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

how did you get 1640

Noor Reply

If auger is pair are the roots of equation x2+5x-3=0

Peter Reply

Wayne and Dennis like to ride the bike path from Riverside Park to the beach. Dennis’s speed is seven miles per hour faster than Wayne’s speed, so it takes Wayne 2 hours to ride to the beach while it takes Dennis 1.5 hours for the ride. Find the speed of both bikers.

MATTHEW Reply

420

Sharon

from theory: distance [miles] = speed [mph] × time [hours] info #1 speed_Dennis × 1.5 = speed_Wayne × 2 => speed_Wayne = 0.75 × speed_Dennis (i) info #2 speed_Dennis = speed_Wayne + 7 [mph] (ii) use (i) in (ii) => [...] speed_Dennis = 28 mph speed_Wayne = 21 mph

George

Let W be Wayne's speed in miles per hour and D be Dennis's speed in miles per hour. We know that W + 7 = D and W * 2 = D * 1.5. Substituting the first equation into the second: W * 2 = (W + 7) * 1.5 W * 2 = W * 1.5 + 7 * 1.5 0.5 * W = 7 * 1.5 W = 7 * 3 or 21 W is 21 D = W + 7 D = 21 + 7 D = 28

Salma

Devon is 32 32 years older than his son, Milan. The sum of both their ages is 54 54. Using the variables d d and m m to represent the ages of Devon and Milan, respectively, write a system of equations to describe this situation. Enter the equations below, separated by a comma.

Aaron Reply

find product (-6m+6) ( 3m²+4m-3)

SIMRAN Reply

-42m²+60m-18

Salma

what is the solution

bill

how did you arrive at this answer?

bill

-24m+3+3mÁ^2

Susan

i really want to learn

Amira

I only got 42 the rest i don't know how to solve it. Please i need help from anyone to help me improve my solving mathematics please

Amira

Hw did u arrive to this answer.

Aphelele

Bajemah

-6m(3mA²+4m-3)+6(3mA²+4m-3) =-18m²A²-24m²+18m+18mA²+24m-18 Rearrange like items -18m²A²-24m²+42m+18A²-18

Salma

complete the table of valuesfor each given equatio then graph. 1.x+2y=3

Jovelyn Reply

x=3-2y

Salma

y=x+3/2

Salma

Enock

given that (7x-5):(2+4x)=8:7find the value of x

Nandala

3x-12y=18

Kelvin

please why isn't that the 0is in ten thousand place

Grace Reply

please why is it that the 0is in the place of ten thousand

Grace

Send the example to me here and let me see

Stephen

A meditation garden is in the shape of a right triangle, with one leg 7 feet. The length of the hypotenuse is one more than the length of one of the other legs. Find the lengths of the hypotenuse and the other leg

Marry Reply

how far

Abubakar

cool u

Enock

state in which quadrant or on which axis each of the following angles given measure. in standard position would lie 89°

Abegail Reply

hello

BenJay

Method

I am eliacin, I need your help in maths

Rood

how can I help

Sir

hmm can we speak here?

Amoon

however, may I ask you some questions about Algarba?

Amoon

Enock

what the last part of the problem mean?

Roger

The Jones family took a 15 mile canoe ride down the Indian River in three hours. After lunch, the return trip back up the river took five hours. Find the rate, in mph, of the canoe in still water and the rate of the current.

cameron Reply

Shakir works at a computer store. His weekly pay will be either a fixed amount, $925, or $500 plus 12% of his total sales. How much should his total sales be for his variable pay option to exceed the fixed amount of $925.

mahnoor Reply

I'm guessing, but it's somewhere around $4335.00 I think

Lewis

12% of sales will need to exceed 925 - 500, or 425 to exceed fixed amount option. What amount of sales does that equal? 425 ÷ (12÷100) = 3541.67. So the answer is sales greater than 3541.67. Check: Sales = 3542 Commission 12%=425.04 Pay = 500 + 425.04 = 925.04. 925.04 > 925.00

Munster

difference between rational and irrational numbers

Arundhati Reply

When traveling to Great Britain, Bethany exchanged $602 US dollars into £515 British pounds. How many pounds did she receive for each US dollar?

Jakoiya Reply

how to reduced echelon form

Solomon Reply

Jazmine trained for 3 hours on Saturday. She ran 8 miles and then biked 24 miles. Her biking speed is 4 mph faster than her running speed. What is her running speed?

Zack Reply

d=r×t the equation would be 8/r+24/r+4=3 worked out

Sheirtina

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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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