Yet because we can compare the set of positive rational numbers exactly to the collection of natural numerals, we may infer that the set of positive rational must have a number of clusters of 0, or n(Q+)=0. Due to the fact that these two numbers must equal one another, 02 = 0.
Since they may be arranged in a one-to-one correspondence with the natural numbers (the integers, such as 1, 2, and 3), Cantor showed in 1873 that the rational numbers, although being infinite, are countable (or denumerable).
How are the rational numbers counted as a group?
To achieve this, we take into account the this double integers grid upon that positive orthogonal. If p and q are positive natural numbers, then p/q may be used to represent any positive valid value.
Simply put, we establish the following values: h(0) = 0, h(2n) = f(n), for all natural numbers n > 0, and h(2n-1) = g(n) = -f(n), for all natural numbers n > 0.
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Ethan is older than Jordan. Their ages are consecutive integers. Find Ethan’s age if the sum of the square of Ethan’s age is 4 times Jordan’s age is 56.
If the square of Ethan's age is added to Jordan's age, the result is [tex]56[/tex], which equals Ethan's age. The answer is that Ethan is [tex]6[/tex] years old.
How can you calculate age using maths?A person's birth date is compared to the day on which their age has to be computed as part of the age calculation process. The age of the individual is calculated by subtracting the person's age from the given date. Provided Date - Birthdate is used to compute age.
How can ageing be quantified?A person's age is merely a measurement of how long they have been living. These metrics don't adequately describe who you are or what you have accomplished. At any age, either old or young, one may do anything.
The solution states that [tex]56[/tex] is equal to [tex]4[/tex] times Jordan's age multiplied by the square of Ethan's age. This may be expressed as an equation:
[tex]x^2 + 4(x-1) = 56[/tex]
Simplifying the equation:
[tex]x^2 + 4x - 4 = 56[/tex]
[tex]x^2 + 4x - 60 = 0[/tex]
We can solve this quadratic equation by factoring:
(x+10)(x-6) = 0
Therefore, [tex]x = -10[/tex] or [tex]x = 6[/tex]. We can discard the negative value since age cannot be negative, so Ethan's age is [tex]6[/tex].
To check, we can substitute x=6 into the original equation:
[tex]6^2 + 4(6-1) = 36 + 20 = 56[/tex]
So the solution is Ethan is [tex]6[/tex] years old.
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F(x)=1/x squared -3x +1 then iind the inverse
The inverse function for the given function F(x)=1/x² -3x +1 is given by
f⁻¹(x) =(3 ± √(9 + 4/(x - 1))) /2.
Function f(x) is equals to,
F(x)=1/x² -3x +1
Inverse of a function, we need to swap the positions of the x and y variables and then solve for y.
Let's start with the original function
f(x) = 1/x^2 - 3x + 1
Now we will swap x and y,
⇒ x = 1/y^2 - 3y + 1
Next, Solve for y in terms of x
⇒ x = 1/y^2 - 3y + 1
⇒ x - 1 = 1/y^2 - 3y
⇒ 1/(x - 1) = y^2 - 3y
⇒ 1/(x - 1) = y(y - 3)
⇒ y(y - 3) = 1/(x - 1)
⇒ y^2 - 3y - 1/(x - 1) = 0
Using the quadratic formula, solve for y to get inverse function we have,
y = (3 ± √(9 + 4/(x - 1))) / 2
Therefore, the inverse function of f(x) is equal to f⁻¹(x) =(3 ± √(9 + 4/(x - 1))) /2.
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The above question is incomplete, the complete question is:
F(x)=1/x squared -3x +1 then Find the inverse
A total of 646 tickets were sold for the school play. They were either adult tickets or student tickets. There were 54 fewer student tickets sold than adult tickets. How many adult tickets were sold?
Answer:
64 im pretty cxsure
Step-by-step explanation:
1/3750 as a percetage
Answer: 37.5
Step-by-step explanation:
z score of 0.28 and corresponding z score
Step-by-step explanation:
Mathematics for Social Science 1011 PDF
Suppose a second, similar study had 1000 teens and that each gender was equally
epresented. Keeping the original conclusion that 17% of the male and 22% of the female
nternet users create their own blog, what percentage of teen bloggers would be female?
What percentage of teen bloggers would be male?
Answer:
If we assume that the proportion of male and female internet users in the second study is the same as the first study (i.e., 60% female and 40% male), and that the percentage of male and female internet users who create their own blog is the same as the first study (i.e., 17% for male and 22% for female), then we can calculate the percentage of teen bloggers who are female and male.
First, we can calculate the number of male and female internet users in the second study:
Number of female internet users = 1000 x 0.5 x 0.6 = 300
Number of male internet users = 1000 x 0.5 x 0.4 = 200
Next, we can calculate the number of female and male teen bloggers in the second study, based on the percentages from the first study:
Number of female teen bloggers = 300 x 0.22 = 66
Number of male teen bloggers = 200 x 0.17 = 34
Finally, we can calculate the percentage of teen bloggers who are female and male:
Percentage of teen bloggers who are female = (66 / 1000) x 100% = 6.6%
Percentage of teen bloggers who are male = (34 / 1000) x 100% = 3.4%
Therefore, in the second study, 6.6% of teen bloggers would be female and 3.4% of teen bloggers would be male.
Kalakaua Middle School is doing a fundraiser by selling Popsicles and Kona Ice. Each Popsicle costs $2 and each Kona Ice costs $4. The school sold a total of $160 from the
fundraiser. It sold a total of 50 Kona Ice and Popsicles combined. How many Kona Ice were sold and how many Popsicles were sold?
Therefore, the school sold 20 Popsicles and 30 Kona Ice.
What is system of equation?A system of equations is a set of two or more equations that are to be solved simultaneously. The variables in the equations are related to each other in such a way that they must satisfy all the equations in the system.
Given by the question.
Let's assume that the number of Popsicles sold is "x" and the number of Kona Ice sold is "y".
From the problem, we know that each Popsicle costs $2 and each Kona Ice costs $4, and the total amount sold is $160. Therefore, we can write two equations based on the given information:
2x + 4y = 160 (equation 1)
x + y = 50 (equation 2)
We can solve this system of equations by substitution or elimination. Let's use substitution:
From equation 2, we can solve for "x" in terms of "y":
x = 50 - y
Substituting this expression for "x" into equation 1, we get:
2(50 - y) + 4y = 160
Simplifying and solving for "y", we get:
100 - 2y + 4y = 160
2y = 60
y = 30
So, the number of Kona Ice sold is 30.
Substituting this value of "y" back into equation 2, we get:
x + 30 = 50
x = 20
So, the number of Popsicles sold is 20.
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Whereas the carbohydrate and fat you eat is used mostly to give you energy, protein is used mostly to build and maintain your body.
True False
The statement "Whereas the carbohydrate and fat you eat is used mostly to give you energy, protein is used mostly to build and maintain your body" is True.
What are carbohydrates, fats, and proteins?
Carbohydrates, fats, and proteins are the three macronutrients that supply our bodies with energy.
All of these nutrients, particularly proteins, are required for the development and maintenance of the human body.In terms of providing energy, carbohydrates and fats are the most effective.
Carbohydrates are the body's primary source of energy. When you consume carbohydrates, they are broken down into glucose, which is transported throughout the body to be used as energy.
Fat is the second-most efficient source of energy, and it is utilized when carbohydrate stores are depleted. Fats are broken down into fatty acids and transported throughout the body, where they are utilized as an energy source. Proteins are not the body's preferred source of energy; instead, they are mostly used to build and repair tissues in the body.
In summary, while carbohydrates and fats are primarily used to produce energy, protein is mostly used for body maintenance and growth.
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Select the conic section represented by the equation. 2x + 5y - 7 = 0 ellipse parabola point hyperbola line circle
The conic section represented by the equation. 2x + 5y - 7 = 0 is a line
Selecting the conic section of the equationThe given equation is 2x + 5y - 7 = 0.
To determine the conic section represented by this equation, we can use the discriminant D:
D = B^2 - 4AC
where A, B, and C are the coefficients of x^2, xy, and y^2, respectively, in the general form of the conic section equation Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0.
In this case, A = 0, B = 0, and C = 0 (since there are no x^2, xy, or y^2 terms), so the discriminant is:
D = 0^2 - 4(0)(0) = 0
Since the discriminant is 0, the conic section represented by the equation 2x + 5y - 7 = 0 is a degenerate case, which is a line.
Therefore, the answer is "line".
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Simplify this Please!!!!!
Answer: 4/9 ≅ 0.4444444 = four ninths. ... Then simplify the result to the lowest terms or a mixed number.
The annual rainfall for Austin, Texas, and San Antonio, Texas, in each of the years from 2002 to 2011 are shown in the tables. Use the data for 9-it.' Annual Rainfall for Austin, Texas (in.) 36.00 21.41 52.27 22.33 34.70 46.95 16.07 31.38 37.76 19,68 Annual Rainfall for San Antonio, Texas (in.) 46.27 28,45 45.32 16.54 21.34 47.25 13.76 30.69 37.39 1759 9. Use a spreadsheet to find the mean for the two cities' annual rainfalls. In which city does it rain more in a year, on average?
In San Antonio, Texas it rain more in a year, on average when compared to Austin, Texas according to the mean of both cities.
Data of Austin, Texas is as follows:
36.00, 21.41, 52.27, 22.33, 34.70, 46.95, 16.07, 31.38, 37.76 and 19.68.
Number of values ( n ) = 10
Sum of the values = 318.23
Average of the rainfall in Austin, Texas is equal to:
Mean = Total sum / No. of Values
Mean = 318.23 / 10
Mean = 31.82 in
Data of San Antonio, Texas is as follows:
46.27, 28.45, 45.32, 16.54, 21.34, 47.25, 13.76, 30.69, 37.39 and 17.59.
Number of values ( n ) = 10
Sum of the values = 304.6
Average of the rainfall in San Antonio, Texas is equal to:
Mean = Total sum / No. of Values
Mean = 304.6 / 10
Mean = 30.46 in
So, comparatively the average of San Antonio is higher.
Therefore in San Antonio, Texas it rain more in a year, on average when compared to Austin, Texas.
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how many strings of 6 letters of the english alphabet contain one vowel? exactly two vowels? at least one vowel? at least two vowels? what is number of strings of length n with exactly k vowels?
Strings of 6 letters of the English alphabet containing one vowel: 2,832,240, exactly two vowels: 2,233,200, at least one vowel: 10,919,736, least two vowels: 8,087,496 and number of strings of length n with exactly k vowels: nCk * 5Ck * 21^(n-k)
There are 26 letters in the English alphabet, out of which 5 are vowels (A, E, I, O, U). To find the number of strings of 6 letters of the English alphabet that contain one vowel, we can choose the position of the vowel in 6C1 ways and fill the remaining positions with any of the 21 consonants in 21C5 ways.
Therefore, the total number of such strings is 6C1 * 21C5 = 2,832,240.
To find the number of strings with exactly two vowels, we can choose the positions of the vowels in 6C2 ways and fill them with any of the 5 vowels in 5C2 ways. We can fill the remaining positions with any of the 21 consonants in 21C4 ways.
Therefore, the total number of such strings is 6C2 * 5C2 * 21C4 = 2,233,200.
To find the number of strings with at least one vowel, we can subtract the number of strings with no vowels from the total number of strings. The number of strings with no vowels is 21^6 (since there are 21 consonants and we can choose any of them for each of the 6 positions).
Therefore, the number of strings with at least one vowel is 26^6 - 21^6 = 10,919,736.
To find the number of strings with at least two vowels, we can subtract the number of strings with one vowel or no vowel from the total number of strings. The number of strings with one vowel is 2,832,240 (as we found earlier) and the number of strings with no vowels is 21^6.
Therefore, the number of strings with at least two vowels is 26^6 - 2,832,240 - 21^6 = 8,087,496.
To find the number of strings of length n with exactly k vowels, we can choose the positions of the k vowels in nCk ways and fill them with any of the 5 vowels in 5Ck ways. We can fill the remaining positions with any of the 21 consonants in 21^(n-k) ways.
Therefore, the total number of such strings is nCk * 5Ck * 21^(n-k).
Hence, the number of strings of 6 letters of the English alphabet containing one vowel is 2,832,240, while the number of strings with exactly two vowels is 2,233,200. The number of strings with at least one vowel is 10,919,736, and the number of strings with at least two vowels is 8,087,496.
Finally, the number of strings of length n with exactly k vowels is nCk * 5Ck * 21^(n-k).
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PLEASE HELP NEED IT DONE RN
Answer:
B) 3
Step-by-step explanation:
219 / 73= 3
3 x 73= 219
Answer: 3
Step-by-step explanation:
y=14-3x domain (-6,5,13)
Answer:
y = {18 x + 14, 14 - 15 x, 14 - 39 x}
Step-by-step explanation:
There is zero context to this, so I have no clue if this helps or even answers your question. Try harder with the question next time.
the admission fee at an amusement park is $4.25 for children and $7.00 for adults. on a certain day, 303 people entered the park, and the admission fees collected totaled 1824 dollars. how many children and how many adults were admitted?
The admission fee at an amusement park is $4.25 for children and $7.00 for adults. on a certain day, 303 people entered the park, and the admission fees collected totaled 1824 dollars. There are 108 children and 195 adults were admitted
Let the number of children admitted = C and the number of adults admitted = A
Total number of people admitted = 303
We can form two equations from the given information.
The first equation is to represent the number of people admitted in terms of children and adults.
So, the equation will be
C + A = 303 ------(1)
The second equation represents the total amount collected from admission fees.
So, the equation will be
4.25C + 7A = 1824 ------(2)
Multiplying equation (1) by 4.25, we get
4.25C + 4.25A = 1289.25 ------(3)
Subtracting equation (3) from equation (2), we get:
7A - 4.25A = 1824 - 1289.25
Simplifying, we get:
2.75A = 534.75
Dividing by 2.75, we get:
A = 195
Putting A = 195 in equation (1), we get:
C + 195 = 303
Simplifying, we get:
C = 108
So, there were 108 children and 195 adults admitted on that day.
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An Adult membership fee is £120
A Junior membership fee is 1/5
of the Adult fee.
Work out the total membership fee for 2 Adults and 3 Juniors.
Step-by-step explanation:
The membership fee for one Adult is £120, and the membership fee for one Junior is 1/5 of the Adult fee, which is:
1/5 x £120 = £24
So the total membership fee for 2 Adults and 3 Juniors is:
2 x £120 + 3 x £24
= £240 + £72
= £312
Therefore, the total membership fee for 2 Adults and 3 Juniors is £312.
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There are 4 toys in the first box and 7 toys in the second box. what is the ratio of the number of toys in the second box to the total of toys in both boxes
Answer:
7:11
Step-by-step explanation:
To find the ratio, we must find the number of toys in the second box and the total number of toys in both boxes. First, we know that there are 7 toys in the second box. So our ratio is looking like this so far: 7:___.
Next, we must find the total. The total is 4+7 because we need to find the total number of toys in both boxes, and 4+7 is 11. The ratio is 7:11. So now, we finished our ratio.
the graph of a certain geometric sequence can be described as a curve increasing from left to right. which sequence would have a similar graph?
The graph of a certain geometric sequence can be described as a curve increasing from left to right. A sequence that would have a similar graph is the geometric sequence whose common ratio is less than 1.
A geometric sequence is defined as a sequence of numbers where each number is the product of the previous number and a fixed constant. The fixed constant is known as the common ratio, and it is denoted by r.The nth term of a geometric sequence is given by the formula an= ar^(n-1) where a is the first term and r is the common ratio of the sequence.
Similar graphIn a similar graph, the shape of the graph is the same, but the size may be different. Thus, the sequence that would have a similar graph to the given sequence is a geometric sequence whose common ratio is less than 1 because in such a sequence, the successive terms will be smaller than the previous term.This means that as we move from left to right, the values will gradually decrease in size, and the graph will look like a curve increasing from left to right.
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The three interior angles of a triangle measure z degrees , (x+5)degrees , and (x+25) degrees. Which is the measure of one of the three angles?
The three interior angles of a triangle measure z degrees , (x+5)degrees , and (x+25) degrees. So, the measure of one of the three angles is x+5.
The sum of the interior angles of a triangle is always 180 degrees.
Therefore, we can write an equation using the given angle measures:
z + (x+5) + (x+25) = 180
Simplifying the equation, we get:
2x + 30 + z = 180
Subtracting 30 and z from both sides, we get:
2x = 150 - z
Dividing both sides by 2, we get:
x = (150 - z) / 2 - 5
So the measure of one of the three angles is x+5, which is:
x + 5 = [(150 - z) / 2 - 5] + 5
x + 5 = (150 - z) / 2
x + 5 = 75 - z/2
Therefore, one of the angles measures (x+5) degrees.
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A group of workers can plant 56 acres in 8days.
What is their rate in acres per day?
Answer: I believe the answer is 7
Step-by-step explanation:
To solve this I just divided 56 by 8, the answer then is 7. Hope this helps!!!
PLS HELP......!
Fill in the blanks to describe each sequence in words. Then write the missing terms.
• Pyramidal Sequence: Each term in the sequence is the (BLANK) of a perfect square and all perfect squares before it.
1, (BLANK) , 14, 30, (BLANK) , 91, ...
• Fibonacci Sequence: Each term in the sequence is the (BLANK) of the 2 terms before it.
1, 1, 2, 3, (BLANK) , 8, 13, (BLANK) , 34, ...
For a t-distribution with sample size 10, P (t > 1.96) 2 0.0408 and P (t < -1.96) 2 0.0408 . Which of the following is a property of the t-distribution illustrated by the probabilities? A. With sample size 10, the tails of the curve of the t-distribution have more area than the tails of the curve of the z-distribution. B. With sample size 10, the tails of the curve of the t-distribution have less area than the tails of the curve of the z-distribution. C. With sample size 10, the middle of the curve of the t-distribution has more area than the middle of the curve of the z-distribution. With sample size 10, the mean of the t-distribution is greater than the mean of the z-distribution. E. With sample size 10, the mean of the t-distribution is less than the mean of the z-distribution.
The correct option is A) With sample size 10, the tails of the curve of the t-distribution have more area than the tails of the curve of the z-distribution.
Explanation:Given:P(t > 1.96) = 0.0408P(t < -1.96) = 0.0408The t-distribution is different from the z-distribution because it has a thicker tail.The tail is thicker since the t-distribution is determined by the sample size, whereas the z-distribution is determined by the population size (the t-distribution is estimated with the standard error of the mean in this case).
A t-distribution with small degrees of freedom has a much thicker tail than a z-distribution with the same degrees of freedom. Because sample size is related to the degrees of freedom, it makes sense that t-distributions with smaller sample sizes will have thicker tails.
The population variance is replaced with the sample variance in the estimation of the t-distribution. As a result, the t-distribution has heavier tails than the z-distribution.
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Duri wants their backpack to weigh less than 45 pounds.
Use w to represent weights where Duri can carry their backpack.
This inequality represents that the weight of Duri's backpack is w < 45
How to represent the backpack as an expressionGiven that
Weight = Less than 45 pounds
We can use the inequality symbol to represent Duri's weight limit as follows:
w < 45
This inequality states that the weight of Duri's backpack, represented by w, must be less than 45 pounds.
Any weight value for w that satisfies this inequality is within Duri's weight limit and can be carried in their backpack.
For example, if Duri's backpack weighs 30 pounds, then w = 30 satisfies the inequality w < 45, so Duri can carry this weight.
However, if the backpack weighs 50 pounds, then w = 50 does not satisfy the inequality w < 45, so Duri cannot carry this weight.
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an led light bulb is 40% efficient, compared to an incandescent, which is 5% efficient. what exactly do these numbers mean?
An LED light bulb is 40% efficient, compared to an incandescent, which is 5% efficient which means that an LED light bulb will convert 40% of the energy used into visible light, while an incandescent light bulb will only convert 5% of the energy used into visible light.
The other 95% of the energy used by an incandescent bulb is lost as heat, making it less efficient. LEDs are much more energy efficient than incandescent bulbs. LEDs are able to produce the same amount of light as an incandescent bulb with only a fraction of the energy. This is because LEDs emit light in a narrow spectrum which reduces energy loss due to the amount of light that is not visible to the human eye. In addition, LEDs do not contain filaments that require power to heat up, meaning that much less energy is wasted as heat.
LEDs also have a longer lifespan than incandescent bulbs. This is due to the fact that LEDs have no filament to burn out, meaning they can last up to 50,000 hours or more. On the other hand, an incandescent bulb typically has a lifespan of around 1,000 hours. This means that an LED light bulb can potentially last up to 50 times longer than an incandescent bulb.
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what is the answer for 239.3 x 55
Answer:
13,161.5
Step-by-step explanation:
5x3= 15 then go over and carry the one then do 5x9= 45+1= 46 then go over to 3 and carry the four and 5x3= 15+4= 19 put 9 down and carry the one then 5x3= 10+1= 11 after x the 0 the repeat what you did and add up the final numbers so 239.5x 55= 13161.5
interpreting a z score from a sample proportion. suppose that you conduct a hypothesis test about a population proportion and calculate the z score to be 0.47. which of the following is the best interpretation of this value? for the problems which are not a good interpretation, indicate the statistical idea being described. a. the probability is 0.47 that the null hypothesis is true. this is not a good interpretation of for the z score what we are computing is the p() b. if the null hypothesis were true, the probability would be 0.47 of obtaining a sample proportion as far as observed from the hypothesized value of the population proportion. c. the sample proportion is 0.47 standard errors greater than the hypothesized value of the population proportion. d. the sample proportion is equal to 0.47 times the standard error. e. the sample proportion is 0.47 away from the hypothesized value of the population. f. the sample proportion is 0.47
The best interpretation of the z score of 0. 47 from the sample proportion is C. The sample proportion is 0.47 standard errors greater than the hypothesized value of the population proportion.
What does the z - score represent ?The z-score, also known as the standard score, represents the number of standard deviations a data point or observation is away from the mean of a distribution.
A sample proportion with a z-score of 0.47 can be best understood as the sample proportion is 0.47 standard errors higher than the population proportion's estimated value.
This means that the observed sample proportion is 0.47 standard deviations above the hypothesized population proportion under the null hypothesis.
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Solve for y.
2y - 8x = 20
y = [ ? ]x + {?}
Answer:
y = 4x + 10
Step-by-step explanation:
2y - 8x = 20 ( add 8x to both sides )
2y = 8x + 20 ( divide through by 2 )
y = 4x + 10
Answer: y=4x+10
Step-by-step explanation:
researchers are studying a population of small plants on an island. before and after a major drought, they dug up a random sample of plants and measured their root length. during the drought, no plant reproduction took place. what can be concluded from the plot below?
The correct conclusion that can be drawn from the plot is that the plant population had a significantly lower mean root length after the major drought occurred.
Explanation:
A frequency polygon is a graphical representation of a frequency distribution using line segments connected to points that indicate the midpoint of each class interval.
The horizontal axis shows the measurement variable, while the vertical axis shows the number of occurrences of the variable for each bin. The midpoint of each bin is used to represent the distribution of data that falls within that bin's range.
Since the values of adjacent bins overlap, the lines are connected. The points are connected by straight lines in a frequency polygon.
In this case, the conclusion that can be drawn from the plot is that the plant population had a significantly lower mean root length after the major drought occurred.
The peak of the frequency polygon has shifted to the left, indicating that the majority of plants have a shorter root length. The decrease in the mean root length of the plant population is due to the impact of the major drought on the island.
As a result, no plant reproduction took place, indicating a decrease in the plant population.
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while shopping, sofia noticed that her favorite bag of chips were on sale for $5 for 2 bags. how much would 7 bags cost?
Sofia's favorite bag of chips were on sale for $5 for two bags. If she wanted to buy seven bags, the cost would be $17.50.
To calculate this, we need to use the formula for finding the total cost of a given number of items: Total Cost = Number of Items x Cost Per Item.In this case, we need to find the total cost of seven bags, so the equation looks like this: Total Cost = 7 x $5.
We can solve this equation by multiplying the two numbers together. 7 multiplied by 5 equals 35. However, the question is asking us to find the total cost of seven bags, which is $17.50. To convert the answer from 35 to 17.50, we need to divide 35 by two: 35 divided by 2 equals 17.50.
Therefore, the cost of seven bags of Sofia's favorite chips would be $17.50.
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Please help!!! I only have a few minutes left!! I've given screenshots of everything
The solution to the system of equations is: x = 1/2 and y = 4. The steps and justification are given below.
How to Solve a System of Equations?Given the equations, -4x + y = 2 and 4x + y = 6, the following shows the justification for each step taken to solve the system.
Step Justification
1. 2y = 8 1. Add the equations
2. y = 4 2. Divide each side by 2
3. 4x + y = 6 3. Write equation 2
4. 4x + (4) = 6 4. Substitute 4 for y
5. 4x = 2 5. Subtract 4 from each side
6. x = 1/2 6. Divide each side by 4
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