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ACT Math : How to find a complex fraction

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Example Questions

Example Question #1 : How To Find A Complex Fraction

Simplify $\frac{x + \frac{1}{x}}{x}$ $\displaystyle \frac{x + \frac{1}{x}}{x}$

Possible Answers:

$\frac{x + 1}{x^{2}}$ $\displaystyle \frac{x + 1}{x^{2}}$

$\frac{x^{2} + 1}{x^{2}}$ $\displaystyle \frac{x^{2} + 1}{x^{2}}$

$\frac{x^{2} + 2x + 1}{x}$ $\displaystyle \frac{x^{2} + 2x + 1}{x}$

$\frac{x^{2} + 1}{x}$ $\displaystyle \frac{x^{2} + 1}{x}$

$\frac{x + 1}{x}$ $\displaystyle \frac{x + 1}{x}$

Correct answer:

$\frac{x^{2} + 1}{x^{2}}$ $\displaystyle \frac{x^{2} + 1}{x^{2}}$

Explanation:

Simplify the complex fraction by multiplying by the complex denominator:

$\frac{x + \frac{1}{x}}{x}\cdot \frac{x}{x}= \frac{x^{2} + 1}{x^{2}}$ $\displaystyle \frac{x + \frac{1}{x}}{x}\cdot \frac{x}{x}= \frac{x^{2} + 1}{x^{2}}$

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Example Question #2 : How To Add And Subtract Fractions

Steven purchased $1\frac{2}{3}\:lbs$ $\displaystyle 1\frac{2}{3}\:lbs$ of vegetables on Monday and $2\frac{3}{4}\:lbs$ $\displaystyle 2\frac{3}{4}\:lbs$ of vegetables on Tuesday. What was the total weight, in pounds, of vegetables purchased by Steven?

Possible Answers:

$4\frac{3}{4}\:lbs$ $\displaystyle 4\frac{3}{4}\:lbs$

$4\frac{7}{12}\:lbs$ $\displaystyle 4\frac{7}{12}\:lbs$

$4\frac{5}{12}\:lbs$ $\displaystyle 4\frac{5}{12}\:lbs$

$4\frac{1}{3}\:lbs$ $\displaystyle 4\frac{1}{3}\:lbs$

$4\frac{1}{2}\:lbs$ $\displaystyle 4\frac{1}{2}\:lbs$

Correct answer:

$4\frac{5}{12}\:lbs$ $\displaystyle 4\frac{5}{12}\:lbs$

Explanation:

To solve this answer, we have to first make the mixed numbers improper fractions so that we can then find a common denominator. To make a mixed number into an improper fraction, you multiply the denominator by the whole number and add the result to the numerator. So, for the presented data:

$1\frac{2}{3}=\frac{(3*1)+2}{3}=\frac{3+2}{3}=\frac{5}{3}$ $\displaystyle 1\frac{2}{3}=\frac{(3*1)+2}{3}=\frac{3+2}{3}=\frac{5}{3}$

and

$2\frac{3}{4}=\frac{(4*2)+3}{4}=\frac{8+3}{4}=\frac{11}{4}$ $\displaystyle 2\frac{3}{4}=\frac{(4*2)+3}{4}=\frac{8+3}{4}=\frac{11}{4}$

Now, to find out how many total pounds of vegetables Steven purchased, we need to add these two improper fractions together:

$\frac{5}{3}+\frac{11}{4}$ $\displaystyle \frac{5}{3}+\frac{11}{4}$

To add these fractions, they need to have a common denominator. We can adjust each fraction to have a common denominator of $\displaystyle 12$ by multiplying $\frac{5}{3}$ $\displaystyle \frac{5}{3}$ by $\frac{4}{4}$ $\displaystyle \frac{4}{4}$ and $\frac{11}{4}$ $\displaystyle \frac{11}{4}$by $\frac{3}{3}$ $\displaystyle \frac{3}{3}$:

$(\frac{5}{3}*\frac{4}{4})+(\frac{11}{4}*\frac{3}{3})$ $\displaystyle (\frac{5}{3}*\frac{4}{4})+(\frac{11}{4}*\frac{3}{3})$

To multiply fractions, just multiply across:

$\frac{20}{12}+\frac{33}{12}$ $\displaystyle \frac{20}{12}+\frac{33}{12}$

We can now add the numerators together; the denominator will stay the same:

$\frac{53}{12}$ $\displaystyle \frac{53}{12}$

Since all of the answer choices are mixed numbers, we now need to change our improper fraction answer into a mixed number answer. We can do this by dividing the numerator by the denominator and leaving the remainder as the numerator:

$53\div12=4\: r.\:5$ $\displaystyle 53\div12=4\: r.\:5$

$\frac{53}{12}=4\frac{5}{12}$ $\displaystyle \frac{53}{12}=4\frac{5}{12}$

This means that our final answer is $4\frac{5}{12}\:lbs$ $\displaystyle 4\frac{5}{12}\:lbs$.

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ACT Math : How to find a complex fraction

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

Example Question #1 : How To Find A Complex Fraction

Example Question #2 : How To Add And Subtract Fractions

All ACT Math Resources

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