# 1.5 Visualize fractions  (Page 3/12)

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Doing the Manipulative Mathematics activity “Model Fraction Multiplication” will help you develop a better understanding of multiplying fractions.

We’ll use a model to show you how to multiply two fractions and to help you remember the procedure. Let’s start with $\frac{3}{4}.$

Now we’ll take $\frac{1}{2}$ of $\frac{3}{4}.$

Notice that now, the whole is divided into 8 equal parts. So $\frac{1}{2}·\frac{3}{4}=\frac{3}{8}.$

To multiply fractions, we multiply the numerators and multiply the denominators.

## Fraction multiplication

If $a,b,c\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}d$ are numbers where $b\ne 0\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}d\ne 0,$ then

$\frac{a}{b}·\frac{c}{d}=\frac{ac}{bd}$

To multiply fractions, multiply the numerators and multiply the denominators.

When multiplying fractions , the properties of positive and negative numbers still apply, of course. It is a good idea to determine the sign of the product as the first step. In [link] , we will multiply negative and a positive, so the product will be negative.

Multiply: $-\phantom{\rule{0.2em}{0ex}}\frac{11}{12}·\frac{5}{7}.$

## Solution

The first step is to find the sign of the product. Since the signs are the different, the product is negative.

$\begin{array}{cccccc}& & & & & -\phantom{\rule{0.2em}{0ex}}\frac{11}{12}·\frac{5}{7}\hfill \\ \text{Determine the sign of the product; multiply.}\hfill & & & & & -\phantom{\rule{0.2em}{0ex}}\frac{11·5}{12·7}\hfill \\ \begin{array}{c}\text{Are there any common factors in the numerator}\hfill \\ \text{and the denominator? No}\hfill \end{array}\hfill & & & & & -\phantom{\rule{0.2em}{0ex}}\frac{55}{84}\hfill \end{array}$

Multiply: $-\phantom{\rule{0.2em}{0ex}}\frac{10}{28}·\frac{8}{15}.$

$-\phantom{\rule{0.2em}{0ex}}\frac{4}{21}$

Multiply: $-\phantom{\rule{0.2em}{0ex}}\frac{9}{20}·\frac{5}{12}.$

$-\phantom{\rule{0.2em}{0ex}}\frac{3}{16}$

When multiplying a fraction by an integer, it may be helpful to write the integer as a fraction. Any integer, a , can be written as $\frac{a}{1}.$ So, for example, $3=\frac{3}{1}.$

Multiply: $-\phantom{\rule{0.2em}{0ex}}\frac{12}{5}\left(-20x\right).$

## Solution

Determine the sign of the product. The signs are the same, so the product is positive.

 $-\phantom{\rule{0.2em}{0ex}}\frac{12}{5}\left(-20x\right)$ Write $20x$ as a fraction. $\frac{12}{5}\left(\frac{20x}{1}\right)$ Multiply. Rewrite 20 to show the common factor 5 and divide it out. Simplify. $48x$

Multiply: $\frac{11}{3}\left(-9a\right).$

$-33a$

Multiply: $\frac{13}{7}\left(-14b\right).$

$-36b$

## Divide fractions

Now that we know how to multiply fractions, we are almost ready to divide. Before we can do that, that we need some vocabulary.

The reciprocal    of a fraction is found by inverting the fraction, placing the numerator in the denominator and the denominator in the numerator. The reciprocal of $\frac{2}{3}$ is $\frac{3}{2}.$

Notice that $\frac{2}{3}·\frac{3}{2}=1.$ A number and its reciprocal multiply to 1.

To get a product of positive 1 when multiplying two numbers, the numbers must have the same sign. So reciprocals must have the same sign.

The reciprocal of $-\phantom{\rule{0.2em}{0ex}}\frac{10}{7}$ is $-\phantom{\rule{0.2em}{0ex}}\frac{7}{10},$ since $-\phantom{\rule{0.2em}{0ex}}\frac{10}{7}\left(-\phantom{\rule{0.2em}{0ex}}\frac{7}{10}\right)=1.$

## Reciprocal

The reciprocal of $\frac{a}{b}$ is $\frac{b}{a}.$

A number and its reciprocal multiply to one $\frac{a}{b}·\frac{b}{a}=1.$

Doing the Manipulative Mathematics activity “Model Fraction Division” will help you develop a better understanding of dividing fractions.

To divide fractions, we multiply the first fraction by the reciprocal of the second.

## Fraction division

If $a,b,c\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}d$ are numbers where $b\ne 0,c\ne 0\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}d\ne 0,$ then

$\frac{a}{b}÷\frac{c}{d}=\frac{a}{b}·\frac{d}{c}$

To divide fractions, we multiply the first fraction by the reciprocal of the second.

We need to say $b\ne 0,c\ne 0\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}d\ne 0$ to be sure we don’t divide by zero!

Divide: $-\phantom{\rule{0.2em}{0ex}}\frac{2}{3}÷\frac{n}{5}.$

## Solution

$\begin{array}{cccccc}& & & & & -\phantom{\rule{0.2em}{0ex}}\frac{2}{3}÷\frac{n}{5}\hfill \\ \begin{array}{c}\text{To divide, multiply the first fraction by the}\hfill \\ \text{reciprocal of the second.}\hfill \end{array}\hfill & & & & & -\phantom{\rule{0.2em}{0ex}}\frac{2}{3}·\frac{5}{n}\hfill \\ \text{Multiply.}\hfill & & & & & -\phantom{\rule{0.2em}{0ex}}\frac{10}{3n}\hfill \end{array}$

Divide: $-\phantom{\rule{0.2em}{0ex}}\frac{3}{5}÷\frac{p}{7}.$

$-\phantom{\rule{0.2em}{0ex}}\frac{21}{5p}$

Divide: $-\phantom{\rule{0.2em}{0ex}}\frac{5}{8}÷\frac{q}{3}.$

$-\phantom{\rule{0.2em}{0ex}}\frac{15}{8q}$

Find the quotient: $-\phantom{\rule{0.2em}{0ex}}\frac{7}{8}÷\left(-\phantom{\rule{0.2em}{0ex}}\frac{14}{27}\right).$

## Solution

 $-\phantom{\rule{0.2em}{0ex}}\frac{7}{18}÷\left(-\phantom{\rule{0.2em}{0ex}}\frac{14}{27}\right)$ To divide, multiply the first fraction by the reciprocal of the second. $-\phantom{\rule{0.2em}{0ex}}\frac{7}{18}\cdot -\phantom{\rule{0.2em}{0ex}}\frac{27}{14}$ Determine the sign of the product, and then multiply.. $\frac{7\cdot 27}{18\cdot 14}$ Rewrite showing common factors. Remove common factors. $\frac{3}{2\cdot 2}$ Simplify. $\frac{3}{4}$

write in this form a/b answer should be in the simplest form 5%
convert to decimal 9/11
August
Equation in the form of a pending point y+2=1/6(×-4)
write in simplest form 3 4/2
August
From Google: The quadratic formula, , is used in algebra to solve quadratic equations (polynomial equations of the second degree). The general form of a quadratic equation is , where x represents a variable, and a, b, and c are constants, with . A quadratic equation has two solutions, called roots.
Melissa
what is the answer of w-2.6=7.55
10.15
Michael
w = 10.15 You add 2.6 to both sides and then solve for w (-2.6 zeros out on the left and leaves you with w= 7.55 + 2.6)
Korin
Nataly is considering two job offers. The first job would pay her $83,000 per year. The second would pay her$66,500 plus 15% of her total sales. What would her total sales need to be for her salary on the second offer be higher than the first?
x > $110,000 bruce greater than$110,000
Michael
Estelle is making 30 pounds of fruit salad from strawberries and blueberries. Strawberries cost $1.80 per pound, and blueberries cost$4.50 per pound. If Estelle wants the fruit salad to cost her $2.52 per pound, how many pounds of each berry should she use? nawal Reply$1.38 worth of strawberries + $1.14 worth of blueberries which=$2.52
Leitha
how
Zaione
is it right😊
Leitha
lol maybe
Robinson
8 pound of blueberries and 22 pounds of strawberries
Melissa
8 pounds x 4.5 = 36 22 pounds x 1.80 = 39.60 36 + 39.60 = 75.60 75.60 / 30 = average 2.52 per pound
Melissa
8 pounds x 4.5 equal 36 22 pounds x 1.80 equal 39.60 36 + 39.60 equal 75.60 75.60 / 30 equal average 2.52 per pound
Melissa
hmmmm...... ?
Robinson
8 pounds x 4.5 = 36 22 pounds x 1.80 = 39.60 36 + 39.60 = 75.60 75.60 / 30 = average 2.52 per pound
Melissa
The question asks how many pounds of each in order for her to have an average cost of $2.52. She needs 30 lb in all so 30 pounds times$2.52 equals $75.60. that's how much money she is spending on the fruit. That means she would need 8 pounds of blueberries and 22 lbs of strawberries to equal 75.60 Melissa good Robinson 👍 Leitha thanks Melissa. Leitha nawal let's do another😊 Leitha we can't use emojis...I see now Leitha Sorry for the multi post. My phone glitches. Melissa Vina has$4.70 in quarters, dimes and nickels in her purse. She has eight more dimes than quarters and six more nickels than quarters. How many of each coin does she have?
10 quarters 16 dimes 12 nickels
Leitha
A private jet can fly 1,210 miles against a 25 mph headwind in the same amount of time it can fly 1,694 miles with a 25 mph tailwind. Find the speed of the jet.
wtf. is a tail wind or headwind?
Robert
48 miles per hour with headwind and 68 miles per hour with tailwind
Leitha
average speed is 58 mph
Leitha
Into the wind (headwind), 125 mph; with wind (tailwind), 175 mph. Use time (t) = distance (d) ÷ rate (r). since t is equal both problems, then 1210/(x-25) = 1694/(×+25). solve for x gives x=150.
bruce
the jet will fly 9.68 hours to cover either distance
bruce
Riley is planning to plant a lawn in his yard. He will need 9 pounds of grass seed. He wants to mix Bermuda seed that costs $4.80 per pound with Fescue seed that costs$3.50 per pound. How much of each seed should he buy so that the overall cost will be $4.02 per pound? Vonna Reply 33.336 Robinson Amber wants to put tiles on the backsplash of her kitchen counters. She will need 36 square feet of tiles. She will use basic tiles that cost$8 per square foot and decorator tiles that cost $20 per square foot. How many square feet of each tile should she use so that the overall cost of the backsplash will be$10 per square foot?
Ivan has $8.75 in nickels and quarters in his desk drawer. The number of nickels is twice the number of quarters. How many coins of each type does he have? mikayla Reply 2q=n ((2q).05) + ((q).25) = 8.75 .1q + .25q = 8.75 .35q = 8.75 q = 25 quarters 2(q) 2 (25) = 50 nickles Answer check 25 x .25 = 6.25 50 x .05 = 2.50 6.25 + 2.50 = 8.75 Melissa John has$175 in $5 and$10 bills in his drawer. The number of $5 bills is three times the number of$10 bills. How many of each are in the drawer?
7-$10 21-$5
Robert
Enrique borrowed $23,500 to buy a car. He pays his uncle 2% interest on the$4,500 he borrowed from him, and he pays the bank 11.5% interest on the rest. What average interest rate does he pay on the total \$23,500? (Round your answer to the nearest tenth of a percent.)
Two sisters like to compete on their bike rides. Tamara can go 4 mph faster than her sister, Samantha. If it takes Samantha 1 hour longer than Tamara to go 80 miles, how fast can Samantha ride her bike?
8mph
michele
16mph
Robert
3.8 mph
Ped
16 goes into 80 5times while 20 goes into 80 4times and is 4mph faster
Robert
what is the answer for this 3×9+28÷4-8
315
lashonna
how do you do xsquard+7x+10=0
What
(x + 2)(x + 5), then set each factor to zero and solve for x. so, x = -2 and x = -5.
bruce
I skipped it
What