# 1.8 The real numbers  (Page 5/13)

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$\begin{array}{ccccccc}\hfill \frac{7}{4}=1\frac{3}{4}\hfill & & & \hfill -\phantom{\rule{0.2em}{0ex}}\frac{9}{2}=-4\frac{1}{2}\hfill & & & \hfill \frac{8}{3}=2\frac{2}{3}\hfill \end{array}$

[link] shows the number line with all the points plotted.

Locate and label the following on a number line: $4,\frac{3}{4},-\phantom{\rule{0.2em}{0ex}}\frac{1}{4},-3,\frac{6}{5},-\phantom{\rule{0.2em}{0ex}}\frac{5}{2},\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\frac{7}{3}.$

Locate and plot the integers, $4,-3.$

Locate the proper fraction $\frac{3}{4}$ first. The fraction $\frac{3}{4}$ is between 0 and 1. Divide the distance between 0 and 1 into four equal parts then, we plot $\frac{3}{4}.$ Similarly plot $-\phantom{\rule{0.2em}{0ex}}\frac{1}{4}.$

Now locate the improper fractions $\frac{6}{5},-\phantom{\rule{0.2em}{0ex}}\frac{5}{2},\frac{7}{3}.$ It is easier to plot them if we convert them to mixed numbers and then plot them as described above: $\frac{6}{5}=1\frac{1}{5},-\phantom{\rule{0.2em}{0ex}}\frac{5}{2}=-2\frac{1}{2},\frac{7}{3}=2\frac{1}{3}.$

Locate and label the following on a number line: $-1,\frac{1}{3},\frac{6}{5},-\phantom{\rule{0.2em}{0ex}}\frac{7}{4},\frac{9}{2},5,-\phantom{\rule{0.2em}{0ex}}\frac{8}{3}.$

Locate and label the following on a number line: $-2,\frac{2}{3},\frac{7}{5},-\phantom{\rule{0.2em}{0ex}}\frac{7}{4},\frac{7}{2},3,-\phantom{\rule{0.2em}{0ex}}\frac{7}{3}.$

In [link] , we’ll use the inequality symbols to order fractions. In previous chapters we used the number line to order numbers.

• a<b a is less than b ” when a is to the left of b on the number line
• a>b a is greater than b ” when a is to the right of b on the number line

As we move from left to right on a number line, the values increase.

Order each of the following pairs of numbers, using<or>. It may be helpful to refer [link] .

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

Be careful when ordering negative numbers.

1. $\begin{array}{cccccc}& & & & & -\phantom{\rule{0.2em}{0ex}}\frac{2}{3}___-1\hfill \\ -\phantom{\rule{0.2em}{0ex}}\frac{2}{3}\phantom{\rule{0.2em}{0ex}}\text{is to the right of}\phantom{\rule{0.2em}{0ex}}-1\phantom{\rule{0.2em}{0ex}}\text{on the number line.}\hfill & & & & & -\phantom{\rule{0.2em}{0ex}}\frac{2}{3}>-1\hfill \end{array}$

2. $\begin{array}{cccccc}& & & & & -3\frac{1}{2}___-3\hfill \\ -3\frac{1}{2}\phantom{\rule{0.2em}{0ex}}\text{is to the left of}\phantom{\rule{0.2em}{0ex}}-3\phantom{\rule{0.2em}{0ex}}\text{on the number line.}\hfill & & & & & -3\frac{1}{2}<-3\hfill \end{array}$

3. $\begin{array}{cccccc}& & & & & -\phantom{\rule{0.2em}{0ex}}\frac{3}{4}___-\phantom{\rule{0.2em}{0ex}}\frac{1}{4}\hfill \\ -\phantom{\rule{0.2em}{0ex}}\frac{3}{4}\phantom{\rule{0.2em}{0ex}}\text{is to the left of}\phantom{\rule{0.2em}{0ex}}-\phantom{\rule{0.2em}{0ex}}\frac{1}{4}\phantom{\rule{0.2em}{0ex}}\text{on the number line.}\hfill & & & & & -\phantom{\rule{0.2em}{0ex}}\frac{3}{4}<-\phantom{\rule{0.2em}{0ex}}\frac{1}{4}\hfill \end{array}$

4. $\begin{array}{cccccc}& & & & & -2___-\phantom{\rule{0.2em}{0ex}}\frac{8}{3}\hfill \\ -2\phantom{\rule{0.2em}{0ex}}\text{is to the right of}\phantom{\rule{0.2em}{0ex}}-\phantom{\rule{0.2em}{0ex}}\frac{8}{3}\phantom{\rule{0.2em}{0ex}}\text{on the number line.}\hfill & & & & & -2>-\phantom{\rule{0.2em}{0ex}}\frac{8}{3}\hfill \end{array}$

Order each of the following pairs of numbers, using<or>:

$-\phantom{\rule{0.2em}{0ex}}\frac{1}{3}___-1$ $-1\frac{1}{2}___-2$ $-\phantom{\rule{0.2em}{0ex}}\frac{2}{3}___-\phantom{\rule{0.2em}{0ex}}\frac{1}{3}$ $-3___-\phantom{\rule{0.2em}{0ex}}\frac{7}{3}.$

> > < <

Order each of the following pairs of numbers, using<or>:

$-1___-\phantom{\rule{0.2em}{0ex}}\frac{2}{3}$ $-2\frac{1}{4}___-2$ $-\phantom{\rule{0.2em}{0ex}}\frac{3}{5}___-\phantom{\rule{0.2em}{0ex}}\frac{4}{5}$ $-4___-\phantom{\rule{0.2em}{0ex}}\frac{10}{3}.$

< < > <

## Locate decimals on the number line

Since decimals are forms of fractions, locating decimals on the number line is similar to locating fractions on the number line.

Locate 0.4 on the number line.

A proper fraction has value less than one. The decimal number 0.4 is equivalent to $\frac{4}{10},$ a proper fraction, so 0.4 is located between 0 and 1. On a number line, divide the interval between 0 and 1 into 10 equal parts. Now label the parts 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0. We write 0 as 0.0 and 1 and 1.0, so that the numbers are consistently in tenths. Finally, mark 0.4 on the number line. See [link] .

Locate on the number line: 0.6.

Locate on the number line: 0.9.

Locate $-0.74$ on the number line.

The decimal $-0.74$ is equivalent to $-\phantom{\rule{0.2em}{0ex}}\frac{74}{100},$ so it is located between 0 and $-1.$ On a number line, mark off and label the hundredths in the interval between 0 and $-1.$ See [link] .

Locate on the number line: $-0.6.$

Locate on the number line: $-0.7.$

Which is larger, 0.04 or 0.40? If you think of this as money, you know that $0.40 (forty cents) is greater than$0.04 (four cents). So,

$0.40>0.04$

Again, we can use the number line to order numbers.

• a<b a is less than b ” when a is to the left of b on the number line
• a>b a is greater than b ” when a is to the right of b on the number line

Where are 0.04 and 0.40 located on the number line? See [link] .

We see that 0.40 is to the right of 0.04 on the number line. This is another way to demonstrate that 0.40>0.04.

How does 0.31 compare to 0.308? This doesn’t translate into money to make it easy to compare. But if we convert 0.31 and 0.308 into fractions, we can tell which is larger.

#### Questions & Answers

DaMarcus and Fabian live 23 miles apart and play soccer at a park between their homes. DaMarcus rode his bike for 34 of an hour and Fabian rode his bike for 12 of an hour to get to the park. Fabian’s speed was 6 miles per hour faster than DaMarcus’s speed. Find the speed of both soccer players.
?
Ann
DaMarcus: 16 mi/hr Fabian: 22 mi/hr
Sherman
Joy is preparing 20 liters of a 25% saline solution. She has only a 40% solution and a 10% solution in her lab. How many liters of the 40% solution and how many liters of the 10% solution should she mix to make the 25% solution?
15 and 5
32 is 40% , & 8 is 10 % , & any 4 letters is 5%.
Karen
It felt that something is missing on the question like: 40% of what solution? 10% of what solution?
Jhea
its confusing
Sparcast
3% & 2% to complete the 25%
Sparcast
because she already has 20 liters.
Sparcast
ok I was a little confused I agree 15% & 5%
Sparcast
8,2
Karen
Jim and Debbie earned $7200. Debbie earned$1600 more than Jim earned. How much did they earned
5600
Gloria
1600
Gloria
Bebbie: 4,400 Jim: 2,800
Jhea
A river cruise boat sailed 80 miles down the Mississippi River for 4 hours. It took 5 hours to return. Find the rate of the cruise boat in still water and the rate of the current.
A veterinarian is enclosing a rectangular outdoor running area against his building for the dogs he cares for. He needs to maximize the area using 100 feet of fencing. The quadratic equation A=x(100−2x) gives the area, A , of the dog run for the length, x , of the building that will border the dog run. Find the length of the building that should border the dog run to give the maximum area, and then find the maximum area of the dog run.
ggfcc
Mike
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?
1,75hrs
Mike
I'm going to guess. Divide Levi's time by 2. Then divide 1 hour by 2. 1.25 + 0.5 = 1.3?
John
Oops I mean 1.75
John
I'm guessing this because since I have divide 1 hour by 2, I have to do the same for the 2.5 hours it takes Levi by himself.
John
1,75 hrs is correct Mike
Emund
How did you come up with the answer?
John
Drew burned 1,800 calories Friday playing 1 hour of basketball and canoeing for 2 hours. On Saturday, he spent 2 hours playing basketball and 3 hours canoeing and burned 3,200 calories. How many calories did he burn per hour when playing basketball?
Brandon has a cup of quarters and dimes with a total value of $3.80. The number of quarters is four less than twice the number of dimes. How many quarters and how many dimes does Brandon have? Kendra Reply Tickets to a Broadway show cost$35 for adults and $15 for children. The total receipts for 1650 tickets at one performance were$47,150. How many adult and how many child tickets were sold?
825
Carol
Arnold invested $64,000, some at 5.5% interest and the rest at 9%. How much did he invest at each rate if he received$4,500 in interest in one year? How do I do this
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