Range = Highest value - Lowest value
= 20 - 3
= 17
Mean =
[tex]\frac{Sum\text{ of all the items}}{total\text{ number of items}}\text{ = }\frac{20\text{ + 13 + 13 + 6 + 3 +12 + 7 + 20 + 20}}{9}=\text{ }\frac{114}{9}\text{ = }\frac{38}{3}[/tex]Median = Middle value when the items are arranged in either increasing or decreasing order
Arranging the items In increasing order we have:
3, 6, 7, 12, 13, 13, 20, 20, 20.
Hence the median is the 5th item = 13
Mode = The most occuring item
The most occuring item or number is 20 (occuring three times)
rewrite the following as an exponential expression in simplest form
The given expression is,
[tex]^5\sqrt[]{\sqrt[]{x}}[/tex]It can be written in exponential terms as,
[tex]\begin{gathered} ^5\sqrt[]{x^{\frac{1}{2}}} \\ =(x^{\frac{1}{2}^{}})^{\frac{1}{5}} \\ =x^{\frac{1}{2}\ast\frac{1}{5}^{}}^{} \\ =x^{\frac{1}{10}} \end{gathered}[/tex]Therefore, the simplest exponential form of the given expression is,
[tex]x^{\frac{1}{10}}[/tex]65 people showed up to the party. There were 7 less men than women present. How many women were there?
Let x be women and y men.
x + y = 65 equation 1.
y = x - 7 equation 2.
Now we need to solve the equation system, as follows:
- on eq. 2 y is already solved, so we can replace y on eq. 1:
x + x - 7 = 65
2x - 7 = 65
2x = 65 + 7
2x = 72
x = 72/2
x = 36 women
Finally, we replace x on eq 2:
y = 36 - 7
y = 29 men
solve the equation b. 3(x + 8) = 21
given that 3(x+8)=12
on multiplying 3 into(x+8) we get
3x+24=21
3x=21-24
x=-3/3
x=-1
According to the provided equation
3x + 24 = 21
3x = 21 - 24
3x = -3
x = -3/3
x = -1
What is the slope of f(x)=3? 3 Undefined
Given the function:
[tex]f(x)=3[/tex]notice that this is a constant function, and its graph looks like this:
therefore, the slope of a constant function is 0
The remains of an ancient ball court include a rectangular playing alley with a perimeter of about 24m. The length of the alley is three times the width. Find the length and the width of the playing alley.The width is ? m and the length is ? m
ANSWER
Length of the playing alley = 9m
Width of the playing alley = 3m
STEP-BY-STEP EXPLANATION:
Given information
The perimeter of a rectangular playing alley = 24 m
The length of the alley is three times the width
Let l represents the length of the alley
Let w represents the width of the alley
Step 1: Write the formula for calculating the perimeter of a rectangle
[tex]\text{Perimeter of a rectangle = 2(l + w)}[/tex]Where l is the length and w is the width of the rectangle
Recall, length = 3 times the width of the alley
Mathematically,
[tex]\begin{gathered} l\text{ = 3 }\times\text{ w} \\ l\text{ = 3w} \end{gathered}[/tex]Step 2: Substitute the value of l = 3w into the above formula
[tex]\begin{gathered} P\text{ = 2(l + w)} \\ p\text{ = 24m} \\ l\text{ = 3w} \\ 24\text{ = 2(3w + w)} \end{gathered}[/tex]Step 3: Solve for w
[tex]\begin{gathered} 24\text{ = 2(4w)} \\ 24\text{ = 8w} \\ \text{Divide both sides by 8} \\ \frac{24}{8}\text{ = }\frac{8w}{8} \\ w\text{ = 3 m} \end{gathered}[/tex]From the calculations above, you will see that the width of the playing alley is 3m
Step 4: Solve for l
[tex]\begin{gathered} \text{Recall, l = 3w} \\ w\text{ = 3} \\ l\text{ = 3 }\times3 \\ l\text{ = 9m} \end{gathered}[/tex]Hence, the length of the playing alley is 9m
I already have the first part what is the value of x?
ANSWER
A) 540º
B) x = 121º
EXPLANATION
PART A
The sum of the interior angles of a polygon is:
[tex](n-2)\cdot180[/tex]Where n is the number of sides of the polygon.
This polygon has 5 sides, so the sum of its interior angles must be:
[tex](5-2)\cdot180=3\cdot180=540º[/tex]PART B
Now that we know the sum of the interior angles we can find x:
[tex]x+87+92+135+105=540[/tex]Solving for x:
[tex]\begin{gathered} x+419=540 \\ x=540-419 \\ x=121º \end{gathered}[/tex]Simplify the expression. Express your answer as a power. 130^6/130^4
Find the equation of the line through (7,2) which is parallel to the line y=-3x-2
The general slope-intercept form of the line: y = m * x + b
The parallel lines have the same slope
Given the equation of the line: y= -3x - 2
the slope of the given line = -3
The required line is parallel to the given line
So, the slope of the required line = m = -3
The equation of the line will be: y = -3x + b
The required line passes through the point (7, 2)
So, when x = 7, y = 2
We will find the value of b as follows:
[tex]\begin{gathered} 2=-3\cdot7+b \\ 2=-21+b \\ b=2+21=23 \end{gathered}[/tex]So, the answer will be the equation of the line is:
[tex]y=-3x+23[/tex]Solve each system by substitution.y = x + 5 4x + y = 201. Substitute the value of from the first equation into the second equation. 4x + = 20 2. Now solve for .3. Substitute the value of in one of the original equations. Solve for .4. Write your answer as an ordered pair .Solution: ( , )
1. Substitute the value of y from the first equation to the second equation
4x + x+5 = 20 (First fill in the blank)
2. Solve for x
4x + x + 5 = 20
5x + 5 = 20
5x = 20 - 5
5x = 15
5x/5 = 15/5
x = 3
3. Substitute x = 3 to the original equation
y = x+5
y = 3 + 5
y = 8
4. Therefore, the solution of the system of equation is (3,8) (second fill in the blank)
Which pair of functions are inverses of each other?A. f(x) = 5 + 6 and g(x) = 5x - 6B. f(x) = { and g(x) = 6x3C. f(x) = 7x - 2 and g(x) = 42D. f(x) = - 2 and g(x) = -2
To find the pair of functions that are inverses of each other, we should check through the options.
Option A
[tex]\begin{gathered} f(x)\text{ = }\frac{x}{5}\text{ + 6} \\ g(x)\text{ = 5x -6} \end{gathered}[/tex]First, we set f(x) = y. Thus:
[tex]y=\text{ }\frac{x}{5}\text{ + 6}[/tex]Then, we swap the variables:
[tex]x\text{ = }\frac{y}{5}\text{ + 6}[/tex]Make y the subject of formula:
[tex]\begin{gathered} x\text{ = }\frac{y}{5}\text{ + 6} \\ \frac{y}{5}\text{ = x -6} \\ y\text{ =5x -30} \end{gathered}[/tex]But:
[tex]g(x)\text{ }\ne\text{ 5x -30}[/tex]Hence, option A is incorrect
Option B
[tex]\begin{gathered} f(x)\text{ =}\frac{\sqrt[3]{x}}{6} \\ g(x)=6x^3 \end{gathered}[/tex]Set f(x) = y. Thus:
[tex]y\text{ = }\frac{\sqrt[3]{x}}{6}[/tex]Swap the variables:
[tex]x\text{ = }\frac{\sqrt[3]{y}}{6}[/tex]Make y the subject of the formula:
[tex]\begin{gathered} 6x\text{ = }\sqrt[3]{y} \\ \text{Cube both sides} \\ (6x)^3\text{ = y} \\ y\text{ = }216x^3 \end{gathered}[/tex]But:
[tex]g(x)\text{ }\ne216x^3[/tex]Hence, Option B is incorrect
Option C:
[tex]\begin{gathered} f(x)\text{ = }7x\text{ -2} \\ g(x)\text{ = }\frac{x\text{ + 2}}{7} \end{gathered}[/tex]Set f(x) = y. Thus:
[tex]y\text{ = 7x - 2}[/tex]Swap the variable:
[tex]x\text{ = 7y -2}[/tex]Make y the subject of formula:
[tex]\begin{gathered} x\text{ + 2 = 7y} \\ 7y\text{ = x +2} \\ \text{Divide both sides by 7} \\ \frac{7y}{7}\text{ = }\frac{x+2}{7} \\ y\text{ = }\frac{x+2}{7} \end{gathered}[/tex]But :
[tex]g(x)\text{ = }\frac{x+2}{7}[/tex]Hence, option C is correct
Option D
[tex]\begin{gathered} f(x)\text{ = }\frac{5}{x}\text{ -2} \\ g(x)\text{ =}\frac{x+2}{5} \end{gathered}[/tex]Set y =f(x). Thus:
[tex]y\text{ = }\frac{5}{x}\text{ -2}[/tex]Swap the variables:
[tex]x\text{ = }\frac{5}{y}\text{ -2}[/tex]Make y the subject of formula:
[tex]undefined[/tex]the table and graph shows the distance Javier and chloe are from a motion detector in terms of time Javier table and chloe table photo in the chatwho is moving away from the motion detector faster.
Chloe is moving away from the motion detector faster
Explanations:The one that has the highest rate of change of distance with time is moving away from the motion detector faster.
Let us calculate the rate of change of Javier's distance from the motion detector with time.
[tex]\begin{gathered} \frac{\Delta d}{\Delta t}=\text{ }\frac{10-5}{4-2} \\ \frac{\Delta d}{\Delta t}\text{ = }\frac{5}{2} \\ \frac{\Delta d}{\Delta t}\text{ = 2.5 m/s} \end{gathered}[/tex]Javier is moving with a speed of 2.5 m/s from the motion detector
Then, let us calculate the rate of change of Chloe's distance from the motion detector with time.
[tex]\begin{gathered} \frac{\Delta d}{\Delta t}=\text{ }\frac{12-9}{4-3} \\ \frac{\Delta d}{\Delta t}=\text{ }\frac{3}{1} \\ \frac{\Delta d}{\Delta t}=\text{ 3 m/s} \end{gathered}[/tex]Chloe is moving with a speed of 3.0 m/s from the motion detector
Chloe is moving away from the motion detector faster than Javier
Describe one possible way to estimate (without paper pencil and calculator) the value of 0.99804 x 32.785
One way of estimating this product is rounding each number to the nearest tenth first:
[tex]\begin{gathered} 0.99804=1.0\\ \\ 32.785=32.8 \end{gathered}[/tex]Then, the product of 32.8 and 1 is equal to 32.8, because every number multiplied by 1 is equal to itself.
Also, since the number 0.99804 is very close to 1, we can just keep the second value of the product, so the estimated result would be 32.785.
PLEASE HELP FAST! The function f (t) = 450(0.945)t/30 šo represents the change in a quantity over t months. What does the constant 0.945 reveal about the rate of change of the quantity?
It shows that the the quantity is smaller than the time (In proportion).
What is the range of the given function?{(-2, 0), (-4, -3), (2, -9), (0, 5), (-5, 7)}{x|x=-5, -4, -2, 0, 2}{y | y=-9, -3, 0, 5, 7}{x|x=-9, -5, -4, -3, -2, 0, 2, 5, 7}{y|y=-9, -5, -4, -3, -2, 0, 2, 5, 7}
The range of a function is the set of all possible y-values the function takes.
We have the function:
[tex]\left\{\left(-2,0\right),(-4,-3),(2,-9),(0,5),(-5,7)\right\}[/tex]Then, the range is:
[tex]Ran={}\lbrace y|y=-9,-3,0,5,7\rbrace[/tex]Answer:B
Step-by-step explanation:
cuz it is
In an inverse variation, y = 2 when x = 8. Write an inverse variation equation that showsthe relationship between x and y
The general equation for the inverse variation is,
[tex]\begin{gathered} y\propto\frac{1}{x} \\ y=\frac{k}{x} \end{gathered}[/tex]Substitute the value of x and y in the equation to obtain the value of k.
[tex]\begin{gathered} 2=\frac{k}{8} \\ k=8\cdot2 \\ =16 \end{gathered}[/tex]Thus, relation between x and y is,
[tex]y=\frac{16}{x}[/tex]The length of each edge of a cube is x centimeters. If xis an integer, why is the volume of the cube not equal to 15 cmº? show work please i want to understand:)
Apparently, there is a typing mistake in the statement of the question:
The length of each edge of a cube is x centimeters. If xis an integer, why is the volume of the cube not equal to 15 cm squared? show work please i want to understand:)
The answer is:
Because the units of volume are cm^3, or m^3, not cm^2 nor m^2.
The measure of the volume is cm * cm * cm, meanwhile the area is cm * cm.
Student became unresponsive.
A corner lot that originally was square lost 185 m² of area when one of the adjacent syreets what is wide by 3 m and the other was widnened by 5 m find the new dimensions of the lot
The new dimensions of the lot is length = 22 meters and width = 20 meters
The corner lot that originally was square
Consider the side of the lot = x
The area of the square = Length × length
Therefore,
x × x = [tex]x^2[/tex]
one of the adjacent streets which is widened by 3 m and the other was widened by 5
Therefore
(x-3) × (x-5) = [tex]x^2[/tex] - 185
open the brackets
[tex]x^2-5x-3x+15=x^2-185[/tex]
The x square term will cancel each other
-5x-3x = -185-15
-8x = -200
x = 25 meter
The new length of the lot = x-3
= 25-3
= 22 meter
The new width of the lot = 25-5
= 20 meters
Hence, the new dimensions of the lot is length = 22 meters and width = 20 meters
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If 1/2 cup of water fills 2/3 of a plastic container, how many containers will 1 cup full?
1 cup will fills double than 1/2 cup, so:
[tex]\frac{2}{3}\cdot2=\frac{4}{3}[/tex]With 1 cup you will be able to fills 4/3, in other words 1 full container and 1/3 of another.
Help me please ?
anyone ?
Mathematics .
The length of the midsegment of trapezoid ABCD is 6.23 units.
In this question, we have been given a trapezoid ABCD with vertices A(0,0), B(2, 5), C(3, 5) and D(8, 0)
We need to find the length of the midsegment of trapezoid.
We know that the length of the midsegment is one-half the sum of the lengths of the bases.
The length of base AB,
AB = √(2 - 0)² + (5 - 0)²
AB = √4 + 25
AB = √29
AB = 5.38
The length of base CD would be,
CD = √(8 - 3)² + (0 - 5)²
CD = √5² + (-5)²
CD = √25 + 25
CD = 5√2
So, the length of the midsegment would be,
l = 1/2[(5.38 + 5√2)]
l = 6.23
Therefore, the length of the midsegment of trapezoid ABCD is 6.23 units.
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1.An integer is 3 less than 5 times another. If the product of the two integers is 36, then find the integers
integer 1 = x
Integer 2 = y
An integer is 3 less than 5 times another
x-3 = 5y
A baseball diamond is square. The distance from home plate to first base is 90 feet. In feet, what is the distance from home plate to second base?
Options: 127, 81, 19, 13
The distance from home plate to second base is 90[tex]\sqrt{2}[/tex] ≈ 127.3 feet so option 1 is correct.
Given :
A baseball diamond is square.
The distance from home plate to first base is 90 feet which is nothing but side of square.
Let x be the distance from home plate to second plate.
we need to find home plate to second plate distance which is nothing but diagonal of square.
with 2 sides of length 90 feet and one diagonal forms a right angled triangle.
According to pythagorean theory:
[tex]90^{2}[/tex] + [tex]90^{2}[/tex] = [tex]X^{2}[/tex]
[tex]X^{2}[/tex] = 8100 + 8100
[tex]X^{2}[/tex] = 16200
[tex]X^{}[/tex] = 90*[tex]\sqrt{2}[/tex] ≈ 127.3 feet
Therefore the distance from home plate to second base is 90[tex]\sqrt{2}[/tex] ≈ 127.3 feet so option 1 is correct.
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Put the following equation of a line into slope-intercept form, simplifying allfractions.4x + 10y = –20
Given the equation of the line:
[tex]4x+10y=-20[/tex]The general slope - intercept form is :
[tex]y=m\cdot x+b[/tex]Where: m is the slope and b is y - intercept
So, we will transform the given equation to the slope - intercept form as following:
[tex]\begin{gathered} 4x+10y=-20 \\ \\ 10y=-4x-20 \end{gathered}[/tex]Divide both sides by ( 10):
[tex]\begin{gathered} \frac{10y}{10}=\frac{-4x-20}{10} \\ \\ y=-\frac{4}{10}x-\frac{20}{10} \\ \\ y=-\frac{2}{5}x-2 \end{gathered}[/tex]So, the slope - intercept form of the given equation is :
[tex]y=-\frac{2}{5}x-2[/tex]Line P parallel to side AB of triangle ABC, intersects AC and BC at points D and E, respectively. Find the area of triangle ABC if area of ABD=a and area of AEC=b
The area of triangle ABC if area of ABD = a and area of AEC = b is; (b + a) sq. units
What is the Area of the Triangle?The formula for the area of a triangle is;
Area = ¹/₂ × base × height
Thus;
Area of Triangle ADB is;
Area of ΔADB = ¹/₂ × AB * h_d
The area of the triangle AEB is;
Area of ΔAEB = ¹/₂ × AB * h_e
Due to the fact that AB and p are parallel lines, then it means we can say that; h_d = h_e
Therefore by substitution property, we can say that;
Area of ΔADB = Area of ΔAEB
Thus;
Area of Triangle ABC = Area of ΔADB + Area of ΔAEB = (b + a) sq. units
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spent $95 buying 13 books to donate to the local elementary school. Activity books cost $5 each and story books cost $11 each, how many of each type of book did purchase?
Answer:
Number of Activity books = 8
Number of Storybooks = 5
Explanation:
Let x represent the number of activity books.
Let y represent the number of storybooks.
Let's go ahead and set up our equations as follows;
[tex]\begin{gathered} x+y=13\ldots\ldots\text{.}\mathrm{}\text{Equation 1} \\ 5x+11y=95\ldots\ldots\ldots\ldots\text{.Equation 2} \end{gathered}[/tex]From equation 1, we can see that x = 13 - y
Let's go ahead and substitute the value of x into equation 2 and solve for y;
[tex]\begin{gathered} 5(13-y)+11y=95 \\ 65-5y+11y=95 \\ 6y=30 \\ y=\frac{30}{6} \\ y=5 \end{gathered}[/tex]Since y = 5, let's substitute the value of y into equation 1 and solve for x;
[tex]\begin{gathered} x+5=13 \\ x=13-5 \\ x=8 \end{gathered}[/tex]9x+23y=10 rewrite the equation so that y is a function of x (explanation pls)
which could be a graph of y=ax^4 + bx^3 + cx^2 + dx + e where a,b,c,d and e are real numbers not equal to 0 and a<0?
y=ax^4 + bx^3 + cx^2 + dx + e
Since a< 0 we know that it has to go to - infinity for x going to - infinity and x going to + infinity. That rules out choices B and C.
The maximum number of turning points a polynomial can have is n-1 where n is the degree of the polynomial
(turning points is change of direction)
The graph D has 5 turning points and the degree of the polynomial is 4 so it is not possible
A is your solution.
Accurately sketch or graph: f(x) = x^4/4 - 2x^3/3 -x^2/2 + 2xon the interval [ -2.5, 2.3] using the graph paper provided.
Graphing the function f(x) over the interval [-2.5, 2.3], we have the following
x-intercepts
Judging on the graph, the x-intercept of the function f(x) are the following
[tex]x=-1.619\text{ and }x=0[/tex]critical points
Finding the first derivative and setting it to zero and solve for x to get the critical points in the interval [-2.5,2.3] we have
[tex]\begin{gathered} f\mleft(x\mright)=\frac{x^4}{4}-\frac{2x^3}{3}-\frac{x^2}{2}+2x \\ f^{\prime}\left(x\right)=x^3-2x^2-x+2 \\ \\ x^3-2x^2-x+2=0 \\ \left(x−2\right)\left(x+1\right)\left(x−1\right)=0 \end{gathered}[/tex]Therefore, the critical points are x = 2, x = 1, x = -1.
relative minimums
Basing on the graph our relative minimum is at x = -1, and x = 2
asymptotes
Since the given function is a polynomial function, the function has no asymptotes
critical numbers
same as critical points with x = 2, x = 1, and x = -1.
relative maximum
observing the graph, the relative maximum of the function is at x = 1.
inflection points
get the second derivative of the function and set it to zero
[tex]\begin{gathered} f^{\prime}\left(x\right)=x^3-2x^2-x+2 \\ f^{\prime}^{\prime}\left(x\right)=3x^2-4x-1 \\ 3x^2-4x-1=0 \end{gathered}[/tex]Using the quadratic formula we have the values at
[tex]x=\frac{2}{3}\pm\sqrt{\frac{7}{3}}[/tex]end behavior
Since the function is a polynomial with a degree of 4, and a positive leading coefficient, the end behavior of the function is increasing in both approaching to -2.5, and 2.3.
find all the zerosf(x)=16x^5-72x^4+137x^3+43x^2-244x+120
For f(1), the given polynomial function f(x) = 16x⁵ - 72x⁴ + 137x³ + 43x² - 244x + 120 is zero.
Given,
The polynomial function;
f(x) = 16x⁵ - 72x⁴ + 137x³ + 43x² - 244x + 120
We have to all the zeros .
Here,
We can use Rational zero theorem;
That is,
f(x) = 16x⁵ - 72x⁴ + 137x³ + 43x² - 244x + 120
Find p/q
Where, p is 16x⁵ - 72x⁴ + 137x³ + 43x² - 244x + 120 and q is 16
So,
p/q = (±16, ±72, ±137, ±43, ±244, ±120) / ±16
= ±1, ±4.6875. ±8.5625, ±2.6875, ±15.25, ±7.5
Now,
f(1) = 1 × 1⁵ - 4.6875 × 1⁴ + 8.5625 × 1³ + 2.6875 × 1² - 15.25 × 1 + 7.5
f(1) = 1 - 4.6875 + 8.5625 + 2.6875 - 15.25 + 7.5
f(1) = 0
That is,
For f(1), the given polynomial function f(x) = 16x⁵ - 72x⁴ + 137x³ + 43x² - 244x + 120 is zero.
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Which function below is the inverse of f(x) = X2 - 25?
Answer: [tex]f^{-1}(x)= \sqrt{x+25}[/tex] and [tex]f^{-1}(x)= - \sqrt{x+25}[/tex]
Step-by-step explanation:
To find the inverse, we will set f(x) equal to y, swap the x and y values, and then solve for y.
Given:
f(x) = x² - 25
Equal to y:
y = x² - 25
Swamp x and y values:
x = y² - 25
Add 25 to both sides of the equation:
y² = x + 25
Square root both sides of the equation:
[tex]y= \sqrt{x+25}[/tex] and [tex]y= -\sqrt{x+25}[/tex]
Interval notation:
[tex]f^{-1}(x)= \sqrt{x+25},\text{and}\;f^{-1}(x)= -\sqrt{x+25}[/tex]
⇒Note for inverses we swap the positions between x and y.
⇒[tex]x=y^{2} -25\\[/tex]
But in this case we know that we write the equation such that the variable for the input is the subject of the equation
[tex]y^{2}= x+25\\y=\sqrt{x+25}[/tex]
Note y in this case was representing f(x)
⇒therefore our final equation now is
[tex]f(x)=\sqrt{x+25}[/tex]
GOODLUCK!!!
Given the numbers -100, -14.5, -2, -2/3, 0, 10,15, 20 1/6, which are rational numbers?
A rational number is a number that can be written in the form of p/q where q is not equal to zero.
The rational numbers are -100, -14.5, -2, -2/3, 0,10, 15, 20 1/6.are rational.
What is a rational number?A rational number is a number that can be written in the form of p/q where q is not equal to zero.
We have,
-100, -14.5, -2, -2/3, 0, 19, 15, 20, 1/6
-100 = -100/1
It is in the form of p/q.
It is a rational number.
-14.5 = -145/10
It is in the form of p/q.
It is a rational number.
-2 = -2/1
It is in the form of p/q.
It is a rational number.
-2/3 = -2/3
It is in the form of p/q.
It is a rational number.
0 = 0/1
It is in the form of p/q.
It is a rational number.
19 = 19/1
It is in the form of p/q.
It is a rational number.
15 = 15/1
It is in the form of p/q.
It is a rational number.
20 = 20/1
It is in the form of p/q.
It is a rational number.
1/6
It is in the form of p/q.
It is a rational number.
We see that all the numbers given are rational numbers.
Thus,
The rational numbers are -100, -14.5, -2, -2/3, 0, 10, 15, 20 1/6.
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