In this case, Cai is right.
Basically, Cai is right because a ratio is a fraction. So, if you divide the numerator and denomirator by the same number, the fraction won't be changed, in that case you would get an equivalent fraction.
For example, if we have 4/6, and we divide both numbers by 2, we get 2/3, these operations are valid because you are dividing both numbers by the same (2).
The diagram shows two similar polygonsN51P224048MR3016.5SFigures not drawn to scale.What is the length of CS?
Notice the correspondence between the vertices of the polygons:
[tex]VQRGX\approx CNPMS[/tex]Corresponding segments of similar polygons are proportional. Then:
[tex]\frac{CS}{VX}=\frac{PM}{RG}[/tex]Substitute VX=48, PM=22 and RG=16.5 and solve for CS:
[tex]\begin{gathered} \Rightarrow\frac{CS}{48}=\frac{22}{16.5} \\ \Rightarrow CS=\frac{22}{16.5}\times48 \\ \Rightarrow CS=64 \end{gathered}[/tex]Therefore, the length of CS is 64.
A right triangle is shown in the figure what is the value of x
So,
We could use the pythagorean theorem as follows:
[tex](3x)^2+x^2=(\sqrt[]{40})^2[/tex]And then solve for x:
[tex]\begin{gathered} 10x^2=40 \\ x^2=\frac{40}{10} \\ x^2=4 \\ x=2 \end{gathered}[/tex]Therefore, x=2.
At a local school, 164 students play soccer and 112 students play baseball. What is the ratio of soccer players to baseball players?41:2828:4113:2828:13
Given
The number of students who play soccer is 164.
The number of students who play baseball is 112
Explanation
To find the ratio of soccer player to baseball players .
Divide the number of soccer player by the number of baseball player.
[tex]\frac{164}{112}=\frac{41}{28}[/tex]Answer
Hence the ratio of soccer players to baseball players is
[tex]41:28[/tex]Expand the polynomial. 1. (m^2-n)(m^2+2n^2)2. (a-2)(4a^3-3a^2)
1)
The given polynomial is
[tex](m^2-n)(m^2+2n^2)[/tex]Multiply as follows:
[tex](m^2-n)(m^2+2n^2)=m^2(m^2+2n^2)-n(m^2+2n^2)[/tex][tex]=m^2\times m^2+m^2\times2n^2-n\times m^2-n\times2n^2[/tex][tex]=m^4+2m^2n^2-m^2n-2n^3[/tex]Hence the required expansion is
[tex](m^2-n)(m^2+2n^2)=m^4+2m^2n^2-m^2n-2n^3[/tex]2)
The given polynomial is
[tex](a-2)(4a^3-3a^2)[/tex]Multiply as follows:
[tex](a-2)(4a^3-3a^2)=a(4a^3-3a^2)-2(4a^3-3a^2)[/tex][tex]=a\times4a^3-a\times3a^2-2\times4a^3-(-2)\times3a^2[/tex][tex]=4a^4-3a^3-8a^3+6a^2[/tex][tex]=4a^4-11a^3+6a^2[/tex]Hence the required expansion is
[tex](a-2)(4a^3-3a^2)=4a^4-11a^3+6a^2[/tex]TRIGONOMETRY Find the length of the longer diagonal of this parallelogram round to the nearest tenth
Given the parallelogram ABCD
As shown: AB = 4 ft
m∠BAC = 30
m∠BDC = 104
We will find the length of the longer diagonal which will be AC
See the following figure:
The point of intersection of the diagonals = O
The opposite sides are parallel
AB || CD
m∠ABD = m∠BDC because the alternate angles are congruent
So, in the triangle AOB, the sum of the angles = 180
m∠AOB = 180 - (30+104) = 46
We will find the length of OA using the sine rule as follows:
[tex]\begin{gathered} \frac{OA}{\sin104}=\frac{AB}{\sin 46} \\ \\ OA=AB\cdot\frac{\sin104}{\sin46}=4\cdot\frac{\sin104}{\sin46}\approx5.3955 \end{gathered}[/tex]The diagonals bisect each other
So,
[tex]AC=2\cdot OA=10.79[/tex]The longer diagonal is AC
Rounding to the nearest tenth
So, the answer will be AC = 10.8 ft
An object was dropped off the top of a building. The function f(x) = -16x2 + 36represents the height of the object above the ground, in feet, X seconds after beingdropped. Find and interpret the given function values and determine an appropriatedomain for the function.
f(x) = -16x^2 + 36
Where:
f(x) = height of the object
x = seconds after being dropped.
f(-1) = -16 (-1)^2 + 36
f(-1) = -16 (1) + 36
f(-1) = 20
-1 seconds after the object was dropped, the object was 20 ft above the ground.
This interpretation does not make sense, because seconds can't be negative.
f(0.5) = -16 (0.5)^2 + 36
f(0.5) = -16 (0.25) +36
f(0.5) = -4 + 36
f(0.5) = 32
0.5 seconds after the object was dropped, the object was 32 ft above the ground.
This interpretation makes sense in the context of the problem.
f(2) = -16 (2)^2 + 36
f(2) = -16 (4) +36
f(2) = -64+36
f(2) = -28
2 seconds after the object was dropped, the object was -28 ft above the ground.
This interpretation does not make sense in the context of the problem, because the height can't be negative.
Based on the observation, the domain of the function is real numbers in a <- x <-b , possible values of x where f(x) is true.
before the object is released x=0
next, calculate x when f(x)=0 ( after the object hits the ground)
0= -16x^2+36
16x^2 = 36
x^2 = 36/16
x^2 = 2.25
x = √2.25
x = 1.5
0 ≤ x ≤ 1.5
Hello I need help with this question as fast as possible please , I am on my last few questions and I have been studying all day for my final exam for math tomorrow. It is past my bed time and I am exhausted . Thank you so much for understanding:))
Notice that:
[tex]665.6=10.4\times64.[/tex]Therefore, if we divide the resulting equation of step 3 by 10.4 we get:
[tex]\begin{gathered} \frac{10.4x}{10.4}=\frac{665.6}{10.4}, \\ x=64. \end{gathered}[/tex]Then the missing step is:
Divide both sides of the equation by 10.4.
Answer: Last option.
heyy could you help me out with this problem I'm stuck
Since congruent angles are equal
Therefore the two figures are similar
we have
9 / 9 = 2x / x + 4
introduce cross multiplication
9 (2x) = 9(x + 4)
18x = 9*x + 9*4
18x = 9x + 36
collect the like terms
18x - 9x = 36
9x = 36
divide boths sides by 9
9x / 9 = 36/9
x = 4
The first missing variable is 2x
2 x 4
= 8
The second is x + 4
we have 4 + 4
= 8
m
The figure below is an iscoceles trapezoid. If m
..Given an isosceles trapezoid
The following are the properties of an isosceles trapezoid
The legs are congruent by definition (From the diagram, the legs are JM and KL)
The lower base angles are congruent. The lower base angles are
[tex]m\angle M\cong m\angle L[/tex]The upper base angles are congruent. The upper base angles are
[tex]m\angle J\cong m\angle K[/tex]Any lower base angle is supplementary to any upper base angle. This means that
[tex]\begin{gathered} m\angle J+m\angle M=180^0 \\ m\angle K+m\angle L=180^0 \end{gathered}[/tex][tex]\begin{gathered} \text{If} \\ m\angle K=61^0 \\ \text{Therefore} \\ m\angle J\cong m\angle K=61^0 \\ m\angle J=61^0 \end{gathered}[/tex]Also,
[tex]\begin{gathered} m\angle L+m\angle K=180^0 \\ m\angle L+61^0=180^0 \\ m\angle L=180^0-61^0 \\ m\angle L=119^0 \end{gathered}[/tex][tex]\begin{gathered} m\angle L\cong m\angle M,m\angle L=119^0 \\ Therefore\colon \\ m\angle M=119^0 \end{gathered}[/tex]Hence
m∠J = 61⁰
m∠L = 119⁰
m∠M = 119⁰
What is the unknown angle b?
Answer:
48°
Explanation:
A line always equals 180°. The angle on the right is a 90° angle (we know this because or the little red box shown) and the angle in the middle is 42°. We would add 42° and 90° to get the combination of both which is 132°
42+90=132
Then subtract 132° from 180° to find unknown angle b.
180-132=48
Unknown angle b= 48°
To produce a textbook, suppose the publisher spent $110,000 for typesetting and $7.50 per book for printingand binding. The total cost to produce and print n books can be written asC = 110,000+ 7.51a. Suppose the number of books printed in the first printing is 10,000. What is the total cost?The total cost is $b. If the average cost is the total cost divided by the number of books printed, find the average cost of producing10,000 textbooks.The average cost of producing 10,000 textbooks is $c. Find the cost to produce one more textbook when you have already produced 10,000 textbooks.If you have already produced 10,000 textbooks, it'll cost you $ to produce one more.
The given equation is
[tex]C=110000+7.5n[/tex]a)
The number of books=10000
Substitute n=10000 in the given equation, we get
[tex]C=110000+7.5\times10000[/tex][tex]C=185000[/tex]The total cost is $185,000.
b)
[tex]\text{Average cost =}\frac{185000}{10000}=18.5[/tex]The average cost of producing 10,000 textbooks is $18.5.
c)
If we need to produce one more after producing 10000 books.
substitute n=10001 in the given equation, we get
[tex]C=110000+7.5(10001)=185007.5[/tex][tex]\text{One book cost after printed 10000 book=cost of 10000 books-cost of 10001 books}[/tex][tex]\text{One book cost after printed 10000 book=1}85000-185007.5=7.5[/tex]
If you have already produced 10,000 textbooks, it'll cost you $7.5 to produce one more.
What is 4 1/10 equal to
We are given the following mixed fraction:
[tex]4\frac{1}{10}[/tex]This is a fraction of the form:
[tex]a\frac{b}{c}[/tex]Any mixed fraction can be rewritten using the following relationship:
[tex]a\frac{b}{c}=a+\frac{b}{c}[/tex]Applying the relationship we get:
[tex]4\frac{1}{10}=4+\frac{1}{10}[/tex]Now, we add the whole number and the fraction using the following relationship:
[tex]a+\frac{b}{c}=\frac{ac+b}{c}[/tex]Applying the relationship we get:
[tex]4+\frac{1}{10}=\frac{40+1}{10}=\frac{41}{10}=4.1[/tex]Therefore, the mixed fraction is equivalent to 4.1
K
Use the data in the following table, which lists drive-thru order accuracy at popular fast food chains. Assume that orders are randomly selected from those included in
the table.
Drive-thru Restaurant D
B C
D
280
245
122
60
32
12
A
Order Accurate
331
Order Not Accurate 38
If one order is selected, find the probability of getting an order that is not accurate or is from Restaurant C. Are the events of selecting an order that is not accurate and
selecting an order from Restaurant C disjoint events?
The probability of getting an order from Restaurant C or an order that is not accurate is
(Round to three decimal places as needed.)
Are the events of selecting an order from Restaurant C and selecting an inaccurate order disjoint events?
disjoint because it
possible to
The events
The probability is 0.236 and the events are not disjoint events
Given,
The data;
A B C D
Order accurate ; 280 245 122 331
Order not accurate; 60 32 12 38
The probability of getting an order that is not accurate or is from Restaurant C
This is illustrative of
P(Not accurate or Restaurant C) (Not accurate or Restaurant C)
The calculation is
P(Not accurate or Restaurant C) is equal to [n(Not accurate) + (Not accurate and Restaurant C) - n(Restaurant C)] /Total
Thus, we have
P(Not accurate or Restaurant C) is calculated as follows: (60 + 32 + 12 + 38 + 122 + 12 - 12)/(280 + 245 + 122 + 331 + 60 + 32 + 12 + 38).
Analyze the difference and the total.
Restaurant C or P(Not accurate) = 264/1120
Assess the quotient.
P(Not accurate or Restaurant C) = 0.236
Last but not least, choosing an incorrect order and choosing an order from Restaurant C are not separate events.
This is because choosing an inaccurate order from restaurant C is a possibility.
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3
Drag each tile to the correct box.
Place the parallelograms in order from least area to greatest area.
3 cm
4 cm
6 cm
3 cm
4 cm
5 cm
4 cm
3 cm
----
4 cm
Submit Test
}
The least area of the parallelogram will be 12cm² and the greatest area will be 20cm².
What will be the area of the parallelogram?The area of a parallelogram is simply calculated thus:
= Base × Height
The least area will be:
= Base × Heights
= 3cm × 4cm
= 12cm²
The greatest area of the parallelogram will be:
= Base × Height
= 4cm × 5cm
= 20cm²
Note that the figures are gotten from the. information given.
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Express y in terms of x. Then find the value of y when x= -1-3 (x + 2) = 5yY in terms of x:Y=
LEt's express y in term of x:
[tex]\begin{gathered} -3(x+2)=5y \\ y=\frac{-3(x+2)}{5} \end{gathered}[/tex]Therefore:
[tex]y=-\frac{3}{5}x-\frac{6}{5}[/tex]Now, if x=-1, then we have:
[tex]\begin{gathered} y=-\frac{3}{5}(-1)-\frac{6}{5} \\ =\frac{3}{5}-\frac{6}{5} \\ =-\frac{1}{5} \end{gathered}[/tex]Therefore, if x=-1 then y=-1/5
The first five multiples for the numbers 4 and 6 are shown below.Multiples of 4: 4, 8, 12, 16, 20Multiples of 6: 6, 12, 18, 24, 30,What is the least common multiple of 4 and 6?241224
We have
Multiples of 4: 4, 8, 12, 16, 20
Multiples of 6: 6, 12, 18, 24, 30,
the least common multiple is the first number share between these numbers as we can see the first number share is 12
Find a polynomial f(x) of degree 4 with real coefficients and the following zeros.3 (multiplicity 2) , -i
We are told that we want a polynomial f(x) with the given zeros.
Recall that if we know the zeros oa polynomial, we can write the polynomial by writing the factors (x - zero of the polynomial) and multiply them all together.
For example, if we want a polynomial of degree 2 with zeros at 2 and 3, then the polynomial would be
[tex](x\text{ -2)}\cdot(x\text{ -3)}[/tex]In this case, we have a polynomial f(x) of degree 4. So far, we know that 3 is a zero and that -i is a zero. So we write the following
[tex]f(x)=(x\text{ -a)}\cdot(x\text{ -b)}\cdot(x\text{ -c) }\cdot\text{ (x -d)}[/tex]where a,b,c and d are the zeros of f(x). We know that 3 is a zero and that -i is a zero. So we have
[tex]f(x)=(x\text{ -3)}\cdot(x\text{ -b)}\cdot(x\text{ -( -i)) }\cdot(x\text{ -d)}[/tex]So to fully describe f(x) we need to find the values of b and d. We are told that 3 is a zero of multiplicity 2. This means that the factor (x -3) appears two times in the factorization of f(x). So we can say that b=3. So we have
[tex]f(x)=(x\text{ -3)}\cdot(x\text{ -3) }\cdot(x\text{ +i ) }\cdot(x-d)=(x-3)^2\cdot(x\text{ +i)}\cdot(x\text{ -d)}[/tex]Now, we need to find the value of d. Note that we are told that -i is a zero of the function. -i is a complex number, so one important property of polynomials is that if a complex number a+bi is a zero of the polynomial, then the number a-bi (which is called the complex conjugate) is also a zero. Note that the complex conjugate of a complex number is calculated by leaving the real part intact and multiplying the imaginary part by -1.
In our case we have the complex number -i. So we can write -i= 0 - 1i . Then, its complex conjugate is i.
So, we have that d=i.
Then our polynomial would look like this
[tex]f(x)=(x-3)^2\cdot(x+i)\cdot(x\text{ -i)}[/tex]Note that
[tex](x+i)\cdot(x-i)=x^2\text{ -i}\cdot x\text{ + i}\cdot x+1=x^2+1[/tex]So our polynomial ends up being
[tex]f(x)=(x-3)^2\cdot(x^2+1)[/tex]Not a timed or graded assignment. Need a quick answer tho. Thank you
ANSWER:
Difference of squares
[tex]8x-7[/tex]STEP-BY-STEP EXPLANATION:
We have the following quotient:
[tex]\frac{64x^2-49}{8x+7}[/tex]We factor, knowing that the numerator is a difference of squares, therefore:
[tex]\begin{gathered} a^2-b^2=(a+b)(a-b) \\ \text{ in this case} \\ a=8x \\ b=7 \\ 64x^2-49=(8x+7)(8x-7) \\ \text{ Replacing:} \\ \frac{(8x+7)(8x-7)}{8x+7}=8x-7 \end{gathered}[/tex]Tell whether the triangle with the given side lengths is a right triangle. 18, 80, 82 Write the pythagorean theorem Substitute the values from the triangle in the equation then solve If both side of the equation is the same then yes the triangle is right triangle. Il both sides are different then no the triangle is not a right triangle
Pythagorean theorem :
c^2 = a^2 + b^2
Where:
c = hypotenuse (the longest side ) = 82
A & b = the other 2 sides (18 and 80)
Replacing:
82^2 = 80^2 + 18^2
Solve:
6,724 = 6,400+ 324
6,724 = 6,724
Both sides of the equations are equal. IT is a right triangle
I wonder what I’m doing wrong ?
P^2-10p+16=1+6p
Answer is (15,1)
But I can’t seem to figure it out.
Steps to Solve:
1. Collect like terms
2. Factor quadratic
3. solve for p
Factoring a Quadratic where a = 1
1. find two numbers that are the product of ac and the sum of b
2. set up the two linear terms with the variable associated with a
2. insert the values found in step 1 into parentheses
1. collect like terms
[tex]p^2-10p-6p+15-1=0[/tex]
[tex]p^2-16p+15 = 0[/tex]
2. Factor quadratic
ac = 15 and b =-16, two numbers that multply to ac and are the sum of b are -15 and 1[tex](p-15)(p-1)=0[/tex]
2 solutions can be found[tex]p-15=0[/tex] OR [tex]p-1=0[/tex]
[tex]p= 15[/tex] [tex]p=1[/tex]
Solving a Debra, Ravi, and Ahmad sent a total of 76 text messages during the weekend. Ahmad sent 2 times as many messages as Ravi. Debra sent 8 more messages than Ravi. How many messages did they each send?
Let the number of messages sent by Ravi be x.
Ahmad sent 2 times as many messages as Ravi. Therefore, Ahmad sent 2x messages.
Debra sent 8 more messages than Ravi. Therefore, Debra sent (x + 8) messages.
The sum of all the messages is 76:
[tex]x+2x+x+8=76[/tex]Solving for x, we have:
[tex]\begin{gathered} 4x+8=76 \\ 4x=76-8 \\ 4x=68 \\ x=\frac{68}{4} \\ x=17 \end{gathered}[/tex]The number of messages Ahmad sent will be:
[tex]2(17)=34[/tex]The number of messages Debra sent will be:
[tex]17+8=25[/tex]ANSWER
[tex]\begin{gathered} Debra\to25\text{ }messages \\ Ravi\to17\text{ }messages \\ Ahmad\to34\text{ }messages \end{gathered}[/tex]Solve for x: 3x + 2 = 11 A : 11/5 B: 3. C : 11/3. D : 13/3. E : 6
Explanation:
The equation is given as:
3x + 2 = 11
The first step is to collect like terms ( Note that if 2 crosses to the other side of the equality sign, it becomes -2)
3x = 11 - 2
3x = 9
The next step is to divide both sides by 3:
3x/3 = 9/3
x = 3
A long distance runner runs 2⁵ miles one week and 2⁷ miles the next week. How many times farther did he run in the second week than the first week?
Answer:
he ran 96 miles farther in the second week.
Explanation:
Given that A long distance runner runs 2⁵ miles one week;
[tex]2^5miles=2\times2\times2\times2\times2=32miles[/tex]And 2⁷ miles the next week;
[tex]2^7miles=2\times2\times2\times2\times2\times2\times2=128\text{ miles}[/tex]The amount of miles farther he run in the second week than the first week is;
[tex]\begin{gathered} 128-32 \\ =96\text{ miles} \end{gathered}[/tex]Therefore, he ran 96 miles farther in the second week.
The world's largest swimming pool is the Orthalieb pool in Casablanca, Morocco the length is 30 m longer then 6 times the width. If the perimeter of the pool is 1110 Meters what are the dimensions of the pool?
The length of the rectangular pool is 30m longer than 6 times the width.
Let "x" represent the length of the width, then you can express the dimensions of the pool as follows:
[tex]\begin{gathered} w=x \\ l=6x+30 \end{gathered}[/tex]The perimeter of the pool is 1110m, this perimeter was obtained using the formula:
[tex]P=2w+2l[/tex]Replace the formula with the expressions determined for the width and length:
[tex]1110=2(x)+2(6x+30)[/tex]From this expression, you can determine the value of x:
-First, distribute the multiplications on the right side of the equation:
[tex]\begin{gathered} 1110=2x+2\cdot6x+2\cdot30 \\ 1110=2x+12x+60 \\ 1110=14x+60 \end{gathered}[/tex]-Second, pass 60 to the left side of the equal sign by applying the opposite operation to both sides of it:
[tex]\begin{gathered} 1110-60=14x+60-60 \\ 1050=14x \end{gathered}[/tex]-Third, divide both sides of the equation by 14 to determine the value of x:
[tex]\begin{gathered} \frac{1050}{14}=\frac{14x}{14} \\ 75=x \end{gathered}[/tex]The width of the pool is w= 75 meters
Now you can determine the length of the pool:
[tex]\begin{gathered} l=6x+30 \\ l=6\cdot75+30 \\ l=480 \end{gathered}[/tex]The length of the pool is l=480 meters
Ms. Morgan is the cafeteria manager. She keeps track of how many students select each type of drink. Today during breakfast, 32 children picked milk while 44 children picked juice. What is the ratio of the numbe of children who picked juice to those who picked milk?
Answer:
ratio of those who picked juice to milk
it refers to division
help me pleaseeeeeeeee
The values of given functions f(-2), f(0) and f(7) when f(x) = 1-6x are 13, 1 and -41 respectively.
According to the question,
We have the following function:
f(x) = 1-6x
Now, we can find the values of each function by putting the numbers in place of x.
Now, in order to find the value of f(-2), we will put -2 in place of x in the given function.
f(-2) = 1-6*(-2)
f(-2) = 1+12
f(-2) = 13
Now, in order to find the value of f(0), we will put 0 in place of x in the given function.
f(0) = 1-6(0)
f(0) = 1-0
(We know that when a number is multiplied with 0 then the result is always 0.)
f(0) = 1
Now, in order to find the value of f(7), we will put 7 in place of x in the given function.
f(7) = 1-6*7
f(7) = 1-42
f(7) = -41
Hence, the values of f(-2), f(0) and f(7) are 13, 1 and -41 respectively.
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plant A produced 3 times as many panels as plant b. two percent of the panels from plant A and 5% of the panels from plant b were defective. how many panels did plant b produce if the two plants together produced 990 defective panels
let x the number of panels that Plant A produced
y the number of panels that Plant B produced
then, we have
x = 3y
0.02x + 0.05y = 990
and solve the system:
[tex]\begin{gathered} 0.02(3y)+0.05y=990 \\ 0.06y+0.05y=990 \\ 0.11y=990 \\ \frac{0.11y}{0.11}=\frac{990}{0.11} \\ y=9000 \end{gathered}[/tex]answer: plant b produced 9000 panels
I need help understanding slope
we know that
the formula to calculate teh slope between two points is equal to
[tex]m=\frac{(y2-y1)}{(x2-x1)}[/tex]where
(x1,y1) is one point
and
(x2,y2) is the other point
substitute the values in the formula and solve for m
Example
you have the points
(1,4) and (3,2)
so
(x1,y1)=(1,4)
(x2,y2)=(3,2)
substitute in te formula
m=(2-4)/(3-1)
m=-2/2
m=-1
the slope is -1
how do i solve the equation?
Answer: 7x=63 and 12x+9= 117
Step-by-step explanation:
add those two equations and set it to 180 degree
7x+12x+9=180
19x=171
x= 9
7x = 7 (9) = 63
12x+9 = 12 (9)+9 = 117
In may 2020, there were 2,119,800 people in the available labor force in oregon. the unemployment rate in oregon for may 2020 was 14.2%. determine the number of people who were unemployed in oregon during may 2020. round your answer to the nearest whole number.
301,012
1) Considering the data, we can write out the following equivalence:
[tex]\begin{gathered} 2,119,800---100\% \\ x--------14.2\% \end{gathered}[/tex]2) Notice that we need to rewrite those figures as 100% =1, and 14.2% as 0.142.
Now we can calculate how many people are equivalent to that rate of unemployment:
[tex]\begin{gathered} \frac{2119800}{x}=\frac{1}{0.142} \\ x=2119800\cdot0.142 \\ x=301,011.6\approx301,012 \end{gathered}[/tex]Notice that we rounded off to the nearest whole number.
3) So, there were approximately 301,012 Oregonians unemployed at that time