1.5 Visualize fractions (Page 4/12)

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Find the quotient: $- \frac{7}{27} \div (- \frac{35}{36}) .$

$\frac{4}{15}$

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Find the quotient: $- \frac{5}{14} \div (- \frac{15}{28}) .$

$\frac{2}{3}$

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There are several ways to remember which steps to take to multiply or divide fractions. One way is to repeat the call outs to yourself. If you do this each time you do an exercise, you will have the steps memorized.

“To multiply fractions, multiply the numerators and multiply the denominators.”
“To divide fractions, multiply the first fraction by the reciprocal of the second.”

Another way is to keep two examples in mind:

$This is an image with two columns. The first column reads “One fourth of two pizzas is one half of a pizza. Below this are two pizzas side-by-side with a line down the center of each one representing one half. The halves are labeled “one half”. Under this is the equation “2 times 1 fourth”. Under this is another equation “two over 1 times 1 fourth.” Under this is the fraction two fourths and under this is the fraction one half. The next column reads “there are eight quarters in two dollars.” Under this are eight quarters in two rows of four. Under this is the fraction equation 2 divided by one fourth. Under this is the equation “two over one divided by one fourth.” Under this is two over one times four over one. Under this is the answer “8”.$

The numerators or denominators of some fractions contain fractions themselves. A fraction in which the numerator or the denominator is a fraction is called a complex fraction .

Complex fraction

A complex fraction is a fraction in which the numerator or the denominator contains a fraction.

Some examples of complex fractions are:

\frac{\frac{6}{7}}{3} \frac{\frac{3}{4}}{\frac{5}{8}} \frac{\frac{x}{2}}{\frac{5}{6}}

To simplify a complex fraction, we remember that the fraction bar means division . For example, the complex fraction $\frac{\frac{3}{4}}{\frac{5}{8}}$ means $\frac{3}{4} \div \frac{5}{8} .$

Simplify: $\frac{\frac{3}{4}}{\frac{5}{8}} .$

Solution

	$\frac{\frac{3}{4}}{\frac{5}{8}}$
Rewrite as division.	$\frac{3}{4} \div \frac{5}{8}$
Multiply the first fraction by the reciprocal of the second.	$\frac{3}{4} \cdot \frac{8}{5}$
Multiply.	$\frac{3 \cdot 8}{4 \cdot 5}$
Look for common factors.
Divide out common factors and simplify.	$\frac{6}{5}$

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Simplify: $\frac{\frac{2}{3}}{\frac{5}{6}} .$

$\frac{4}{5}$

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Simplify: $\frac{\frac{3}{7}}{\frac{6}{11}} .$

$\frac{11}{14}$

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Simplify: $\frac{\frac{x}{2}}{\frac{x y}{6}} .$

Solution

	$\frac{\frac{x}{2}}{\frac{x y}{6}}$
Rewrite as division.	$\frac{x}{2} \div \frac{x y}{6}$
Multiply the first fraction by the reciprocal of the second.	$\frac{x}{2} \cdot \frac{6}{x y}$
Multiply.	$\frac{x \cdot 6}{2 \cdot x y}$
Look for common factors.
Divide out common factors and simplify.	$\frac{3}{y}$

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Simplify: $\frac{\frac{a}{8}}{\frac{a b}{6}} .$

$\frac{3}{4 b}$

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Simplify: $\frac{\frac{p}{2}}{\frac{p q}{8}} .$

$\frac{4}{2 q}$

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Simplify expressions with a fraction bar

The line that separates the numerator from the denominator in a fraction is called a fraction bar. A fraction bar acts as grouping symbol. The order of operations then tells us to simplify the numerator and then the denominator. Then we divide.

To simplify the expression $\frac{5 - 3}{7 + 1},$ we first simplify the numerator and the denominator separately. Then we divide.

\frac{5 - 3}{7 + 1}

\frac{2}{8}

\frac{1}{4}

Simplify an expression with a fraction bar.

Simplify the expression in the numerator. Simplify the expression in the denominator.
Simplify the fraction.

Simplify: $\frac{4 - 2 (3)}{2^{2} + 2} .$

Solution

$\begin{matrix} \frac{4 - 2 (3)}{2^{2} + 2} \\ \begin{matrix} Use the order of operations to simplify the \\ numerator and the denominator. \end{matrix} & \frac{4 - 6}{4 + 2} \\ Simplify the numerator and the denominator. & \frac{−2}{6} \\ \begin{matrix} Simplify. A negative divided by a positive is \\ negative. \end{matrix} & - \frac{1}{3} \end{matrix}$

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Simplify: $\frac{6 - 3 (5)}{3^{2} + 3} .$

$- \frac{3}{4}$

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Simplify: $\frac{4 - 4 (6)}{3^{2} + 3} .$

$- \frac{2}{3}$

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Where does the negative sign go in a fraction? Usually the negative sign is in front of the fraction, but you will sometimes see a fraction with a negative numerator, or sometimes with a negative denominator. Remember that fractions represent division. When the numerator and denominator have different signs, the quotient is negative.

\begin{matrix} \frac{−1}{3} = - \frac{1}{3} & \frac{negative}{positive} = negative \\ \frac{1}{−3} = - \frac{1}{3} & \frac{positive}{negative} = negative \end{matrix}

For any positive numbers a and b ,

\frac{− a}{b} = \frac{a}{− b} = - \frac{a}{b}

Simplify: $\frac{4 (−3) + 6 (−2)}{−3 (2) - 2} .$

Solution

The fraction bar acts like a grouping symbol. So completely simplify the numerator and the denominator separately.

$\begin{matrix} \frac{4 (−3) + 6 (−2)}{−3 (2) - 2} \\ Multiply. & \frac{−12 + (−12)}{−6 - 2} \\ Simplify. & \frac{−24}{−8} \\ Divide. & 3 \end{matrix}$

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Simplify: $\frac{8 (−2) + 4 (−3)}{−5 (2) + 3} .$

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Simplify: $\frac{7 (−1) + 9 (−3)}{−5 (3) - 2} .$

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Translate phrases to expressions with fractions

Now that we have done some work with fractions, we are ready to translate phrases that would result in expressions with fractions.

Questions & Answers

I'm interested in biological psychology and cognitive psychology

Tanya Reply

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sammie Reply

physiological Psychology

Nwosu Reply

How can I develope my cognitive domain

Amanyire Reply

why is communication effective

Dakolo Reply

Communication is effective because it allows individuals to share ideas, thoughts, and information with others.

effective communication can lead to improved outcomes in various settings, including personal relationships, business environments, and educational settings. By communicating effectively, individuals can negotiate effectively, solve problems collaboratively, and work towards common goals.

it starts up serve and return practice/assessments.it helps find voice talking therapy also assessments through relaxed conversation.

miss

Every time someone flushes a toilet in the apartment building, the person begins to jumb back automatically after hearing the flush, before the water temperature changes. Identify the types of learning, if it is classical conditioning identify the NS, UCS, CS and CR. If it is operant conditioning, identify the type of consequence positive reinforcement, negative reinforcement or punishment

Wekolamo Reply

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Wekolamo

because it helps many people around the world to understand how to interact with other people and understand them well, for example at work (job).

Manix Reply

Agreed 👍 There are many parts of our brains and behaviors, we really need to get to know. Blessings for everyone and happy Sunday!

ARC

A child is a member of community not society elucidate ?

JESSY Reply

Isn't practices worldwide, be it psychology, be it science. isn't much just a false belief of control over something the mind cannot truly comprehend?

Simon Reply

compare and contrast skinner's perspective on personality development on freud

namakula Reply

Skinner skipped the whole unconscious phenomenon and rather emphasized on classical conditioning

war

explain how nature and nurture affect the development and later the productivity of an individual.

Amesalu Reply

nature is an hereditary factor while nurture is an environmental factor which constitute an individual personality. so if an individual's parent has a deviant behavior and was also brought up in an deviant environment, observation of the behavior and the inborn trait we make the individual deviant.

Samuel

I am taking this course because I am hoping that I could somehow learn more about my chosen field of interest and due to the fact that being a PsyD really ignites my passion as an individual the more I hope to learn about developing and literally explore the complexity of my critical thinking skills

Zyryn Reply

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Jonathan

and having a good philosophy of the world is like a sandwich and a peanut butter 👍

Jonathan

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Kelu Reply

interpersonal relationships

Abdulfatai Reply

What would be the best educational aid(s) for gifted kids/savants?

Heidi Reply

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Saurabh

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Source: OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2

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1.5 Visualize fractions (Page 4/12)

Complex fraction

Solution

Solution

Simplify expressions with a fraction bar

Simplify an expression with a fraction bar.

Solution

Placement of negative sign in a fraction

Solution

Translate phrases to expressions with fractions

Questions & Answers

Read also:

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