Find the solution: $\frac{1}{3} \cdot \frac{2}{1} \cdot \frac{3}{5} \cdot \frac{5}{8} \cdot \frac{8}{9} \cdot \frac{9}{1} \cdot \frac{2}{3}$

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Answer

Multiply all the nominators and the denominators separately.

Then, simplify the solution untill you get an answer that is an option on the exam.

$\frac{1}{3} \cdot \frac{2}{1} \cdot \frac{3}{5} \cdot \frac{5}{8} \cdot \frac{8}{9} \cdot \frac{9}{1} \cdot \frac{2}{3}=\frac{4320}{3240}=\frac{4}{3}$

An alternate, quicker solution would be to cancel out any matching denominators and nominators.

Once that is done, the equation simpifies to $2 \cdot \frac{2}{3}=\frac{4}{3}$

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Question

Suppose a recipe calls for $\frac{2}{3}$ cup of flour, $\frac{7}{12}$ cup of cinnamon, and $\frac{3}{4}$ cup of sugar.

How many cups of ingredients total does the recipe call for?

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Answer

To solve, we must first find a common denominator between 3, 4, and 12. All 3 numbers are factors of 12 and therefore 12 can be the common denominator. Then the numerators must be multiplied by the same factor as the demominator was.

This gives us our new fractions:

$\frac{2}{3}\bigg(\frac{4}{4}\bigg) + \frac{3}{4}\bigg(\frac{3}{3}\bigg)+ \frac{7}{12}$

Next, we add the numerators, but our denominators stay the same:

$\frac{8+9+7}{12}=\frac{24}{12}$ .

Lastly, we simplify to get as our answer.

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Question

Simplify the following fraction:

$\frac{12}{39}$

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Answer

Determine the factors of both the numerator and the denominator:

$12 = 2 \times 2 \times 3$

$39 = 3 \times 13$

We notice that 3 is a factor of both 12 and 39 so we can simplify by dividing both 12 and 39 by 3.

$\frac{12}{3}=4$

$\frac{39}{3}=13$

The result is therefore,

$\frac{4}{13}$

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Question

Find the simplified result:

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

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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

Find the result. (It does not need to be simplified).

$\frac{9}{10}-\frac{4}{5}$

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Answer

We start by adjusting both equations to the same denominator.

Notice that the denominator of the second term is a factor of the denominator of the first term. Therefore we only need to adjust the second term by multiplying both numerator and denominator by 2:

$\frac{9}{10}-\frac{4 \cdot 2}{5 \cdot 2}= \frac{9}{10}-\frac{8}{10}$

Now we need to subtract the numerators:

$=\frac{1}{10}$

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Question

What is the sum of $\frac{2}{5}$ and ${}1\frac{3}{5}$ ?

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Answer

Convert the mixed number to an improper fraction.

Then add the numerators together and keep the common denominator.

Finally, simplify.

$\frac{2}{5}+1\frac{3}{5}=\frac{2}{5}+\frac{8}{5}=\frac{10}{5}=2$

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Question

Add:

$\frac{1}{x}+ \frac{4}{x-1}$

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Answer

To add rational expressions, you must find the common denominator. In this case, it's .

Next, you must change the numerators to offset the new denominator.

$\frac{1}{x}$ becomes $\frac{x-1}{x(x-1)}$ and $\frac{4}{x-1}$ becomes $\frac{4(x)}{(x-1)x}$ .

Now you can combine the numerators: .

Put that over the denomiator and see if you can simplify/factor further. In this case, you can't.

Therefore, your final answer is:

$\frac{5x-1}{x^2-1}$ .

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Question

Subtract:

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

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Answer

To subtract rational expressions, you must first find the common denominator, which in this case is . That means we only have to adjust the first fraction since the second fraction has that denominator already.

Therefore, the first fraction now looks like:

$\frac{(x-1)(x)}{x(x)}$ .

Now that the denominators are the same, combine numerators:

.

Now, put that over the denominator and see if you can simplify any further.

In this case, you can't, so your final answer is:
$\frac{x^2-x-4}{x^2}$ .

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Question

Simplify:

$\frac{1}{4a}+\frac{b}{2ac}$

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Answer

To add or subtract fractions, the fractions must have the same denominator.

The smallest common denominator in this case is .

After multiply both terms so that they have the same denominator.

$\frac{1(c)}{4a(c)}+\frac{b(4)}{2ac(4)}=\frac{c}{4ac}+\frac{4b}{8ac}$

$\frac{c}{4ac}+\frac{4b}{8ac}=\frac{c}{4ac}+\frac{2b}{4ac}$

Simply add these two terms together.

$\frac{c}{4ac}+\frac{2b}{4ac}=\frac{2b+c}{4ac}$

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Question

Add these fractions:

$\frac{2}{3}+\frac{4}{5}$

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Answer

To solve this we need to first find common denominators. We do that by multiplying the first fraction by 5 on the top and bottom and the second fraction by 3 on the top and bottom.

$\frac{2}{3}\frac{5}{5}=\frac{10}{15}$

$\frac{4}{5}\frac{3}{3}=\frac{12}{15}$

Add these fractions together, then write it as a mixed number

$\frac{10}{15}+\frac{12}{15}=\frac{22}{15}=1\frac{7}{15}$

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Question

Add these fractions:

$\frac{1}{2}+\frac{1}{6}$

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Answer

To solve this we need to first find common denominators. We do that by multiplying the first fraction by 3 over 3.

$\frac{1}{2}*\frac{3}{3}=\frac{3}{6}$

Add these fractions together, then reduce it to get the final answer.

$\frac{3}{6}+\frac{1}{6}=\frac{4}{6}=\frac{2\cdot2}{2\cdot 3}=\frac{2}{3}$

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Question

Add these fractions:

$\frac{3}{5}+\frac{2}{7}$

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Answer

To solve this we need to first find common denominators. We do that by multiplying the first fraction by 7 over 7 and the second fraction by 5 over 5.

$\frac{3}{5}\frac{7}{7}=\frac{21}{35}$

$\frac{2}{7}\frac{5}{5}=\frac{10}{35}$

Add these fractions together, to get the final answer.

$\frac{21}{35}+\frac{10}{35}=\frac{31}{35}$

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Question

Subtract these fractions:

$\frac{2}{3}-\frac{1}{2}$

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Answer

To solve this we need to first find common denominators. We do that by multiplying the first fraction by 2 over 2 and the second fraction by 3 over 3.

$\frac{2}{3}\frac{2}{2}=\frac{4}{6}$

$\frac{1}{2}\frac{3}{3}=\frac{3}{6}$

Subtract these fractions to get the final answer.

$\frac{4}{6}-\frac{3}{6}=\frac{1}{6}$

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Question

Subtract these fractions:

$\frac{4}{5}-\frac{1}{3}$

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Answer

To solve this we need to first find common denominators. We do that by multiplying the first fraction by 3 over 3 and the second fraction by 5 over 5.

$\frac{4}{5}\frac{3}{3}=\frac{12}{15}$

$\frac{1}{3}\frac{5}{5}=\frac{5}{15}$

Subtract the numerators of the fractions to get the final answer.

$\frac{12}{15}-\frac{5}{15}=\frac{7}{15}$

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Question

Add these fractions:

$\frac{1}{7}+\frac{5}{6}$

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Answer

To solve this we need to first find common denominators. We do that by multiplying the first fraction by 6 over 6 and the second fraction by 7 over 7.

$\frac{1}{7}\frac{6}{6}=\frac{6}{42}$

$\frac{5}{6}\frac{7}{7}=\frac{35}{42}$

Add the numerators of the fractions to get the final answer.

$\frac{6}{42}+\frac{35}{42}=\frac{41}{42}$

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Question

Add these fractions:

$\frac{1}{12}+\frac{1}{2}$

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Answer

To solve this we need to first find common denominators. We do that by multiplying the second fraction by 6 over 6.

$\frac{1}{2}*\frac{6}{6}=\frac{6}{12}$

Add the numerators of the fractions to get the final answer.

$\frac{1}{12}+\frac{6}{12}=\frac{7}{12}$

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Question

Add these fractions:

$\frac{3}{7}+\frac{2}{5}$

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Answer

To solve this we need to first find common denominators. We do that by multiplying the first fraction by 5 over 5 and the second fraction by 7 over 7.

$\frac{3}{7}\frac{5}{5}=\frac{15}{35}$

$\frac{2}{5}\frac{7}{7}=\frac{14}{35}$

Add the numerators of the fractions to get the final answer.

$\frac{15}{35}+\frac{14}{35}=\frac{29}{35}$

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