One of the forms we can write the equation of a line is like this:
y - y1 = m(x - x1)
Where (x1, y1) is a point where the line passes through, and the value of m, the slope of a line is given by the following formula:
[tex]m=\frac{y2-y1}{x2-x1}[/tex]Then, for the first line (line q), we can take the points (-9, 0) and (-3, 18), then we get:
[tex]mq=\frac{18-0}{-3-(-9)}=\frac{18}{-3+9}=\frac{18}{6}=3[/tex]By replacing the value of the slope and the coordinates of the point (-9, 0), we get:
y - 0 = 3(x - (-9))
y = 3(x + 9)
y = 3x + 27
y - y = 3x + 27 - y
0 = 3x + 27 - y
-27 = 3x + 27 - 27 - y
-27 = 3x - y
For line v, we can take the points (-4, 10) and (0, 8), then we get:
[tex]mv=\frac{8-10}{0-(-4)}=\frac{-2}{4}=-\frac{1}{2}[/tex]By taking -1/2 for the slope and the coordinates of the point (0,8), we gat:
y - 0 = -1/2(x - 8)
y = -1/2x + 4, multiplying both sides by 2:
2y = -x + 16
2y + x = -x + x + 16
2y + x = 16
Then, the system represented by the lines q and v is option G
Suppose the population of a small town increases from 12,334 to 16,049. What is the percentincrease in the population, rounded to the nearest tenth of a percent?
First, we compute the population increase:
[tex]16049-12334=3715.[/tex]Now we divided the above result by the initial population:
[tex]\frac{3715}{16049}\approx0.23147.[/tex]Finally, we multiply by 100 and round to the nearest tenth of percent:
[tex]100\cdot0.23147=23.147\approx23.1.[/tex]Answer: 23.1%
Condense the left-hand side into a single logarithm.log(4) + log(2) + log(y) = log(____)
Condensation of left-hand side of the given expression into a single logarithm is :log(4) + log(2) + log(y) = log(8y)
How to condense log?To evaluate logarithmic expressions, methods for condensing logarithms in order to rewrite multiple logarithmic terms into one can be used. it is a useful tool for the simplification of logarithmic terms. To condense logarithms we use the rules of logarithms: the product rule, the quotient rule and the power rule.
According to the product laws of logarithm,
log x+log y = log (xy)
For the given expression,
log(4) + log(2) + log(y) = log(____)
So, after applying the product rule:
log(4) + log(2) + log(y) = log(4×2×y)
log(4) + log(2) + log(y) = log(8y)
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1) Movie Checkout for the week Days 2 3 4 A. 10 How many movies were checked out on Day 1? Number of Movies 30 45 60 C. 20 B. 15 D. 25
A barrel of oil is filled at a constant rate of 7.5 gal/min. The barrel had 12 gallons before filling began. Write the equation in standard form to model the linear situation
HELP ASAP 100 POINTS!!!!!!!!!
The points (6,2) and (10,4) fall on a particular line. What is its equation in point-slope form?
Use one of the specified points in your equation. Write your answer using integers, proper fractions, and improper fractions. Simplify all fractions.
Answer:
[tex]y-2=\dfrac{1}{2}(x-6)[/tex]
Step-by-step explanation:
To find the equation of a line that passes through two given points, first find its slope by substituting the given points into the slope formula.
[tex]\boxed{\begin{minipage}{4.4cm}\underline{Slope Formula}\\\\Slope $(m)=\dfrac{y_2-y_1}{x_2-x_1}$\\\\where $(x_1,y_1)$ and $(x_2,y_2)$ \\are two points on the line.\\\end{minipage}}[/tex]
Define the points:
(x₁, y₁) = (6, 2)(x₂, y₂) = (10, 4)Substitute the points into the slope formula:
[tex]\implies m=\dfrac{4-2}{10-6}=\dfrac{2}{4}=\dfrac{1}{2}[/tex]
Therefore, the slope of the line is ¹/₂.
[tex]\boxed{\begin{minipage}{5.8 cm}\underline{Point-slope form of a linear equation}\\\\$y-y_1=m(x-x_1)$\\\\where:\\ \phantom{ww}$\bullet$ $m$ is the slope. \\ \phantom{ww}$\bullet$ $(x_1,y_1)$ is a point on the line.\\\end{minipage}}[/tex]
To find the equation in point-slope form, substitute the found slope and one of the given points into the point-slope formula:
[tex]\implies y-2=\dfrac{1}{2}(x-6)[/tex]
I don't understand how to write the equations using this diagram.
Given the segment on the picture, we can write the following 3 equations:
[tex]\begin{gathered} PR=PQ+QR \\ QS=QR+RS \\ PT=PR+RT \end{gathered}[/tex]Hi, can you help me to solve this problem, please !!!
the y intercept of the curve of the function is 0.
y intercept is the point at which the curve of the function intersects the y axis.
here,
the curve intersects the Y axis at the origin.
coordinates of the origin are:
(0, 0)
So, the y intercept will be:
y = 0
Therefore, we get that, the y intercept of the curve of the function is 0.
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In a triangle ABC, a line BP is drawn from B so that P lies on the side AC. For the triangle, AP=27mm, BP=30mm, BC=82mm and the angle APB=76∘ apply. Determine the area of triangle ABC
SOLUTION
The diagram for this is shown below
From the diagram above, considering triangle BPC, let us find angle C
Using sine rule, we have
[tex]\begin{gathered} \frac{sinC\degree}{30}=\frac{sin104\degree}{82} \\ sinC=\frac{30\times sin104\degree}{82} \\ C=sin^{-1}\frac{30sin104}{82} \\ C=20.79\degree \end{gathered}[/tex]From the same triangle BPC to get angle B, we have
[tex]\begin{gathered} B+P+C=180\degree \\ B+104+20.79=180 \\ B+124.79=180 \\ B=55.21\degree \end{gathered}[/tex]From the same triangle BPC, using sine rule to get the side PC, which I called x, we have
[tex]\begin{gathered} \frac{sinB}{x}=\frac{sinP}{82} \\ \frac{sin55.21\degree}{x}=\frac{sin104\degree}{82} \\ x=\frac{sin55.21\times82}{sin104\degree} \\ x=69.404mm \end{gathered}[/tex]This makes the side AC to become
[tex]27+69.404=96.404mm[/tex]So, to get the area of the triangle ABC, we have that
[tex]Area=\frac{1}{2}|AC|\times|BC|\times sinC[/tex]Applying we have
[tex]\begin{gathered} Area=\frac{1}{2}|96.404|\times|82|\times sin20.79\degree \\ =1402.94mm^2 \end{gathered}[/tex]Hence the answer is approximately 1402.94 square-millimeters to the nearest hundredth
Hi I think I am doing this wrong. If I can get an explanation that would help a bunch!
Investment = $5,000
Price of the product or sale price = $3.49
Production cost = $2.16
Profit = 3.49 - 2.16 = $1.33
We need to sell X items to get our investment back, therefore
X * Profit = Investment
1.33 * X = 5,000
X = 5,000 / 1.33 =3759.4
Therefore you need to sell at least 3760 articles to break even
Additionally, if you want to make a $10,000 profit, then you need to sell those items plus Y more or in other words, you want to sell as many items to get your $5,000 investment plus $10,000
This is...
5,000 + 10,000 = K * (1.33)
15,000 = 1.33 * K
K = 15,000 / 1.33 = 11,278.2 ;
Therefore, you need to sell 11,279 items
Sammy has $125 saved from his summer lifeguarding job. Sammy has to take the bus to get to school this semester. Each week, Sammy uses $10.50 for bus fare. A. Function s represents the remaining amount, in dollars, in his savings account, as a function of the number of weeks, w, since school started. Complete the table: 6 weeks = 10 weeks = w = = is the dollars remaining in savings B. Use function notation to write a rule that defines function s C. Graph function s on a coordinate plane. D. Solve the equation s(w) = 41 and explain what the solution means.
The solution to the equations are
Equation: s = 125 - 10.5wTableWeeks (w) Amount remaining
6 62
10 20
See attachment for graphFunction notation: s(w) = 125 - 10.5ws(w) = 41 means that the amount remaining after 8 weeks is $41How to determine the equation of the scenario?From the question, we have the following parameters
Savings = $125
Bus fare = $10.5
The equation of the amount remaining is represented as
Amount remaining = Savings - Bus fare x Number of weeks
Substitute the known values in the above equation
So, we have the following equation
s = 125 - 10.5w
Where w represents the number of weeks
Also, we have
w = 6 and w = 10
So, we have
s = 125 - 10.5 * 6 = 62
s = 125 - 10.5 * 10 = 20
So, the table is
Weeks (w) Amount remaining
6 62
10 20
The function notationIn (a), we have
s = 125 - 10.5w
Express as a function notation
s(w) = 125 - 10.5w
The graph of the functionSee attachment for the graph
The solution and the interpretation of s(w) = 41We have
s(w) = 125 - 10.5w
This gives
125 - 10.5w = 41
So, we have
10.5w = 84
Evaluate
w = 8
This means that the amount remaining after 8 weeks is $41
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y = 3x – 5 y=-x-5Parallel, perpendicular or neither ?
The given equations are-
[tex]\begin{gathered} y=3x-5 \\ y=-x-5 \end{gathered}[/tex]These equations represent straight lines, where the coefficients of x are the slope of each one.
When two lines are parallel, they have the same slope, which is not the case here because the first one has a slope of 3, and the second one has a slope of -1.
Additionally, if two lines are perpendicular, their slopes are totally opposite in number and sign, which is not the case here because 3 is not opposite to -1.
Therefore, the answer is neig
The rent in an urban neighborhood is normally distributed. The mean is $650 per month with a standarddeviation of $50. Estimate the percent of apartment residents who pay from $600 to $750 per month.Show Your Work
Solution
The z-score, also referred to as standard score, z-value, and normal score, among other things, is a dimensionless quantity that is used to indicate the signed, fractional, number of standard deviations by which an event is above the mean value being measured. Values above the mean have positive z-scores, while values below the mean have negative z-scores.
Mean is $650
Standard deviation = $50
[tex]\begin{gathered} \mu=650 \\ \sigma=50 \end{gathered}[/tex][tex]Z=\frac{x-\mu}{\sigma}=\frac{600-650}{50}=-\frac{50}{50}=-1[/tex][tex]Z=\frac{x-\mu}{\sigma}=\frac{750-650}{50}=\frac{100}{50}=2[/tex][tex]P(-1Therefore the estimated percent of apartment residents = 81.86%
A box contains different-colored marbles. If P(blue) = P(green) = and P(blue and green) = whic
statement is true?
O The events are independent because P(blue) P(green) = P(blue and green).
Answer:
p(blue and green)
Step-by-step explanation:
the events are independent because p(blue) p(green)=p(blue and green).
A machine at the candy factory dispensed different numbers of lemon-flavored candies into various bags. Number of lemon-flavored candies Number of bags 29 1 69 4 108 5 4 118 5 162 4 167 2 186 X is the number of lemon-flavored candies that a randomly chosen bag had. What is the expected va of X? Write your answer as a decimal.
In order to calculate the expected value of x, we just need to divide the total number of lemon-flavored candies by the total number of bags, so we have:
[tex]\begin{gathered} E(x)=\frac{29\cdot1+69\cdot4+108\cdot5+118\cdot4+162\cdot5+167\cdot4+186\cdot2}{1+4+5+4+5+4+2} \\ E(x)=\frac{3167}{25} \\ E(x)=126.68 \end{gathered}[/tex]So the expected value of x is 126.68.
second drop-down options are can not be proven congruent and can be proven congruent
Solution
For this case the answer is:
Side-Angle-Side (SAS)
Since we have two sides congruent and also one angle
Find the value of x:a) 2+3−6=−5b) 2(2+1)=5−1
In order to find the value of x in the expression, we need to isolate the terms with the variable 'x' in one side of the equation, and the variables without 'x' in the other side.
Then, we divide the whole equation by the number multiplying 'x', so we have just 'x' in one side of the equation.
So we have that:
a)
[tex]\begin{gathered} 2+3x-6x=-5x \\ 3x-6x+5x=-2 \\ 2x=-2 \\ x=\frac{-2}{2}=-1 \end{gathered}[/tex]So the value of x is -1.
b)
[tex]\begin{gathered} 2\left(2+1\right)=5-1 \\ 4x+2=5x-1 \\ 4x-5x=-1-2 \\ -x=-3 \\ x=3 \end{gathered}[/tex]So the value of x is 3
can someone help me with this problem a triangle has a base of 2 meters and a height of 8 meters what is the area in square feet round your answer to the nearest tenth
Answer:
8.
Step-by-step explanation:
A little trick is to multiply 2 and 8, then divide by two because a triangle is half a square.
2 x 8 = 16
16 divided by 2 is 8.
Answer = 8
The Pederson family needed to cut expenses on their cell and landline phone service, cable television service, and Internet service. The best prices they found for basicservice, using different service providers, are listed below. They found a company with a package deal with all 4 services for $143.66. What is the percentage decrease bypurchasing the package deal? Cell Phone Service: $34.15 per month Landline Phone Service: $31.36 per month Cable TV Service: $56.25 per month Internet Service:$41.48 per month9.7%10.9%11.4%12.0%None of these choices are correct.
From the information given, the original costs per month were
Cell phone service $34.15
Landline phone service = $31.36
Cable TV service = $56.25
Internet service = 41.48
The total cost per month would be
34.15 + 31.36 + 56.25 + 41.48 = 163.24
For the new company that they found, the total cost is $1431
What is the perimeter of a rectangle with a length of:
3+√3
and a width of:
4+ √5
Answer:
7√8
Step-by-step explanation:
3+√3+4√5
3+4+√3+√5
7+√8
write the equation of the line that goes through points (1, 1) and (3, 7)
Given the points:
(x1, y1) ==> (1, 1)
(x2, y2) ==> (3, 7)
To find an equation of the line that goes through the points, first find the slope using the slope formula below:
[tex]m=\frac{y2-y1}{x2-x1}[/tex]Substitue values into the formula and solve for the slope, m:
[tex]\begin{gathered} m=\frac{7-1}{3-1} \\ \\ m=\frac{6}{2} \\ \\ m=3 \end{gathered}[/tex]The slope of the line, m is = 3.
Now, use the point slope form:
(y - y1) = m(x - x1)
Substitute 1 for y1, 1 for x1, and 3 for m:
(y - 1) = 3(x - 1)
Let's rewrite the equation to slope intercept form: y = mx + b
Where m is the slope and b is the y-intercept
y - 1 = 3(x) + 3(-1)
y - 1 = 3x - 3
Add 1 to both sides:
y - 1 + 1 = 3x - 3 + 1
y + 0 = 3x - 2
y = 3x - 2
Therefore, the equation of the line in slope intercept form is:
y = 3x - 2
ANSWER:
y = 3x - 2
Find the average rate of change of f(x)=x²-4x+1 from x=3 to x=7.
Simplify your answer as much as possible.
( explain pls )
At x=3 to x=7, the average rate of change of the given function
f(x) = x2- 4x + 1 is 6.
Solution:Given function,
f(x) = x² - 4x + 1
given x=3 and x=7
To find the average rate of change of a function, find its slope(m),
To find m ,
m= (y2 - y1) / (x2 - x1)
here y1 and y2 not given.
then substitute the given values of x into f. (x)
when x=3 ,
y1 = (3² - 4×3 +1)
= (9 - 12 +1)
= -2
so (x1,y1) = (3,-2)
similarly to find y2,
substitute x = 7 to f(x)
y2 = (7² - 4×7 + 1)
= (49 - 28 +1)
= 22
so (x2,y2) = (7,22)
To find m, substitute the values.
m= (y2 - y1) / (x2 - x1)
= (22 -( - 2)) / (7 - 3)
= 24/4
= 6
As a result, the average rate of change of the function x2-4x + 1 is 6.
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What is a rule for the nth term of Geometric sequence -3,-6,-12,-24,-48
The sequence is given as,
[tex]-3,\text{ -6, -12, -24, -48.......}[/tex]For the given sequence,
[tex]\begin{gathered} First\text{ term\lparen a\rparen = -3} \\ Common\text{ ratio\lparen r\rparen = }\frac{-6}{-3}=\text{ 2} \end{gathered}[/tex]nth term of a geometric progression is given as,
[tex]a_n\text{ = ar}^{n-1}[/tex]Where,
a = First term
r = common ratio
Therefore the nth term is calculated as,
[tex]a_n=\text{ -3\lparen2\rparen}^{n-1}[/tex]Thus the required answer is,
[tex]a_n=\text{ -3\lparen2\rparen}^{n-1}[/tex]Gretchen runs backwards at a speed of -4m/s. Mika continues running at a speed of 3 m/a. If Gretchen dropped her baton-48 meters from her current position how far ahead will Mika be?
The distance by which Mika will be ahead is equal to 36 m from present location.
According to the question Gretchen and Mika are at the same point when Gretchen runs backward at the speed of 4 m/s in opposite direction to pick her baton and Mika runs in the direction of the race at the speed of 3 m/s. We need to find the distance covered by Mika form the current location in the time when Gretchen picks her baton up. Let us assume that the time taken by Gretchen to pick up her baton be equal to t. It is given that 48 meters from her current location she dropped her baton. Since she is running at the speed of 4 m/s then the time will be equal to t = 48/4 = 12 seconds. Now, Gretchen will be able to pick up the baton in 12 seconds. Thus, the distance covered by Mika in 12 seconds will be equal to 3×12 = 36m.
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Complete Question:
Gretchen and Mika are involved in a relay race. Both runners are at the same point on the course when Gretchen realizes that she dropped her baton and has to turn around to retrieve it. Gretchen runs backwards at a speed of −4 m/s, relative to the direction of the race. Mika continues running at a speed of 3 m/s. If Gretchen dropped her baton −48 meters from her current location, relative to the direction of the race, how far ahead will Mika be when Gretchen picks her baton up?
1 point2. Consider the inequality below. Which value of x is a solution to theinequality?*2x + 39O x=-10х+13x = -60 x = 0O x = 6
We will solve as follows:
[tex]\frac{2x+3}{9}>\frac{x}{3}+1\Rightarrow\frac{2}{9}x+\frac{1}{3}>\frac{x}{3}+1[/tex][tex]\Rightarrow\frac{2}{9}x-\frac{x}{3}>1-\frac{1}{3}\Rightarrow-\frac{x}{9}>\frac{2}{3}[/tex][tex]\Rightarrow x<-6[/tex]So, the value that is a solution for the inequality is:
[tex]x=-10[/tex][Option A]
Points B and C lie on a circle with center O and a radius of 15 units. If the length of arc BC is 21 units, what is m *BOC in radians?
Given:
[tex]\begin{gathered} r=15\text{ }units \\ \hat{BC}=21\pi\text{ }units \end{gathered}[/tex]To find:
The angle BOC.
Explanation:
Using the length of the arc formula,
[tex]\begin{gathered} l=r\theta \\ 21\pi=15\theta \\ \theta=\frac{21\pi}{15} \\ \theta=\frac{7}{5}\pi \end{gathered}[/tex]The angle is
[tex]\angle BOC=\frac{7}{5}\pi[/tex]Final answer:
The angle is
[tex]\angle BOC=\frac{7}{5}\pi[/tex]Purchases of streamed music can be modeled by f (t) = 3.7 + 0.84 ln(t +1), where f (t) is measured in billions of dollars and t is measured in years, with t=0 corresponding to 2012.
a. What was the amount spent by consumers on streamed music in 2014? Use the correct units.
b. How fast was the amount spent by consumers on streamed music changing in 2014?
Use the correct units.
If the purchases of streamed music can be modeled by function f(t) = 3.7+0.84 ln(t+1)
Part a
The amount spent by consumers on streamed music in 2014 is 4.62 billion
Part b
The amount spent by consumers on streamed music changing in 2014 at the rate of 0.28 billions per year
The purchases of streamed music model
f(t) = 3.7+0.84 ln(t+1)
Where f(t) is measure in billion of dollar
t is the measured in years
Part a
We have to find the amount spent by consumer on streamed music in 2014.
t = 0 corresponding to the year 2012
Then 2014 corresponding to t = 2
Substitute the value of t in the equation
f(2) = 3.7+0.84 ln(2+1)
f(2) = 3.7+0.922
f(2) = 4.62 billion
Part b
Differentiate the model with respect to x
f'(t) = 0.84×[tex]\frac{1}{t+1}[/tex]
t = 2
Substitute the value in the equation
f'(2) = 0.84×[tex]\frac{1}{2+1}[/tex]
f'(2) = 0.28 billions per year
Hence, if the purchases of streamed music can be modeled by f(t) = 3.7+0.84 ln(t+1)
Part a
The amount spent by consumers on streamed music in 2014 is 4.62 billion
Part b
The amount spent by consumers on streamed music changing in 2014 at the rate of 0.28 billions per year
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Suppose you want to have $700,000 for retirement in 30 years. Your account earns 5% interest. How much would you need to deposit in the account each month?$_________________
The formula for a final amount A by depositing a regular amount R each month for n years with a interest rate r is given by:
[tex]A=R\frac{\lbrack(1+\frac{r}{12})^{^{12n}}-1\rbrack}{\frac{r}{12}}[/tex]For A = $700,000, r = 0.05, n = 30, we have:
[tex]\begin{gathered} 700000=R\cdot\frac{\lbrack(1+\frac{0.05}{12})^{12\cdot30}-1\rbrack}{\frac{0.05}{12}} \\ 700000=832.26R \\ R=\text{ \$841.08} \end{gathered}[/tex]Consider the parabola given by the equation:
f
(
x
)
=
−
2
x
2
−
12
x
−
9
Find the following for this parabola:
A) The vertex:
B) The vertical intercept is the point
C) Find the coordinates of the two
x
intercepts of the parabola and write them as a list of points of form (x, y) separated by commas:
It is OK to round your value(s) to to two decimal places.
The features of the given parabola are as follows:
a) Vertex: (3,9).
b) Vertical intercept: y = -9.
c) x-intercepts: (-5.12, 0) and (-0.88, 0).
Features of the quadratic functionThe equation of the parabola is given by:
f(x) = -2x² - 12x - 9.
Hence the coefficients are:
a = -2, b = -12, c = -9.
The formula for the coordinates of the vertex, considering the coefficients of a quadratic equation, is given as follows:
x = -b/2a.y = -(b² - 4ac)/(4a).The coordinates of the vertex are given as follows:
[tex]x_v = -\frac{-12}{2(-4)} = -3[/tex]
[tex]y_v = -\frac{(-12)^2 - 4(-2)(-9)}{4(-2)} = 9[/tex]
Hence (3,9).
The vertical intercept is given as follows:
f(0) = -2(0)² - 12(0) - 9 = -9.
The roots are given as follows:
[tex]x_{1} = \frac{-b + \sqrt{b^2 - 4ac}}{2a} = \frac{12 + \sqrt{(-12)^2 - 4(-2)(-9)}}{2(-2)} = -5.12[/tex][tex]x_{2} = \frac{-b - \sqrt{b^2 - 4ac}}{2a} = \frac{12 - \sqrt{(-12)^2 - 4(-2)(-9)}}{2(-2)} = -0.88[/tex]Hence the points are:
(-5.12, 0) and (-0.88, 0).
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Which inequality is equivalent to this one?y-85-2y-8+82-2+ 8y-8+8 <-2+8y-8+25-2+8y-8+25-2+2
Explanation
[tex]y+8=2[/tex]solve for y :
The subtraction property of equality tells us that if we subtract from one side of an equation, we also must subtract from the other side of the equation to keep the equation the same
hence,
subtract 8 in both sides
[tex]\begin{gathered} y+8=2 \\ y+8-8=2-8 \\ y=-6 \end{gathered}[/tex]so, the answer is
y=-6
I hope this helps you
Royal Furniture bought a sofa for $820. The sofa had a $1,420 list price. What was the trade discount rate Royal received? (Round your answer to the nearest hundredth percent.)
The received trade discount amount is $600 and percentage is 42.25%.
The trade discount in amount is = Original price - New price
Keep the values in formula to find the trade discount
The trade discount in amount is = 1420 - 820
Performing subtraction on Right Hand Side of the equation
The trade discount amount is = $600
Finding the trade discount in percentage
Trade discount in percentage = discount ÷ original price × 100
Keep the values in formula to find the trade discount in percentage
Trade discount in percentage = (600/1420)× 100
Performing multiplication and division
Trade discount in percentage = 42.25%
The trade discount is $600 and 42.25%.
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