# 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?

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? Cecilia Reply 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 Heidi Reply 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. Elbert Reply 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
Darrin is hanging 200 feet of Christmas garland on the three sides of fencing that enclose his rectangular front yard. The length is five feet less than five times the width. Find the length and width of the fencing.
Mario invested $475 in$45 and $25 stock shares. The number of$25 shares was five less than three times the number of $45 shares. How many of each type of share did he buy? Jawad Reply let # of$25 shares be (x) and # of $45 shares be (y) we start with$25x + $45y=475, right? we are told the number of$25 shares is 3y-5) so plug in this for x. $25(3y-5)+$45y=$475 75y-125+45y=475 75y+45y=600 120y=600 y=5 so if #$25 shares is (3y-5) plug in y.
Joshua
will every polynomial have finite number of multiples?
a=# of 10's. b=# of 20's; a+b=54; 10a + 20b=$910; a=54 -b; 10(54-b) + 20b=$910; 540-10b+20b=$910; 540+10b=$910; 10b=910-540; 10b=370; b=37; so there are 37 20's and since a+b=54, a+37=54; a=54-37=17; a=17, so 17 10's. So lets check. $740+$170=$910. David Reply . A cashier has 54 bills, all of which are$10 or $20 bills. The total value of the money is$910. How many of each type of bill does the cashier have?
whats the coefficient of 17x
the solution says it 14 but how i thought it would be 17 im i right or wrong is the exercise wrong
Dwayne
17
Melissa
wow the exercise told me 17x solution is 14x lmao
Dwayne
thank you
Dwayne
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
Washing his dad’s car alone, eight-year-old Levi takes 2.5 hours. If his dad helps him, then it takes 1 hour. How long does it take the Levi’s dad to wash the car by himself?
Ethan and Leo start riding their bikes at the opposite ends of a 65-mile bike path. After Ethan has ridden 1.5 hours and Leo has ridden 2 hours, they meet on the path. Ethan’s speed is 6 miles per hour faster than Leo’s speed. Find the speed of the two bikers.
Nathan walked on an asphalt pathway for 12 miles. He walked the 12 miles back to his car on a gravel road through the forest. On the asphalt he walked 2 miles per hour faster than on the gravel. The walk on the gravel took one hour longer than the walk on the asphalt. How fast did he walk on the gravel?
Mckenzie
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