# 1.5 Visualize fractions  (Page 4/12)

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Find the quotient: $-\phantom{\rule{0.2em}{0ex}}\frac{7}{27}÷\left(-\phantom{\rule{0.2em}{0ex}}\frac{35}{36}\right).$

$\frac{4}{15}$

Find the quotient: $-\phantom{\rule{0.2em}{0ex}}\frac{5}{14}÷\left(-\phantom{\rule{0.2em}{0ex}}\frac{15}{28}\right).$

$\frac{2}{3}$

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: 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:

$\frac{\frac{6}{7}}{3}\phantom{\rule{1em}{0ex}}\frac{\frac{3}{4}}{\frac{5}{8}}\phantom{\rule{1em}{0ex}}\frac{\frac{x}{2}}{\frac{5}{6}}$

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 $\frac{3}{4}÷\frac{5}{8}.$

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

## Solution

 $\frac{\frac{3}{4}}{\frac{5}{8}}$ Rewrite as division. $\frac{3}{4}÷\frac{5}{8}$ 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. $\frac{6}{5}$

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

$\frac{4}{5}$

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

$\frac{11}{14}$

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

## Solution

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

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

$\frac{3}{4b}$

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

$\frac{4}{2q}$

## 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.

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

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

## Solution

$\begin{array}{cccccc}& & & & & \hfill \frac{4-2\left(3\right)}{{2}^{2}+2}\hfill \\ \begin{array}{c}\text{Use the order of operations to simplify the}\hfill \\ \text{numerator and the denominator.}\hfill \end{array}\hfill & & & & & \hfill \frac{4-6}{4+2}\hfill \\ \text{Simplify the numerator and the denominator.}\hfill & & & & & \hfill \frac{-2}{6}\hfill \\ \begin{array}{c}\text{Simplify. A negative divided by a positive is}\hfill \\ \text{negative.}\hfill \end{array}\hfill & & & & & \hfill -\phantom{\rule{0.2em}{0ex}}\frac{1}{3}\hfill \end{array}$

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

$-\phantom{\rule{0.2em}{0ex}}\frac{3}{4}$

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

$-\phantom{\rule{0.2em}{0ex}}\frac{2}{3}$

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.

$\begin{array}{cccccc}\frac{-1}{3}=-\phantom{\rule{0.2em}{0ex}}\frac{1}{3}\hfill & & & & & \frac{\text{negative}}{\text{positive}}=\text{negative}\hfill \\ \frac{1}{-3}=-\phantom{\rule{0.2em}{0ex}}\frac{1}{3}\hfill & & & & & \frac{\text{positive}}{\text{negative}}=\text{negative}\hfill \end{array}$

## Placement of negative sign in a fraction

For any positive numbers a and b ,

$\frac{\text{−}a}{b}=\frac{a}{\text{−}b}=-\phantom{\rule{0.2em}{0ex}}\frac{a}{b}$

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

## Solution

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

$\begin{array}{cccccc}& & & & & \hfill \phantom{\rule{5em}{0ex}}\frac{4\left(-3\right)+6\left(-2\right)}{-3\left(2\right)-2}\hfill \\ \text{Multiply.}\hfill & & & & & \hfill \phantom{\rule{5em}{0ex}}\frac{-12+\left(-12\right)}{-6-2}\hfill \\ \text{Simplify.}\hfill & & & & & \hfill \phantom{\rule{5em}{0ex}}\frac{-24}{-8}\hfill \\ \text{Divide.}\hfill & & & & & \hfill \phantom{\rule{5em}{0ex}}3\hfill \end{array}$

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

4

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

2

## 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

Larry and Tom were standing next to each other in the backyard when Tom challenged Larry to guess how tall he was. Larry knew his own height is 6.5 feet and when they measured their shadows, Larry’s shadow was 8 feet and Tom’s was 7.75 feet long. What is Tom’s height?
6.25
Ciid
Wayne is hanging a string of lights 57 feet long around the three sides of his patio, which is adjacent to his house. the length of his patio, the side along the house, is 5 feet longer than twice it's width. Find the length and width of the patio.
Ciid
tyler
(sin=opp/adj) (tan= opp/adj) cos=hyp/adj dont quote me on it look it up
tyler
(sin=opp/adj) (tan= opp/adj) cos=hyp/adj dont quote me on it look it up
tyler
(sin=opp/adj) (tan= opp/adj) cos=hyp/adj dont quote me on it look it up
tyler
SOH = Sine is Opposite over Hypotenuse. CAH= Cosine is Adjacent over Hypotenuse. TOA = Tangent is Opposite over Adjacent.
tyler
H=57 and O=285 figure out what the adjacent?
tyler
Amara currently sells televisions for company A at a salary of $17,000 plus a$100 commission for each television she sells. Company B offers her a position with a salary of $29,000 plus a$20 commission for each television she sells. How many televisions would Amara need to sell for the options to be equal?
what is the quantity and price of the televisions for both options?
karl
Amara has to sell 120 televisions to make 29,000 of the salary of company B. 120 * TV 100= commission 12,000+ 17,000 = of company a salary 29,000 17000+
Ciid
Amara has to sell 120 televisions to make 29,000 of the salary of company B. 120 * TV 100= commission 12,000+ 17,000 = of company a salary 29,000
Ciid
I'm mathematics teacher from highly recognized university.
here a question professor How many soldiers are there in a group of 27 sailors and soldiers if there are four fifths as many sailors as soldiers? can you write out the college you went to with the name of the school you teach at and let me know the answer I've got it to be honest with you
tyler
is anyone else having issues with the links not doing anything?
Yes
Val
chapter 1 foundations 1.2 exercises variables and algebraic symbols
June needs 45 gallons of punch for a party and has 2 different coolers to carry it in. The bigger cooler is 5 times as large as the smaller cooler. How many gallons can each cooler hold? Enter the answers in decimal form.
Joseph would like to make 12 pounds of a coffee blend at a cost of $6.25 per pound. He blends Ground Chicory at$4.40 a pound with Jamaican Blue Mountain at $8.84 per pound. How much of each type of coffee should he use? Samer 4x6.25=$25 coffee blend 4×4.40= $17.60 ground chicory 4x8.84= 35.36 blue mountain. In total they will spend for 12 pounds$77.96 they will spend in total
tyler
DaMarcus and Fabian live 23 miles apart and play soccer at a park between their homes. DaMarcus rode his bike for three-quarters of an hour and Fabian rode his bike for half an hour to get to the park. Fabian’s speed was six miles per hour faster than DaMarcus’ speed. Find the speed of both soccer players.
i need help how to do this is confusing
what kind of math is it?
Danteii
help me to understand
huh, what is the algebra problem
Daniel
How many soldiers are there in a group of 27 sailors and soldiers if there are four fifths many sailors as soldiers?
tyler
What is the domain and range of heaviside
What is the domain and range of Heaviside and signum
Christopher
25-35
Fazal
The hypotenuse of a right triangle is 10cm long. One of the triangle’s legs is three times the length of the other leg. Find the lengths of the three sides of the triangle.
Tickets for a show are $70 for adults and$50 for children. For one evening performance, a total of 300 tickets were sold and the receipts totaled \$17,200. How many adult tickets and how many child tickets were sold?
A 50% antifreeze solution is to be mixed with a 90% antifreeze solution to get 200 liters of a 80% solution. How many liters of the 50% solution and how many liters of the 90% solution will be used?
June needs 45 gallons of punch for a party and has 2 different coolers to carry it in. The bigger cooler is 5 times as large as the smaller cooler. How many gallons can each cooler hold?   By     By David Martin  