# 1.4 Multiply and divide integers  (Page 4/4)

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The temperature in Anchorage, Alaska one morning was 15 degrees. By mid-afternoon the temperature had dropped to 30 degrees below zero. What was the difference in the morning and afternoon temperatures?

The difference in temperatures was 45 degrees.

The temperature in Denver was $-6$ degrees at lunchtime. By sunset the temperature had dropped to $-15$ degrees. What was the difference in the lunchtime and sunset temperatures?

The difference in temperatures was 9 degrees.

## Apply a strategy to solve applications with integers.

1. Read the problem. Make sure all the words and ideas are understood
2. Identify what we are asked to find.
3. Write a phrase that gives the information to find it.
4. Translate the phrase to an expression.
5. Simplify the expression.
6. Answer the question with a complete sentence.

The Mustangs football team received three penalties in the third quarter. Each penalty gave them a loss of fifteen yards. What is the number of yards lost?

## Solution

$\begin{array}{cccc}\begin{array}{c}\mathbf{\text{Step 1.}}\phantom{\rule{0.2em}{0ex}}\text{Read the problem. Make sure all the words and}\hfill \\ \text{ideas are understood.}\hfill \end{array}\hfill & & & \\ \mathbf{\text{Step 2.}}\phantom{\rule{0.2em}{0ex}}\text{Identify what we are asked to find.}\hfill & & & & & \text{the number of yards lost}\hfill \\ \mathbf{\text{Step 3.}}\phantom{\rule{0.2em}{0ex}}\text{Write a phrase that gives the information to find it.}\hfill & & & & & \text{three times a 15-yard penalty}\hfill \\ \mathbf{\text{Step 4.}}\phantom{\rule{0.2em}{0ex}}\text{Translate the phrase to an expression.}\hfill & & & & & 3\left(-15\right)\hfill \\ \mathbf{\text{Step 5.}}\phantom{\rule{0.2em}{0ex}}\text{Simplify the expression.}\hfill & & & & & -45\hfill \\ \mathbf{\text{Step 6.}}\phantom{\rule{0.2em}{0ex}}\text{Answer the question with a complete sentence.}\hfill & & & & & \text{The team lost}\phantom{\rule{0.2em}{0ex}}45\phantom{\rule{0.2em}{0ex}}\text{yards.}\hfill \end{array}$

The Bears played poorly and had seven penalties in the game. Each penalty resulted in a loss of 15 yards. What is the number of yards lost due to penalties?

The Bears lost 105 yards.

Bill uses the ATM on campus because it is convenient. However, each time he uses it he is charged a $2 fee. Last month he used the ATM eight times. How much was his total fee for using the ATM? A$16 fee was deducted from his checking account.

## Key concepts

• Multiplication and Division of Two Signed Numbers
• Same signs—Product is positive
• Different signs—Product is negative
• Strategy for Applications
1. Identify what you are asked to find.
2. Write a phrase that gives the information to find it.
3. Translate the phrase to an expression.
4. Simplify the expression.
5. Answer the question with a complete sentence.

## Practice makes perfect

Multiply Integers

In the following exercises, multiply.

$-4·8$

$-32$

$-3·9$

$9\left(-7\right)$

$-63$

$13\left(-5\right)$

$-1.6$

$-6$

$-1.3$

$-1\left(-14\right)$

14

$-1\left(-19\right)$

Divide Integers

In the following exercises, divide.

$-24÷6$

$-4$

$35÷\left(-7\right)$

$-52÷\left(-4\right)$

13

$-84÷\left(-6\right)$

$-180÷15$

$-12$

$-192÷12$

Simplify Expressions with Integers

In the following exercises, simplify each expression.

$5\left(-6\right)+7\left(-2\right)-3$

$-47$

$8\left(-4\right)+5\left(-4\right)-6$

${\left(-2\right)}^{6}$

64

${\left(-3\right)}^{5}$

$\text{−}{4}^{2}$

$-16$

$\text{−}{6}^{2}$

$-3\left(-5\right)\left(6\right)$

90

$-4\left(-6\right)\left(3\right)$

$\left(8-11\right)\left(9-12\right)$

9

$\left(6-11\right)\left(8-13\right)$

$26-3\left(2-7\right)$

41

$23-2\left(4-6\right)$

$65÷\left(-5\right)+\left(-28\right)÷\left(-7\right)$

$-9$

$52÷\left(-4\right)+\left(-32\right)÷\left(-8\right)$

$9-2\left[3-8\left(-2\right)\right]$

$-29$

$11-3\left[7-4\left(-2\right)\right]$

${\left(-3\right)}^{2}-24÷\left(8-2\right)$

5

${\left(-4\right)}^{2}-32÷\left(12-4\right)$

Evaluate Variable Expressions with Integers

In the following exercises, evaluate each expression.

$y+\left(-14\right)$ when
$y=-33$
$y=30$

$-47$ 16

$x+\left(-21\right)$ when
$x=-27$
$x=44$

$a+3$ when $a=-7$
$\text{−}a+3$ when $a=-7$

$-4$ 10

$d+\left(-9\right)$ when $d=-8$
$\text{−}d+\left(-9\right)$ when $d=-8$

$m+n$ when
$m=-15,n=7$

$-8$

$p+q$ when
$p=-9,q=17$

$r+s$ when $r=-9,s=-7$

$-16$

$t+u$ when $t=-6,u=-5$

${\left(x+y\right)}^{2}$ when
$x=-3,y=14$

121

${\left(y+z\right)}^{2}$ when
$y=-3,z=15$

$-2x+17$ when
$x=8$
$x=-8$

1 33

$-5y+14$ when
$y=9$
$y=-9$

$10-3m$ when
$m=5$
$m=-5$

$-5$ 25

$18-4n$ when
$n=3$
$n=-3$

$2{w}^{2}-3w+7$ when
$w=-2$

21

$3{u}^{2}-4u+5$ when $u=-3$

$9a-2b-8$ when
$a=-6\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}b=-3$

$-56$

$7m-4n-2$ when
$m=-4\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}n=-9$

Translate English Phrases to Algebraic Expressions

In the following exercises, translate to an algebraic expression and simplify if possible.

the sum of 3 and $-15,$ increased by 7

$\left(3+\left(-15\right)\right)+7;-5$

the sum of $-8$ and $-9,$ increased by 23

the difference of 10 and $-18$

$10-\left(-18\right);28$

subtract 11 from $-25$

the difference of $-5$ and $-30$

$-5-\left(-30\right);25$

subtract $-6$ from $-13$

the product of $\text{−3 and 15}$

$-3·15;-45$

the product of $\text{−4 and 16}\phantom{\rule{0.2em}{0ex}}$

the quotient of $-60$ and $-20$

$-60÷\left(-20\right);3$

the quotient of $-40$ and $-20$

the quotient of $-6$ and the sum of a and b

$\frac{-6}{a+b}$

the quotient of $-7$ and the sum of m and n

the product of $-10$ and the difference of $p\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}q$

$-10\left(p-q\right)$

the product of $-13$ and the difference of $c\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}d$

Use Integers in Applications

In the following exercises, solve.

Temperature On January $15,$ the high temperature in Anaheim, California, was $84\text{°}.$ That same day, the high temperature in Embarrass, Minnesota was $-12\text{°}.$ What was the difference between the temperature in Anaheim and the temperature in Embarrass?

$96\text{°}$

Temperature On January $21,$ the high temperature in Palm Springs, California, was $89\text{°},$ and the high temperature in Whitefield, New Hampshire was $-31\text{°}.$ What was the difference between the temperature in Palm Springs and the temperature in Whitefield?

Football At the first down, the Chargers had the ball on their 25 yard line. On the next three downs, they lost 6 yards, gained 10 yards, and lost 8 yards. What was the yard line at the end of the fourth down?

21

Football At the first down, the Steelers had the ball on their 30 yard line. On the next three downs, they gained 9 yards, lost 14 yards, and lost 2 yards. What was the yard line at the end of the fourth down?

Checking Account Mayra has $124 in her checking account. She writes a check for$152. What is the new balance in her checking account?

$\text{−}28$

Checking Account Selina has $165 in her checking account. She writes a check for$207. What is the new balance in her checking account?

Checking Account Diontre has a balance of $\text{−}38$ in his checking account. He deposits $225 to the account. What is the new balance?$187

Checking Account Reymonte has a balance of $\text{−}49$ in his checking account. He deposits $281 to the account. What is the new balance? ## Everyday math Stock market Javier owns 300 shares of stock in one company. On Tuesday, the stock price dropped$12 per share. What was the total effect on Javier’s portfolio?

$\text{−}3600$

Weight loss In the first week of a diet program, eight women lost an average of 3 pounds each. What was the total weight change for the eight women?

## Writing exercises

In your own words, state the rules for multiplying integers.

In your own words, state the rules for dividing integers.

Why is $\text{−}{2}^{4}\ne {\left(-2\right)}^{4}?$

Why is $\text{−}{4}^{3}={\left(-4\right)}^{3}?$

## Self check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

On a scale of 1–10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?

#### Questions & Answers

hello, I have algebra phobia. Subtracting negative numbers always seem to get me confused.
what do you need help in?
Felix
subtracting a negative....is adding!!
Heather
look at the numbers if they have different signs, it's like subtracting....but you keep the sign of the largest number...
Felix
for example.... -19 + 7.... different signs...subtract.... 12 keep the sign of the "largest" number 19 is bigger than 7.... 19 has the negative sign... Therefore, -12 is your answer...
Felix
—12
Thanks Felix.l also get confused with signs.
Esther
Thank you for this
Shatey
Bruce drives his car for his job. The equation R=0.575m+42 models the relation between the amount in dollars, R, that he is reimbursed and the number of miles, m, he drives in one day. Find the amount Bruce is reimbursed on a day when he drives 220 miles
168.50=R
Heather
john is 5years older than wanjiru.the sum of their years is27years.what is the age of each
46
mustee
j 17 w 11
Joseph
john is 16. wanjiru is 11.
Felix
27-5=22 22÷2=11 11+5=16
Joyce
I don't see where the answers are.
Ed
Cindy and Richard leave their dorm in Charleston at the same time. Cindy rides her bicycle north at a speed of 18 miles per hour. Richard rides his bicycle south at a speed of 14 miles per hour. How long will it take them to be 96 miles apart?
3
Christopher
18t+14t=96 32t=96 32/96 3
Christopher
show that a^n-b^2n is divisible by a-b
What does 3 times your weight right now
Use algebra to combine 39×5 and the half sum of travel of 59+30
Cherokee
What is the segment of 13? Explain
Cherokee
my weight is 49. So 3 times is 147
Cherokee
kg to lbs you goin to convert 2.2 or one if the same unit your going to time your body weight by 3. example if my body weight is 210lb. what would be my weight if I was 3 times as much in kg. that's you do 210 x3 = 630lb. then 630 x 2.2= .... hope this helps
tyler
How to convert grams to pounds?
paul
What is the lcm of 340
Yes
Cherokee
How many numbers each equal to y must be taken to make 15xy
15x
Martin
15x
Asamoah
15x
Hugo
1y
Tom
1y x 15y
Tom
find the equation whose roots are 1 and 2
(x - 2)(x -1)=0 so equation is x^2-x+2=0
Ranu
I believe it's x^2-3x+2
NerdNamedGerg
because the X's multiply by the -2 and the -1 and than combine like terms
NerdNamedGerg
find the equation whose roots are -1 and 4
Ans = ×^2-3×+2
Gee
find the equation whose roots are -2 and -1
(×+1)(×-4) = x^2-3×-4
Gee
yeah
Asamoah
Quadratic equations involving factorization
there's a chatting option in the app wow
Nana
That's cool cool
Nana
Nice to meet you all
Nana
you too.
Joan
😃
Nana
Hey you all there are several Free Apps that can really help you to better solve type Equations.
Debra
Debra, which apps specifically. ..?
Nana
am having a course in elementary algebra ,any recommendations ?
samuel
Samuel Addai, me too at ucc elementary algebra as part of my core subjects in science
Nana
me too as part of my core subjects in R M E
Ken
at ABETIFI COLLEGE OF EDUCATION
Ken
ok great. Good to know.
Joan
5x + 1/3= 2x + 1/2
sanam
Plz solve this
sanam
5x - 3x = 1/2 - 1/3 2x = 1/6 x = 1/12
Ranu
Thks ranu
sanam
Erica
the previous equation should be 3x = 1/6 x=1/18
Sriram
for the new one 10x + 2x = 38 - 14
Sriram
12x = 24 x=2
Sriram
10x + 14 = -2x +38 10x + 2x = 38 - 14 12x = 24 divide both sides by the coefficient of x, which is 12 therefore × = 2
vida
a trader gains 20 rupees loses 42 rupees and then gains ten rupees Express algebraically the result of his transactions
a trader gains 20 rupees loses 42 rupees and then gains 10 rupees Express algebraically the result of his three transactions
vinaya
a trader gains 20 rupees loses 42 rupees and then gains 10 rupees Express algebraically the result of his three transactions
vinaya
a trader gains 20 rupees loses 42 rupees and then gains 10 rupees Express algebraically the result of his three transactions
vinaya
Kim is making eight gallons of punch from fruit juice and soda. The fruit juice costs $6.04 per gallon and the soda costs$4.28 per gallon. How much fruit juice and how much soda should she use so that the punch costs \$5.71 per gallon?
(a+b)(p+q+r)(b+c)(p+q+r)(c+a) (p+q+r)
really
Asamoah
4x-7y=8 2x-7y=1 what is the answer?
x=7/2 & y=6/7
Pbp
x=7/2 & y=6/7 use Elimination
Debra
true
bismark
factoriz e
usman
4x-7y=8 X=7/4y+2 and 2x-7y=1 x=7/2y+1/2
Peggie
Ok cool answer peggie
Frank
thanks
Ramil
x=7/2 y=6/7
Asamoah
Thanks , all of you are correct.
Joseph