# 1.9 Properties of real numbers  (Page 6/10)

 Page 6 / 10

## Practice makes perfect

Use the Commutative and Associative Properties

In the following exercises, use the associative property to simplify.

3(4 x )

12 x

4(7 m )

$\left(y+12\right)+28$

$y+40$

$\left(n+17\right)+33$

In the following exercises, simplify.

$\frac{1}{2}+\frac{7}{8}+\left(-\phantom{\rule{0.2em}{0ex}}\frac{1}{2}\right)$

$\frac{7}{8}$

$\frac{2}{5}+\frac{5}{12}+\left(-\phantom{\rule{0.2em}{0ex}}\frac{2}{5}\right)$

$\frac{3}{20}·\frac{49}{11}·\frac{20}{3}$

$\frac{49}{11}$

$\frac{13}{18}·\frac{25}{7}·\frac{18}{13}$

$-24.7·\frac{3}{8}$

$-63$

$-36·11·\frac{4}{9}$

$\left(\frac{5}{6}+\frac{8}{15}\right)+\frac{7}{15}$

$1\frac{5}{6}$

$\left(\frac{11}{12}+\frac{4}{9}\right)+\frac{5}{9}$

17(0.25)(4)

17

36(0.2)(5)

[2.48(12)](0.5)

14.88

[9.731(4)](0.75)

7(4 a )

28 a

9(8 w )

$-15\left(5m\right)$

$-75m$

$-23\left(2n\right)$

$12\left(\frac{5}{6}p\right)$

10 p

$20\left(\frac{3}{5}q\right)$

$43m+\left(-12n\right)+\left(-16m\right)+\left(-9n\right)$

$27m+\left(-21n\right)$

$-22p+17q+\left(-35p\right)+\left(-27q\right)$

$\frac{3}{8}g+\frac{1}{12}h+\frac{7}{8}g+\frac{5}{12}h$

$\frac{5}{4}g+\frac{1}{2}h$

$\frac{5}{6}a+\frac{3}{10}b+\frac{1}{6}a+\frac{9}{10}b$

$6.8p+9.14q+\left(-4.37p\right)+\left(-0.88q\right)$

$2.43p+8.26q$

$9.6m+7.22n+\left(-2.19m\right)+\left(-0.65n\right)$

Use the Identity and Inverse Properties of Addition and Multiplication

In the following exercises, find the additive inverse of each number.

$\frac{2}{5}$
4.3
$-8$
$-\phantom{\rule{0.2em}{0ex}}\frac{10}{3}$

$-\phantom{\rule{0.2em}{0ex}}\frac{2}{5}$ $-4.3$ 8 $\frac{10}{3}$

$\frac{5}{9}$
2.1
$-3$
$-\phantom{\rule{0.2em}{0ex}}\frac{9}{5}$

$-\phantom{\rule{0.2em}{0ex}}\frac{7}{6}$
$-0.075$
23
$\frac{1}{4}$

$\frac{7}{6}$ 0.075 $-23$ $-\phantom{\rule{0.2em}{0ex}}\frac{1}{4}$

$-\phantom{\rule{0.2em}{0ex}}\frac{8}{3}$
$-0.019$
52
$\frac{5}{6}$

In the following exercises, find the multiplicative inverse of each number.

6 $-\phantom{\rule{0.2em}{0ex}}\frac{3}{4}$ 0.7

$\frac{1}{6}$ $-\phantom{\rule{0.2em}{0ex}}\frac{4}{3}$ $\frac{10}{7}$

12 $-\phantom{\rule{0.2em}{0ex}}\frac{9}{2}$ 0.13

$\frac{11}{12}$ $-1.1$ $-4$

$\frac{12}{11}$ $-\phantom{\rule{0.2em}{0ex}}\frac{10}{11}$ $-\phantom{\rule{0.2em}{0ex}}\frac{1}{4}$

$\frac{17}{20}$ $-1.5$ $-3$

Use the Properties of Zero

In the following exercises, simplify.

$\frac{0}{6}$

0

$\frac{3}{0}$

$0÷\frac{11}{12}$

0

$\frac{6}{0}$

$\frac{0}{3}$

0

$0·\frac{8}{15}$

$\left(-3.14\right)\left(0\right)$

0

$\frac{\frac{1}{10}}{0}$

Mixed Practice

In the following exercises, simplify.

$19a+44-19a$

44

$27c+16-27c$

$10\left(0.1d\right)$

id

$100\left(0.01p\right)$

$\frac{0}{u-4.99},$ where $u\ne 4.99$

0

$\frac{0}{v-65.1},$ where $v\ne 65.1$

$0÷\left(x-\phantom{\rule{0.2em}{0ex}}\frac{1}{2}\right),$ where $x\ne \frac{1}{2}$

0

$0÷\left(y-\phantom{\rule{0.2em}{0ex}}\frac{1}{6}\right),$ where $x\ne \frac{1}{6}$

$\frac{32-5a}{0},$ where
$32-5a\ne 0$

undefined

$\frac{28-9b}{0},$ where
$28-9b\ne 0$

$\left(\frac{3}{4}+\frac{9}{10}m\right)÷0$ where
$\frac{3}{4}+\frac{9}{10}m\ne 0$

undefined

$\left(\frac{5}{16}n-\phantom{\rule{0.2em}{0ex}}\frac{3}{7}\right)÷0$ where
$\frac{5}{16}n-\phantom{\rule{0.2em}{0ex}}\frac{3}{7}\ne 0$

$15·\frac{3}{5}\left(4d+10\right)$

$36d+90$

$18·\frac{5}{6}\left(15h+24\right)$

Simplify Expressions Using the Distributive Property

In the following exercises, simplify using the distributive property.

$8\left(4y+9\right)$

$32y+72$

$9\left(3w+7\right)$

$6\left(c-13\right)$

$6c-78$

$7\left(y-13\right)$

$\frac{1}{4}\left(3q+12\right)$

$\frac{3}{4}q+3$

$\frac{1}{5}\left(4m+20\right)$

$9\left(\frac{5}{9}y-\phantom{\rule{0.2em}{0ex}}\frac{1}{3}\right)$

$5y-3$

$10\left(\frac{3}{10}x-\phantom{\rule{0.2em}{0ex}}\frac{2}{5}\right)$

$12\left(\frac{1}{4}+\frac{2}{3}r\right)$

$3+8r$

$12\left(\frac{1}{6}+\frac{3}{4}s\right)$

$r\left(s-18\right)$

$rs-18r$

$u\left(v-10\right)$

$\left(y+4\right)p$

$yp+4p$

$\left(a+7\right)x$

$-7\left(4p+1\right)$

$-28p-7$

$-9\left(9a+4\right)$

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

$-3x+18$

$-4\left(q-7\right)$

$\text{−}\left(3x-7\right)$

$-3x+7$

$\text{−}\left(5p-4\right)$

$16-3\left(y+8\right)$

$-3y-8$

$18-4\left(x+2\right)$

$4-11\left(3c-2\right)$

$-33c+26$

$9-6\left(7n-5\right)$

$22-\left(a+3\right)$

$\text{−}a+19$

$8-\left(r-7\right)$

$\left(5m-3\right)-\left(m+7\right)$

$4m-10$

$\left(4y-1\right)-\left(y-2\right)$

$5\left(2n+9\right)+12\left(n-3\right)$

$22n+9$

$9\left(5u+8\right)+2\left(u-6\right)$

$9\left(8x-3\right)-\left(-2\right)$

$72x-25$

$4\left(6x-1\right)-\left(-8\right)$

$14\left(c-1\right)-8\left(c-6\right)$

$6c+34$

$11\left(n-7\right)-5\left(n-1\right)$

$6\left(7y+8\right)-\left(30y-15\right)$

$12y+63$

$7\left(3n+9\right)-\left(4n-13\right)$

## Everyday math

Insurance copayment Carrie had to have 5 fillings done. Each filling cost $80. Her dental insurance required her to pay 20% of the cost as a copay. Calculate Carrie’s copay: 1. First, by multiplying 0.20 by 80 to find her copay for each filling and then multiplying your answer by 5 to find her total copay for 5 fillings. 2. Next, by multiplying [5(0.20)](80) 3. Which of the properties of real numbers says that your answers to parts (a), where you multiplied 5[(0.20)(80)] and (b), where you multiplied [5(0.20)](80), should be equal?$80 $80 answers will vary Cooking time Helen bought a 24-pound turkey for her family’s Thanksgiving dinner and wants to know what time to put the turkey in to the oven. She wants to allow 20 minutes per pound cooking time. Calculate the length of time needed to roast the turkey: 1. First, by multiplying $24·20$ to find the total number of minutes and then multiplying the answer by $\frac{1}{60}$ to convert minutes into hours. 2. Next, by multiplying $24\left(20·\frac{1}{60}\right).$ 3. Which of the properties of real numbers says that your answers to parts (a), where you multiplied $\left(24·20\right)\frac{1}{60},$ and (b), where you multiplied $24\left(20·\frac{1}{60}\right),$ should be equal? Buying by the case Trader Joe’s grocery stores sold a bottle of wine they called “Two Buck Chuck” for$1.99. They sold a case of 12 bottles for $23.88. To find the cost of 12 bottles at$1.99, notice that 1.99 is $2-0.01.$

1. Multiply 12(1.99) by using the distributive property to multiply $12\left(2-0.01\right).$
2. Was it a bargain to buy “Two Buck Chuck” by the case?

$23.88 no, the price is the same Multi-pack purchase Adele’s shampoo sells for$3.99 per bottle at the grocery store. At the warehouse store, the same shampoo is sold as a 3 pack for $10.49. To find the cost of 3 bottles at$3.99, notice that 3.99 is $4-0.01.$

1. Multiply 3(3.99) by using the distributive property to multiply $3\left(4-0.01\right).$
2. How much would Adele save by buying 3 bottles at the warehouse store instead of at the grocery store?

## Writing exercises

What is the difference between the additive inverse and the multiplicative inverse of a number?

Simplify $8\left(x-\phantom{\rule{0.2em}{0ex}}\frac{1}{4}\right)$ using the distributive property and explain each step.

Explain how you can multiply 4($5.97) without paper or calculator by thinking of$5.97 as $6-0.03$ and then using the distributive property.

## Self check

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

After reviewing this checklist, what will you do to become confident for all objectives?

how to square
easiest way to find the square root of a large number?
Jackie
the accompanying figure shows known flow rates of hydrocarbons into and out of a network of pipes at an oil refinery set up a linear system whose solution provides the unknown flow rates (b) solve the system for the unknown flow rates (c) find the flow rates and directions of flow if x4=50and x6=0
What is observation
I'm confused by the question. Can you describe or explain the math question it pertains to?
Melissa
there is no math to it because all you use is your vision or gaze to the sorrounding areas
Cesarp
Teegan likes to play golf. He has budgeted $60 next month for the driving range. It costs him$10.55 for a bucket of balls each time he goes. What is the maximum number of times he can go to the driving range next month?
5 times max
Anton
Felecia left her home to visit her daughter, driving 45mph. Her husband waited for the dog sitter to arrive and left home 20 minutes, or 1/3 hour later. He drove 55mph to catch up to Felecia. How long before he reaches her?
35 min
Debra
Carmen wants to tile the floor of his house. He will need 1,000 square feet of tile. He will do most of the floor with a tile that costs $1.50 per square foot, but also wants to use an accent tile that costs$9.00 per square foot. How many square feet of each tile should he plan to use if he wants the overall cost to be $3 per square foot? Parker Reply what you wanna get Cesar 800 sq. ft @$1.50 & 200 sq. ft @ $9.00 Marco Geneva treated her parents to dinner at their favorite restaurant. The bill was$74.25. Geneva wants to leave 16 % of the total - bill as a tip. How much should the tip be?
74.25 × .16 then get the total and that will be your tip
David
$74.25 x 0.16 =$11.88 total bill: $74.25 +$11.88 = $86.13 ericka yes and tip 16% will be$11.88
David
what is the shorter way to do it
Priam has dimes and pennies in a cup holder in his car. The total value of the coins is $4.21. The number of dimes is three less than four times the number of pennies. How many dimes and how many pennies are in the cup? Alexandra Reply Uno de los ángulos suplementario es 4° más que 1/3 del otro ángulo encuentra las medidas de cada uno de los angulos Enith Reply June needs 48 gallons of punch for a party and has two different coolers to carry it in. The bigger cooler is five times as large as the smaller cooler. How many gallons can each cooler hold? Alexandra Reply I hope this is correct, x=cooler 1 5x=cooler 2 x + 5x = 48 6x=48 ×=8 gallons 5×=40 gallons ericka Priam has pennies and dimes in a cup holder in his car. The total value of the coins is$4.21 . The number of dimes is three less than four times the number of pennies. How many pennies and how many dimes are in the cup?
Arnold invested $64,000 some at 5.5% interest and the rest at 9% interest how much did he invest at each rate if he received$4500 in interest in one year
List five positive thoughts you can say to yourself that will help youapproachwordproblemswith a positive attitude. You may want to copy them on a sheet of paper and put it in the front of your notebook, where you can read them often.
Avery and Caden have saved \$27,000 towards a down payment on a house. They want to keep some of the money in a bank account that pays 2.4% annual interest and the rest in a stock fund that pays 7.2% annual interest. How much should they put into each account so that they earn 6% interest per year?
324.00
Irene
1.2% of 27.000
Irene
i did 2.4%-7.2% i got 1.2%
Irene
I have 6% of 27000 = 1620 so we need to solve 2.4x +7.2y =1620
Catherine
I think Catherine is on the right track. Solve for x and y.
Scott
next bit : x=(1620-7.2y)/2.4 y=(1620-2.4x)/7.2 I think we can then put the expression on the right hand side of the "x=" into the second equation. 2.4x in the second equation can be rewritten as 2.4(rhs of first equation) I write this out tidy and get back to you...
Catherine