# 1.8 The real numbers  (Page 7/13)

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## Practice makes perfect

Simplify Expressions with Square Roots

In the following exercises, simplify.

$\sqrt{36}$

6

$\sqrt{4}$

$\sqrt{64}$

8

$\sqrt{169}$

$\sqrt{9}$

3

$\sqrt{16}$

$\sqrt{100}$

10

$\sqrt{144}$

$\text{−}\sqrt{4}$

$-2$

$\text{−}\sqrt{100}$

$\text{−}\sqrt{1}$

$-1$

$\text{−}\sqrt{121}$

Identify Integers, Rational Numbers, Irrational Numbers, and Real Numbers

In the following exercises, write as the ratio of two integers.

5 3.19

$\frac{5}{1}$ $\frac{319}{100}$

8 1.61

$\text{−}12\phantom{\rule{0.2em}{0ex}}$ 9.279

$\frac{-12}{1}$ $\frac{9297}{1000}$

$\text{−}16$ 4.399

In the following exercises, list the rational numbers, irrational numbers

$0.75,0.22\stackrel{\text{–}}{3},1.39174$

$0.75,0.22\stackrel{\text{–}}{3}$ $1.39174\text{…}$

$0.36,0.94729\text{…},2.52\stackrel{\text{–}}{8}$

$0.4\stackrel{\text{–}}{5},1.919293\text{…},3.59$

$0.4\stackrel{\text{–}}{5},3.59$ $1.919293\text{…}$

$0.1\stackrel{\text{–}}{3},0.42982\text{…},1.875$

In the following exercises, identify whether each number is rational or irrational.

$\sqrt{25}$ $\sqrt{30}$

rational irrational

$\sqrt{44}$ $\sqrt{49}$

$\sqrt{164}$ $\sqrt{169}$

irrational rational

$\sqrt{225}$ $\sqrt{216}$

In the following exercises, identify whether each number is a real number or not a real number.

$\text{−}\sqrt{81}$ $\sqrt{-121}$

real number not a real number

$\text{−}\sqrt{64}$ $\sqrt{-9}$

$\sqrt{-36}$ $\text{−}\sqrt{144}$

not a real number real number

$\sqrt{-49}$ $\text{−}\sqrt{144}$

In the following exercises, list the whole numbers, integers, rational numbers, irrational numbers, real numbers for each set of numbers.

$-8,0,1.95286\text{…},\frac{12}{5},\sqrt{36},9$

$0,\sqrt{36},9$ $-8,\sqrt{36},9$ $-8,0,\frac{12}{5},\sqrt{36},9$ $1.95286\text{…}$ $-8,0,1.95286\text{…},\frac{12}{5},\sqrt{36},9$

$-9,-3\frac{4}{9},\text{−}\sqrt{9},0.40\stackrel{\text{–}}{9},\frac{11}{6},7$

$\text{−}\sqrt{100},-7,-\phantom{\rule{0.2em}{0ex}}\frac{8}{3},-1,0.77,3\frac{1}{4}$

none $\text{−}\sqrt{100},-7,-1$ $\text{−}\sqrt{100},-7,-\phantom{\rule{0.2em}{0ex}}\frac{8}{3},-1,0.77,3\frac{1}{4}$ none $\text{−}\sqrt{100},-7,-\phantom{\rule{0.2em}{0ex}}\frac{8}{3},-1,0.77,3\frac{1}{4}$

$-6,-\phantom{\rule{0.2em}{0ex}}\frac{5}{2},0,0.\stackrel{\text{———}}{714285},2\frac{1}{5},\sqrt{14}$

Locate Fractions on the Number Line

In the following exercises, locate the numbers on a number line.

$\frac{3}{4},\frac{8}{5},\frac{10}{3}$

$\frac{1}{4},\frac{9}{5},\frac{11}{3}$

$\frac{3}{10},\frac{7}{2},\frac{11}{6},4$

$\frac{7}{10},\frac{5}{2},\frac{13}{8},3$

$\frac{2}{5},-\phantom{\rule{0.2em}{0ex}}\frac{2}{5}$

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

$\frac{3}{4},-\phantom{\rule{0.2em}{0ex}}\frac{3}{4},1\frac{2}{3},-1\frac{2}{3},\frac{5}{2},-\phantom{\rule{0.2em}{0ex}}\frac{5}{2}$

$\frac{1}{5},-\phantom{\rule{0.2em}{0ex}}\frac{2}{5},1\frac{3}{4},-1\frac{3}{4},\frac{8}{3},-\phantom{\rule{0.2em}{0ex}}\frac{8}{3}$

In the following exercises, order each of the pairs of numbers, using<or>.

$-1___-\phantom{\rule{0.2em}{0ex}}\frac{1}{4}$

<

$-1___-\phantom{\rule{0.2em}{0ex}}\frac{1}{3}$

$-2\frac{1}{2}___-3$

>

$-1\frac{3}{4}___-2$

$-\phantom{\rule{0.2em}{0ex}}\frac{5}{12}___-\phantom{\rule{0.2em}{0ex}}\frac{7}{12}$

>

$-\phantom{\rule{0.2em}{0ex}}\frac{9}{10}___-\phantom{\rule{0.2em}{0ex}}\frac{3}{10}$

$-3___-\phantom{\rule{0.2em}{0ex}}\frac{13}{5}$

<

$-4___-\phantom{\rule{0.2em}{0ex}}\frac{23}{6}$

Locate Decimals on the Number Line In the following exercises, locate the number on the number line.

0.8

$-0.9$

$-1.6$

3.1

In the following exercises, order each pair of numbers, using<or>.

$0.37___0.63$

<

$0.86___0.69$

$0.91___0.901$

>

$0.415___0.41$

$-0.5___-0.3$

<

$-0.1___-0.4$

$-0.62___-0.619$

<

$-7.31___-7.3$

## Everyday math

Field trip All the 5th graders at Lincoln Elementary School will go on a field trip to the science museum. Counting all the children, teachers, and chaperones, there will be 147 people. Each bus holds 44 people.

How many busses will be needed?
Why must the answer be a whole number?
Why shouldn’t you round the answer the usual way, by choosing the whole number closest to the exact answer?

Child care Serena wants to open a licensed child care center. Her state requires there be no more than 12 children for each teacher. She would like her child care center to serve 40 children.

How many teachers will be needed?
Why must the answer be a whole number?
Why shouldn’t you round the answer the usual way, by choosing the whole number closest to the exact answer?

## Writing exercises

In your own words, explain the difference between a rational number and an irrational number.

Explain how the sets of numbers (counting, whole, integer, rational, irrationals, reals) are related to each other.

## Self check

After completing the exercises, use this checklist to evaluate your mastery of the objective 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?

Nancy took a 3 hour drive. She went 50 miles before she got caught in a storm. Then she drove 68 miles at 9 mph less than she had driven when the weather was good. What was her speed driving in the storm?
Mr Hernaez runs his car at a regular speed of 50 kph and Mr Ranola at 36 kph. They started at the same place at 5:30 am and took opposite directions. At what time were they 129 km apart?
90 minutes
Melody wants to sell bags of mixed candy at her lemonade stand. She will mix chocolate pieces that cost $4.89 per bag with peanut butter pieces that cost$3.79 per bag to get a total of twenty-five bags of mixed candy. Melody wants the bags of mixed candy to cost her $4.23 a bag to make. How many bags of chocolate pieces and how many bags of peanut butter pieces should she use? Jake Reply 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
13.5
Pervaiz
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? Bridget Reply The equation P=28+2.54w models the relation between the amount of Randy’s monthly water bill payment, P, in dollars, and the number of units of water, w, used. Find the payment for a month when Randy used 15 units of water. Bridget help me understand graphs Marlene Reply what kind of graphs? bruce function f(x) to find each value Marlene I am in algebra 1. Can anyone give me any ideas to help me learn this stuff. Teacher and tutor not helping much. Marlene Given f(x)=2x+2, find f(2) so you replace the x with the 2, f(2)=2(2)+2, which is f(2)=6 Melissa if they say find f(5) then the answer would be f(5)=12 Melissa I need you to help me Melissa. Wish I can show you my homework Marlene How is f(1) =0 I am really confused Marlene what's the formula given? f(x)=? Melissa It shows a graph that I wish I could send photo of to you on here Marlene Which problem specifically? Melissa which problem? Melissa I don't know any to be honest. But whatever you can help me with for I can practice will help Marlene I got it. sorry, was out and about. I'll look at it now. Melissa Thank you. I appreciate it because my teacher assumes I know this. My teacher before him never went over this and several other things. Marlene I just responded. Melissa Thank you Marlene -65r to the 4th power-50r cubed-15r squared+8r+23 ÷ 5r WENDY Reply State the question clearly please Rich write in this form a/b answer should be in the simplest form 5% August Reply convert to decimal 9/11 August 0.81818 Rich 5/100 = .05 but Rich is right that 9/11 = .81818 Melissa Equation in the form of a pending point y+2=1/6(×-4) Jose Reply write in simplest form 3 4/2 August definition of quadratic formula Ahmed Reply 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 What Reply 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? Mckenzie Reply 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?
$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