Answer:
Step-by-step explanation:
Find the GCF:
12, 24, 36
8, 16, 24, 32
GCF is 24
The lowest amount of boxes the teacher could buy would be 2 boxes of Rulers and 3 boxes of Protractors. I hope this helped
suppose that random sample in the previous question yielded these observed scores: 6 12 5 10 6 9 12 20 5 23 10 20 11 28 13 18 find the 99% confidence interval
If random sample yielded these observed scores: 6 12 5 10 6 9 12 20 5 23 10 20 11 28 13 18. The 99% confidence interval for the mean height of both groups combined is approximately (3.92, 20.58) cm.
To calculate the 99% confidence interval for the mean height of both groups combined, we can use the t-distribution, since the sample size is relatively small (less than 30) and the population standard deviation is unknown.
First, we need to calculate the sample mean and sample standard deviation. The sample mean can be calculated by adding up all the observed scores and dividing by the total number of scores. In this case, the sample mean is (6+12+5+10+6+9+12+20+5+23+10+20+11+28+13+18) / 16 = 12.25 cm.
The sample standard deviation can be calculated using the formula:
s = √((∑(x - x')^2) / (n - 1))
where x is each observed score, x' is the sample mean, and n is the sample size. In this case, the sample standard deviation is approximately 7.05 cm.
Next, we need to calculate the t-value for a 99% confidence level, with 15 degrees of freedom (16 - 1). Using a t-table or a statistical software, we can find that the t-value is approximately 2.947.
Finally, we can calculate the confidence interval using the formula:
Confidence interval = sample mean ± (t-value * (sample standard deviation / √(sample size)))
Plugging in the values, we get:
Confidence interval = 12.25 ± (2.947 * (7.05 / √(16))) = 12.25 ± 8.33
In summary, to calculate the 99% confidence interval for the mean height of both groups combined, we can use the t-distribution and the formula for confidence intervals. We need to calculate the sample mean, sample standard deviation, t-value for the desired confidence level, and then plug these values into the formula to get the confidence interval.
This interval represents the range of values within which we can be 99% confident that the true mean height of both groups combined falls.
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lThe table below shows the price of different quantities
of standard-sized lemons at Joe's Fruit Stand. What is the
least amount of money needed to purchase exactly 20
standard-sized lemons if the bags must be sold intact
and there is no tax charged for lemons?
Number of lemons:
Total price:
$3.60
1
$0.30
bag of 6
$1.20
bag of 12
$2.10
The least amount of money needed to purchase exactly 20 standard-sized lemons is $1.50, by buying three bags of 6 lemons each and two individual lemons.
What is Combination?
In mathematics, a combination is a way of selecting items from a larger set without regard to the order in which the items are selected. In other words, a combination is a selection of a subset of items from a larger set, where the order of the selected items does not matter.
The number of possible combinations of r items selected from a set of n items is denoted by the notation C(n, r), or sometimes written as nCr. The formula for calculating the number of combinations is:
C(n, r) = n! / (r! * (n - r)!)
To purchase exactly 20 standard-sized lemons at Joe's Fruit Stand, we need to find the combination of bags of lemons that adds up to a total of 20. We want to minimize the cost, so we should choose the combination of bags with the lowest cost.
One possible combination is to buy two bags of 12 lemons each, for a total of 24 lemons, and then discard 4 lemons. This would cost $2.10 x 2 = $4.20.
Another possible combination is to buy three bags of 6 lemons each, for a total of 18 lemons, and then buy two individual lemons for a total of 20 lemons. This would cost $0.30 x 3 + $0.30 x 2 = $1.50.
Therefore, the least amount of money needed to purchase exactly 20 standard-sized lemons is $1.50, by buying three bags of 6 lemons each and two individual lemons.
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Skye’s grandparents invest $1500 in a college account. It earns 4. 9% interest that is compounded annually. How much money will be in the account after 10 years?
Round to the nearest cent, if necessary
The total amount accrued, principal plus interest, with compound interest on a principal of $ 1,500.00 at a rate of 4.9 % per year compounded 1 times per year over 10 years is $2,420.17.
Using the compound interest formula is
Compound interest is calculated by multiplying the initial principal amount (P) by one plus the annual interest rate (R) raised to the number of compound periods (nt) minus one.
[tex]$\mathrm{A}=\mathrm{P}\left(1+\frac{\mathrm{r}}{\mathrm{n}}\right)^{\mathrm{nt}}$[/tex]A= Total amount =?
P= Principal amount = 1500
r=Rate of interest =4.9 %=0.049
t= time =10years
n= compounding = annually =1
Using the above data and formula, we get:
First, convert R as a percent to r as a decimal
[tex]& \mathrm{r}=\frac{\mathrm{R}}{100} \\[/tex]
[tex]& \mathrm{r}=\frac{4.9}{100} \\[/tex]
[tex]& \mathrm{r}=0.049 \text { rate per year, }[/tex]
Then solve the equation for A
[tex]& \mathrm{A}=1,500.00\left(1+\frac{0.049}{1}\right)^{(1)(10)} \\[/tex]
[tex]& \mathrm{A}=1,500.00(1+0.049)^{10} \\[/tex]
[tex]& \mathrm{~A}=\$ 2,420.17[/tex]
The total amount accrued after 10 years is $2,420.17.
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3+2y equals square root of -y
The solution to the given quadratic equation formed is; y = -9/4 and y = -1
How to find the solution to the quadratic equation?The given equation is expressed as;
(3 + 2y) = √(-y)
Taking the square of both sides gives us;
(3 + 2y)² = √(-y)]²
→ 9 + 12y + 4y² = -y
4y² + 13y + 9 = 0
Factoring the expression gives us;
4y² + 4y + 9y + 9 = 0
4y(y + 1) + 9(y + 1) = 0
Thus;
(4y + 9) = 0
(y + 1) = 0
y = -9/4
y = -1
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The stock price of Company A decreased by $1.27 every day for 4
days. The fifth day, the stock price increased by $3.47. What was
the total change in the stock's price over 5 days?
Answer:
Step-by-step explanation:
The stock price of Company A decreased by $1.27 for 4 days, so the total decrease in the stock's price would be $1.27 * 4 = $5.08.
On the fifth day, the stock price increased by $3.47, so the total change in the stock's price over 5 days would be $3.47 - $5.08 = -$1.61.
So the total change in the stock's price over 5 days was a decrease of $1.61.
consider the following regression model: which of the following is an algebraic property of ols estimates of this model and its associated statistics? group of answer choices the sum, and therefore the sample average of the ols residuals, is positive. the sum of the ols residuals is negative. the sample covariance between the regressors and the ols residuals is zero. ols sample regression line always passes through the sample average of x and sample average of y both a and b are correct both a and d are correct both c and d are correct none of the above
Option (c) is correct: the sample covariance between the regressors and the OLS residuals is zero. This is an important property because it implies that the OLS estimator is not biased and the estimated coefficients of the independent variables are consistent.
The algebraic property of OLS (Ordinary Least Squares) estimates is that the sample covariance between the regressors and the OLS residuals is zero. This means that the OLS estimates are unbiased and not correlated with the independent variables in the regression model.
Option (a) is incorrect: the sum, and therefore the sample average of the OLS residuals, can be either positive or negative depending on the distribution of the data and the fit of the model.
Option (b) is also incorrect: the sum of the OLS residuals can be either positive or negative depending on the distribution of the data and the fit of the model.
Option (d) is incorrect: the OLS sample regression line does not always pass through the sample average of x and the sample average of y. However, it does pass through the point of averages when the model includes a constant term (i.e., intercept).
Therefore, the correct answer is (c): the sample covariance between the regressors and the OLS residuals is zero.
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In a particular hospital, 6 newborn babies were delivered yesterday. Here are their weights (in ounces).
94, 113, 93, 123, 93, 102
Send data to calculator
Assuming that these weights constitute an entire population, find the standard deviation of the population. Round your answer to two decimal places.
The standard deviation of the population is approximately 11.8 ounces.
What is Statistics?Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data.
The formula to find the standard deviation is given
σ = √( Σ(x - μ)² / N )
where μ is the mean.
μ = (94 + 113 + 93 + 123 + 93 + 102) / 6 = 100
Next, we can calculate the deviations of each value from the mean:
94 - 100 = -6
113 - 100 = 13
93 - 100 = -7
123 - 100 = 23
93 - 100 = -7
102 - 100 = 2
σ = √( -6)²+( 13)²+( -7)²+( 23)² +( -7)²+( 2)²/ 6 )
=√36+169+49+529+49+4/6
=√836/6=11.8
Hence, the standard deviation of the population is approximately 11.8 ounces.
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PROMPT: A spherical toy of diameter 75 cm must have a triangular sticker of height 12 cm and base of 10 cm. What is the surface area not covered by the sticker?
CLAIM:
EVIDENCE:
REASONING:
The surface area not covered by the sticker on the spherical toy is approximately 5614.1 cm².
CLAIM: The surface area not covered by the sticker on the spherical toy is approximately 5614.1 cm².
EVIDENCE: The total surface area of the spherical toy can be calculated using the formula A = 4πr², where r is the radius of the sphere. Since the diameter of the sphere is given as 75 cm, the radius is half of the diameter, which is 37.5 cm. Therefore, the total surface area of the spherical toy is:
A = 4π(37.5)²
A = 4π(1406.25)
A ≈ 17593.29 cm²
The surface area covered by the triangular sticker can be calculated by finding the area of the triangle, which is given by the formula A = 1/2 bh, where b is the base and h is the height. Therefore, the surface area covered by the sticker is:
A = 1/2 (10)(12)
A = 60 cm²
The surface area not covered by the sticker is the difference between the total surface area of the spherical toy and the surface area covered by the sticker:
A_not_covered = A - A_sticker
A_not_covered ≈ 17593.29 - 60
A_not_covered ≈ 17533.29 cm²
Therefore, the surface area not covered by the sticker on the spherical toy is approximately 5614.1 cm².
REASONING: By calculating the total surface area of the spherical toy and subtracting the surface area covered by the triangular sticker, we can determine the surface area not covered by the sticker. Since the sticker covers a relatively small area compared to the total surface area of the sphere, the surface area not covered by the sticker is almost equal to the total surface area of the sphere, minus the area of the sticker.
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Which procedure correctly simplifies the expression 2(4x + 3) + (x - 3)?
To simplify the expression 2(4x + 3) + (x - 3), we need to use the distributive property of multiplication over addition, and then combine like terms.
The expression 2(4x + 3) + (x - 3) contains parentheses and addition and multiplication operations, so we need to follow the order of operations to simplify it. The order of operations is:
1. Simplify any expressions inside parentheses.
2. Evaluate any exponents.
3. Perform multiplication and division, from left to right.
4. Perform addition and subtraction, from left to right.
Let's follow these steps to simplify the expression,
1. Simplify any expressions inside parentheses.
There are two sets of parentheses in the expression: (4x + 3) and (x - 3).
We can use the distributive property of multiplication over addition to eliminate the parentheses:
2(4x + 3) + (x - 3) = 8x + 6 + x - 3
2. Evaluate any exponents,
There are no exponents in the expression.
3. Perform multiplication and division, from left to right.
There is no multiplication or division to perform.
4. Perform addition and subtraction, from left to right.
We can now simplify the expression by combining like terms:
8x + 6 + x - 3 = (8x + x) + (6 - 3) = 9x + 3
Therefore, the simplified expression is 9x + 3.
In summary, the correct procedure to simplify the expression 2(4x + 3) + (x - 3) is to use the distributive property to eliminate the parentheses, and then combine like terms to obtain the simplified expression 9x + 3.
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What’s the answer for this I’m not sure
Answer:
8√57/57
Step-by-step explanation:
a² + b² = c²
a² + 8² = 11²
a² + 64 = 121
a² = 57
a = √57
For angle Θ, opp = 8, adj = √57, hyp = 11.
tan Θ = opp/adj = 8/√57 = 8√57/57
Which angles are corresponding angles? Se
lect all that apply
The angles that are corresponding angles:
∠1 and ∠3
∠5 and ∠7
∠6 and ∠8
We know that when a transversal intersects parallel lines or non-parallel lines then the angles formed at the same relative position at each intersection are nothing but the corresponding angles.
If a transversal intersects parallel lines, then the corresponding angles formed have equal measure.
Wheras if a transversal intersects non-parallel lines, then the corresponding angles formed doesn’t have any relation with each other.
In the following figure we can observe that line p and line q are non-parallel lines and are intersected by transversal r.
From definition of corresponding angles, the pair of corresponding angles:
∠1 and ∠3
∠5 and ∠7
∠6 and ∠8
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Find the complete question below.
I NEED HELP ON THIS ASAP!!!!
In mathematics, a line is a set of points that extend indefinitely in two directions.
How to explain the linePoints on the line: These are points that lie exactly on the line. They are equidistant from the two endpoints of the line and are considered to be part of the line itself.
Points above the line: Points that are vertically above the line are referred to as points above the line. These points are located higher than the line and have a greater y-coordinate than any point on the line.
Points below the line: Points that are vertically below the line are referred to as points below the line. These points are located lower than the line and have a smaller y-coordinate than any point on the line.
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Given f(x)=4x^2+9x+9, find f(−1)
The value of the function f(-1) is -4
What is a function?A function is simply described as an equation that explains the relationship between the independent variable and the dependent variable.
From the information given, we have the function;
f(x)=4x^2+9x+9
To determine the value of f(-1)
substitute the value of x as- 1, we get;
f(-1) = 4(-1)² + 9(-1) + 9
expand the bracket, we get;
f(-1) = -4 -9 + 9
Add or subtract the values, we have;
f(-1) = -4
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The graph shows the total points earned for several pairs of shoes. The initial number of points earned when signing up is ________. The number of points earned per pair of shoes purchased is _________.
The initial number of points earned when signing up is 30 and earned per pair of shoes purchased is 15.
What is the equation of a line passing through two points?Let the equation of the line pass through (x₁, y₁) and (x₂, y₂). Then the equation of the line is given as,
[tex]\rm (y - y_2) = \left (\dfrac{y_2 - y_1}{x_2 - x_1} \right ) (x - x_2)[/tex]
Let 'x' be the number of shoes purchased and 'y' be the total points earned.
From the graph, the two points are (2, 60) and (4, 90). Then the equation is given as,
[tex]\rm (y - 60) = \left (\dfrac{90 - 60}{4 - 2} \right) (x - 2)[/tex]
y - 60 = 15(x - 2)
y - 60 = 15x - 30
y = 15x + 30
The initial number of points earned when signing up is 30 and earned per pair of shoes purchased is 15.
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How many moles are in 5.3 x 1024 atoms of Kr?
Answer: 8.8 moles
Step-by-step explanation:
Fun Fun! Chemistry :)
So, 5.3 x [tex]10^{24}[/tex] atoms can be converted to moles by dividing by Avogadros number, (6.02 x 10^23). This is because Avogadros number of atoms form one mole.
So the answer is about 8.8 moles of Kr
Sketch and graph please
The graph of the linear inequality y > -4 is given below.
What is linear inequality?Linear inequalities are expressions that use inequality symbols to compare two linear expressions. When a polynomial of degree 1 is contrasted with another algebraic expression of a degree less than or equal to 1, this comparison results in a linear inequality, which is an inequality including at least one linear algebraic expression. There are many different ways to represent different types of linear inequalities.
Given linear inequality is y > -4 which represents y region will be in the positive range or greater than -4.
The graph of this linear inequality is given below.
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What is the value of c in the equation?
15.3 = Start Fraction c Over 2.1 End -Fraction
Each of four friends-Jennifer, Rebecca, Derek, and
Matthew-had the goal of raising $50. How much
money did each person raise? Show or explain your
answer
a. Jennifer raised 50% of her goal.
b. Rebecca raised 100% of her goal.
c. Derek raised 75% of his goal.
d. Matthew raised 150% of his goal.
Answer:
Jennifer raised $25
Rebecca raised £50
Derek raised $37.50
Matthew raised $75
Step-by-step explanation:
J: 50% = dividing by 2, 50/2= 25
R: 50 is 100%, because that was the total
D: 75% = dividing by 4, timsing by three, so 50/4 is 12.5 x 3 is 37.5, put it into currency $37.50
M: We add another 25% onto 50 so 25+50=75.
(hope this works!!)
Use the Slope Formula to calculate the slope of a line with these two points. Find the slope of the line that passes through (5, 11) and (9, 2).
given that x=a,y=b is the solution to the system of equations 2x+y=4, x-y=5 then what is the value of b to the power of a
Answer:
The answer is -8
Step-by-step explanation:
[tex]2x+y=4\\x-y=5\\\\[/tex]
after combining these into a singular equation, we get:
[tex]3x=9\\x=3[/tex]
substitute the value of [tex]x[/tex] into one of the equations:
[tex]x-y=5\\3-y=5\\y=-2[/tex]
now that we found the [tex]x[/tex] and [tex]y[/tex] value, we can substitute it into the term [tex]b^a[/tex] since we know that [tex]x=a[/tex] and [tex]y=b[/tex].
[tex]a^b\\a=x, b=y\\x=3, a=3\\y=-2, b=-2\\-2^3\\-8[/tex]
Shawna joined the bowling league at Bronze Lanes, a bowling alley. She scores a strike 75% of the time. For a perfect game, she needs to bowl a strike 12 times in a row. How likely is it that Shawna will bowl a perfect game at her next game?
Shawna simulates the situation by using a computer to randomly generate 12 numbers from 1 to 4. Each time 1, 2, or 3 appears, it represents a strike
The probability of Shawna bowling a perfect game is 7.23%.
Assuming that Shawna's chance of getting a strike is independent on each roll (which may not be entirely accurate in real life), the probability of her getting a strike on any given roll is 0.75.
To calculate the probability of Shawna bowling a perfect game (i.e., getting a strike on all 12 rolls), we can use the multiplication rule of probability, which states that the probability of two independent events occurring together is the product of their individual probabilities.
So, the probability of Shawna getting a strike on the first roll is 0.75. The probability of her getting a strike on the second roll, assuming she got a strike on the first roll, is also 0.75. Similarly, the probability of her getting a strike on the third roll, assuming she got strikes on the first two rolls, is also 0.75. We can continue this pattern for all 12 rolls.
Therefore, the probability of Shawna bowling a perfect game is:
0.75 x 0.75 x 0.75 x ... (12 times)
which is approximately equal to 0.0723, or about 7.23%.
Using a computer to randomly generate 12 numbers from 1 to 4 and treating 1, 2, or 3 as a strike is a reasonable way to simulate the situation and estimate the probability of Shawna bowling a perfect game. By repeating this process many times, we can get a better idea of the distribution of possible outcomes and the likelihood of different results.
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The following are the distances (in miles) to the nearest airport for 16 families. 9, 12, 18, 18, 19, 20, 26, 28, 29, 31, 31, 32, 32, 34, 35, 37. Notice that the numbers are ordered from least to greatest. Make a box-and-whisker plot for the data.
The average of the middle two figures, which are 18 and 19, is the median of the lower half. Q1 = (18+19)/2, which equals 18.5. The average of the middle two values—32 and 32—represents the median of the upper half. Q3 = (32+32)/2 = 32 as a result.
How far is that?Distance is the sum of an object's movements, regardless of direction. Distance can be defined as the amount of space an object has covered, regardless of its starting or ending position.
First, we calculate the median for the lower and higher halves of the data. The values up to the median are found in the lower half and are as follows:
9, 12, 18, 18, 19, 20, 26, 28
The upper half consists of the values from the median onwards, which are:
29, 31, 31, 32, 32, 34, 35, 37
The median of the lower half is the average of the middle two values, which are 18 and 19. Therefore,
Q1 = (18+19)/2
= 18.5.
The median of the upper half is the average of the middle two values, which are 32 and 32. Therefore,
Q3 = (32+32)/2
= 32.
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What is the area of a right triangle with a height of seven and three fourths yards and a base of 20 yards?
If the height of seven and three fourths yards and a base of 20 yards, then the area of the right triangle is 77.5 square yards.
To calculate the area of a right triangle, we can use the formula:
Area = (base * height) / 2
where the base is one of the sides of the triangle, and the height is the perpendicular distance from that side to the opposite vertex.
In this case, the height of the right triangle is 7 and 3/4 yards, and the base is 20 yards. Plugging these values into the formula, we get:
Area = (20 * 7.75) / 2 = 155 / 2 = 77.5 square yards
In summary, to calculate the area of a right triangle, we can use the formula that relates the base and height to the area. In this case, we needed to use the given values of the height and base of the right triangle to compute the area in square yards.
The resulting area represents the amount of surface inside the triangle and is useful in many applications, such as geometry, physics, and engineering.
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in how many ways can a president, vice-president, and treasurer be chosen from a group of 4 freshmen and 4 sophomores so that at least one freshman and at least one sophomore holds at least one of these three positions? on person cannot serve in more than one position
The total number of ways to choose a president, vice-president, and treasurer from a group of 4 freshmen and 4 sophomores so that at least one freshman is 314 ways.
We can approach this problem using the principle of inclusion-exclusion.
First, let's find the total number of ways to choose a president, vice-president, and treasurer from a group of 8 people without any restrictions. We can do this by using the permutation formula:
8P₃ = 8 x 7 x 6 = 336
Next, let's find the number of ways to choose the three positions where only freshmen are chosen. There are 4 freshmen to choose from, so we can select the president, vice-president, and treasurer in 4P₃ = 4 x 3 x 2 = 24 ways.
Similarly, there are 4 sophomores to choose from, so we can select the three positions where only sophomores are chosen in 4P₃ = 4 x 3 x 2 = 24 ways.
However, we have overcounted the cases where only freshmen or only sophomores are chosen, since the problem statement requires that at least one freshman and at least one sophomore hold at least one of the three positions. Therefore, we need to subtract the number of cases where only freshmen or only sophomores are chosen.
There is only one way to select the president, vice-president, and treasurer from the 4 freshmen, and likewise, there is only one way to select them from the 4 sophomores. Therefore, there are a total of 2 ways to select the three positions where only one class is represented.
Finally, we need to add back the cases where no freshman or no sophomore is chosen. This occurs only if all three positions are filled by juniors or seniors. There are 4 juniors and seniors to choose from, so we can select the three positions in 4P₃ = 4 x 3 x 2 = 24 ways.
Therefore, the total number of ways to choose a president, vice-president, and treasurer from a group of 4 freshmen and 4 sophomores so that at least one freshman and at least one sophomore holds at least one of these three positions is:
336 - 24 - 24 + 2 + 24 = 314 ways.
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The function
y
=
f
(
x
)
y=f(x) is graphed below. Plot a line segment connecting the points on
f
f where
x
=
−
4
x=−4 and
x
=
1.
x=1. Use the line segment to determine the average rate of change of the function
f
(
x
)
f(x) on the interval
−
4
≤
x
≤
1.
−4≤x≤1.
The average rate of change of the function is 1.6
How to to determine the average rate of change of the functionFrom the question, we have the following parameters that can be used in our computation:
The graph
The interval is given as
−4≤x≤1.
From the graph, we have
f(-4) = -4
f(1) = 4
The average rate of change at this interval is calculated as
Rate = Change in function values/Change in x values
So, we have
Rate = (-4 - 4)/(-4 - 1)
Evaluate
Rate = 1.6
Hence, the rate is 1.6
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3
David and Alec are comparing the international calling plans on their cell phones. On his plan,
David pays $4 just to place a call and $1 for each minute. When Alec makes an international
call, he pays $1 to place the call and $2 for each minute. A call of a certain duration would
cost exactly the same under both plans. What is the duration?
Write a system of equations, graph them, and type the solution.
x4-8x2y2+16y4-256 Factorise the equation
Answer:
it's like this the answer
Step-by-step explanation:
x⁴-8x²y²+16y²-256
(x²-8xy)²+(4y-16)²
Answer:
x⁴-8x²y²+16y²-256
(x²-8xy)²+(4y-16)²
Step-by-step explanation:
Help ASAP please. !!!!
Which property of real numbers is shown below?
3 + ((-5) + 6) = (3 + (-5)) + 6
O commutative property of addition
O identity property of multiplication
associative property of addition
commutative property of multiplication
Kim bought a birdhouse that originally cost $36. She used a coupon that took 25% off the price. How much money did Kim save?
Kim saved $9 by using the coupon.
How to find Original price?
The MSRP (i.e., Manufacturer's Suggested Retail Price) set the original price, which is the cost. In the majority of cases, the original price would have been less expensive than the current price, though this is not always the case. Finding an item's original price and knowing the price after a discount are both made possible by original price.
Original price = Sales price/(100%-Discount%)
To find out how much money Kim saved, we need to first find out how much the birdhouse cost after the coupon was applied.
The coupon took 25% off the original price, so the birdhouse cost 100% - 25% = 75% of its original price.
So, the new price of the birdhouse would be:
36 * 75% = $27
Kim saved 36 - 27 = $9 by using the coupon.
Therefore, Kim saved $9 using the coupon.
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