factorise completely.
3x²-12xy

The number of positive integers less than or equal to 200 that are relatively prime to either 15 or 24 but not both is 48 + 64 - 4 = 108.

To solve this problem, we need to count the number of positive integers less than or equal to 200 that are relatively prime to either 15 or 24 but not both.

Let A be the set of positive integers less than or equal to 200 that are relatively prime to 15, and let B be the set of positive integers less than or equal to 200 that are relatively prime to 24. We want to count the number of elements in A union B but not in A intersect B.

To do this, we can use the principle of inclusion-exclusion. The number of elements in A union B is the sum of the number of elements in A and the number of elements in B, minus the number of elements in A intersect B.

The number of elements in A is phi(15) times the number of multiples of 15 less than or equal to 200, which is phi(15) times floor(200/15) = 48, where phi denotes Euler's totient function. Similarly, the number of elements in B is phi(24) times floor(200/24) = 64.

To find the number of elements in A intersect B, we need to find the number of positive integers less than or equal to 200 that are relatively prime to both 15 and 24.

Note that since 15 and 24 are relatively prime, a positive integer is relatively prime to both 15 and 24 if and only if it is relatively prime to their product 15 x 24 = 360. Thus, the number of elements in A intersect B is phi(360) times floor(200/360) = 4.

Therefore, the number of positive integers less than or equal to 200 that are relatively prime to either 15 or 24 but not both is 48 + 64 - 4 = 108.

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The math method that returns the nearest whole number that is greater than or equal to its argument is the "ceiling" function, denoted by ⌈x⌉ in mathematics.

The Ceiling function is denoted by ⌈x⌉, is a mathematical function that takes a real number x as an input and returns the smallest integer that is greater than or equal to x.

The ceiling function takes a real number x as an argument and returns the smallest integer that is greater than or equal to x.

For example, if x = 3.7, then ⌈x⌉ = 4, since 4 is the smallest integer that is greater than or equal to 3.7.

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Answer:

[tex]v = \frac{7}{2}[/tex]

Step-by-step explanation:

Therefore, the length of segment AB is approximately 7.4 units.

Area is a mathematical concept that describes the size of a two-dimensional surface. It is a measure of the amount of space inside a closed shape, such as a rectangle, circle, or triangle, and is typically expressed in square units, such as square feet or square meters. The area of a shape is calculated by multiplying the length of one side or dimension by the length of another side or dimension. For example, the area of a rectangle can be found by multiplying its length by its width.

Here,

To find the radius of the circle, we can use the formula for the area of a sector:

Area of sector = (θ/360) x π x r²

where θ is the central angle of the sector in degrees, r is the radius of the circle, and π is approximately 3.14.

We're given that the area of sector COD is 50.24 square units and the central angle of the sector is 90°. So we can plug in these values and solve for r:

50.24 = (90/360) x 3.14 x r²

50.24 = 0.25 x 3.14 x r²

r² = 50.24 / (0.25 x 3.14)

r² = 201.28

r = √201.28

r ≈ 14.2

Therefore, the radius of the circle is approximately 14.2 units.

Next, we need to find the length of segment AB. Since AB is a chord of the circle, we can use the formula:

AB = 2 x r x sin(θ/2)

where θ is the central angle of the sector in degrees, r is the radius of the circle, and sin() is the sine function.

We're given that the central angle of sector COD is 30°. So we can plug in this value and the radius we found earlier to solve for AB:

AB = 2 x 14.2 x sin(30/2)

AB = 2 x 14.2 x sin(15)

AB ≈ 7.4

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The smallest integer n such that a_n < 1 is n = -2.

Let the common ratio of the geometric progression be denoted by r. Then we have

a_2 = a_1 × r

a_3 = a_2 × r = a_1 × r^2

a_4 = a_3 × r = a_1 × r^3

a_5 = a_4 × r = a_1 × r^4

So in general, we have

a_n = a_1 × r^(n-1)

Now, we can use the given equation

(a_1357)^3 = a_34

Substituting the expressions above for a_34 and a_1357, we get

(a_1 × r^33)^3 = a_1 × r^3

Simplifying this equation by dividing both sides by a_1×r^3 and taking the cube root, we get

r^10 = 1/ (a_1^2)

Now, we need to find the smallest integer n such that a_n < 1. Using the expression for a_n above, we get

a_n < 1

a_1 × r^(n-1) < 1

r^(n-1) < 1/a_1

Taking the logarithm of both sides (with base r), we get

n-1 < log_r (1/a_1)

n < log_r (1/a_1) + 1

We know that r^10 = 1/ (a_1^2), so

1/a_1 = r^(10/2) = r^5

Substituting this into the expression above for n, we get

n < log_r (1/r^5) + 1

n < -5 + 1

n < -4

Since n is an integer, the smallest possible value for n is -3. However, this does not make sense since we cannot have a negative index for a term in the geometric progression. Therefore, the smallest integer n such that a_n < 1 is n = -2.

To verify this, we can substitute n = -2 into the expression for a_n and see if it is less than 1

a_n = a_1 × r^(n-1)

a_{-2} = a_1 × r^(-3)

Since a_1 > 1, we just need to show that r^3 > 1 to prove that a_{-2} < 1. From the equation r^10 = 1/ (a_1^2), we have

r^3 = (r^10)^(3/10) = (1/a_1^2)^(3/10) > 1

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The given question is incomplete, the complete question is:

Let a_1, a_2, a_3, a_4, a_5, . . . be a geometric progression with positive ratio such that a_1 > 1 and

(a_1357)^3 = a_34. Find the smallest integer n such that a_n < 1.

The graph illustrates the linear inequality [tex]y > x - 2[/tex] and [tex]x - 2y < 4[/tex].

The equation-like form of the formula 5x 4 > 2x + 3 has an arrowhead in lieu of the equals sign. That is an illustration of inequity. This shows that the left half, 5x 4, is bigger than the right part, 2x + 3. Finding the x numbers where the inequality holds true is what we are most interested in.

In mathematics, "inequality" means the connection between two reactions or values that is not equal to one another. As either an outcome, inequality occurs because of an imbalance.

We can see that the shaded region is above the line [tex]y = x - 2[/tex], which represents the inequality [tex]y > x - 2[/tex]. Additionally, the shaded region is below the line [tex]x - 2y = 4[/tex], which represents the inequality [tex]x - 2y < 4[/tex].

As a result, the graph's representation of a linear inequality arrangement is as follows:

[tex]y > x - 2[/tex] and [tex]x - 2y < 4[/tex]

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Answer:

d

Step-by-step explanation:

The second decile of the given data set, "24, 64, 25, 40, 45, 34, 14, 26, 28, 24, 58, 51"  is 24.

To find the second decile of a data set, we first need to arrange the data in order from lowest to highest, that is :

⇒ 14, 24, 24, 25, 26, 28, 34, 40, 45, 51, 58, 64

The second decile represents the value that divides the data into two parts, where 20% of the data is below the value and 80% of the data is above the value.

Since there are 12 data points in this set,

So, 20% of the data is equal to 0.2 × 12 = 2.4.

Since we cannot have a fractional data point, we round up to 3.

So, the second decile is the third value in the ordered data set.

which is 24.

Therefore, The second decile is 24.

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Answer:

0.048 in %

Step-by-step explanation:

firstly: remove the decimal point

= 48/1000

secondly : Simplify

48/1000*100

=48/10

=4.8%

Answer:j+309=313

Step-by-step explanation:

313-309=4

J=4

Step-by-step explanation:

(-1, 1) is a solution.

because

-(-1) + 6×1 = 7

1 + 6 = 7

7 = 7

correct.

every ordered pair of x and y values that make the equation true is a solution.

(5, 2) would be another solution. and so on.

Answer: -3/2

Step-by-step explanation:

FIrst rearrange the equation in y = mx + b form.

4x - 6y = -24

-6y = -4x - 24

y = 2/3x + 4

If the line is perpendicular, the slope must be the negative reciprocal of the current line.

The negative reciprocal of 2/3 is -3/2.

A company manufactures rubber balls, random variable X in words is diameter of the rubber ball, standard deviation is -1.5 and z-score of the x = 2 is 2.123.

A random variable is a variable with an unknown value or a function that gives values to each of the results of an experiment. Random variables are frequently identified by letters and fall into one of two categories: continuous variables, which can take on any value within a continuous range, or discrete variables, which have specified values.

In probability and statistics, random variables are used to measure outcomes of a random event, and hence, can take on various values. Real numbers are often used as random variables since they must be quantifiable.

1) X denotes the diameter of the rubber ball.

So the correct option was A. (option A)

Therefore,  the random variable X in words is diameter of the rubber ball.

2) For 1.5 Standard deviations left to the mean , Z score will be -1.5

option(A)

So, standard deviation to the left of the mean is -1.5.

3) [tex]Z=\frac{(x-\mu)}{\sigma}[/tex]

x=2

sigma = √2

Z = 2-(-1)/ √2

Z = 3/√2

Z = 2.123

Hence, the z-score of the x = 2 is 2.123.

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Complete question:

A company manufactures rubber balls. The mean diameter of a rubber ball is 12 cm with a standard deviation of 0.2 cm. Define the random variable X in words. X

diameter of a rubber ball

rubber balls

mean diameter of a rubber ball

12 cm

Question 2 What is the z-score of x=9, if it is 1.5 standard deviations to the left of the mean? Hint: the z-score of the mean is =0 −1.5 1.5 9 Question 3 Suppose X∼N(−1,2). What is the z-score of x=2 ? Hint: z=(x−μ)/σ 1.5 −1.5 0.2222

Answer:

34. (c) 12

35. (a) -12

36. (a) 51

37. (a) 13

Step-by-step explanation:

34.)

[tex] \implies \: \sf\dfrac{4}{xx + 2} = \dfrac{6}{2xx - 3} \\ \\ \implies \: \sf4(2xx - 3) = 6(xx + 2) \\ \\ \implies \: \sf 8xx - 12 = 6xx + 12 \\ \\ \implies \: \sf 8xx - 6xx = 12 + 12 \sf \\ \\ \implies \: \sf 2xx = 24 \\ \\ \implies \: \sf xx = \dfrac{24}{2} \\ \\ \implies \: \sf xx = 12 \\ [/tex]

Hence, Required answer is option (c) 12.

35.)

[tex] \implies \: \sf \dfrac{xx - 2}{2} = \dfrac{3xx + 8}{4} \\ \\ \implies \: \sf2(3xx + 8) = 4(xx - 2) \\ \\ \sf 6xx + 16 = 4xx - 8 \\ \\ \implies \: \sf 6xx - 4xx = - 8 - 16 \\ \\ \implies \: \sf 2xx = - 24 \\ \\ \implies \: \sf xx = \dfrac{ - 24}{2} \\ \\ \implies \: \sf xx = - 12 \\ [/tex]

Hence, Required answer is option (a) -12.

36.)

[tex] \implies \: \sf\sqrt{xx - 2} = 7 \\ \\ \implies \: \sf xx - 2 = {(7)}^{2} \\ \\ \implies \: \sf xx - 2 = 49 \\ \\ \implies \: \sf xx = 49 + 2 \\ \\ \implies \: \sf xx = 51[/tex]

Hence, Required answer is option (a) 51.

37.)

[tex] \implies \: \sf \sqrt{2xx - 10} = 4 \\ \\ \implies \: \sf 2xx - 10 = {(4)}^{2} \\ \\ \implies \: \sf 2xx - 10 = 16 \\ \\ \implies \: \sf 2xx = 16 + 10 \\ \\ \implies \: \sf 2xx = 26 \\ \\ \implies \: \sf xx = \dfrac{26}{2} \\ \\ \implies \: \sf xx = 13 \\ [/tex]

Hence, Required answer is option (a) 13.

Answer:

Step-by-step explanation:

The statement that is true about an angle drawn in standard position is that one side is always aligned with the positive x-axis. The other side of the angle can be aligned with either the positive y-axis or the negative y-axis. The vertex of the angle does not necessarily have to be at point (1,1) and positive angles are measured counterclockwise.

Answer: 565.92 square inches.

Step-by-step explanation:

To find the area of the paperboard that remains, we need to subtract the area of the semicircle from the area of the rectangle.

The rectangle has a length of 33 inches and a width of 24 inches, so its area is:

A_rect = length x width

A_rect = 33 in x 24 in

A_rect = 792 sq in

To find the area of the semicircle, we need to first find its radius. The diameter of the semicircle is the same as the width of the rectangle, which is 24 inches. So, the radius is:

r = 1/2 x diameter

r = 1/2 x 24 in

r = 12 in

The area of the semicircle is:

A_semicircle = 1/2 x pi x r^2

A_semicircle = 1/2 x 3.14 x 12^2

A_semicircle = 1/2 x 3.14 x 144

A_semicircle = 226.08 sq in

To find the area of the paperboard that remains, we subtract the area of the semicircle from the area of the rectangle:

A_remaining = A_rect - A_semicircle

A_remaining = 792 sq in - 226.08 sq in

A_remaining = 565.92 sq in

Therefore, the area of the paperboard that remains is 565.92 square inches.

If △ABC is congruent to △DEF, then it must also be a right triangle. Thus, the two triangles have congruent sides and are congruent.

A key idea in geometry known as the Pythagorean theorem explains the relationship between the sides of a right triangle. The square of the hypotenuse, or side opposite the right angle, is said to be equal to the sum of the squares of the other two sides. It can be expressed mathematically as: a² + b² = c²

given

By defining a right triangle, DEF, with sides a, b, and x, we can demonstrate the opposite of the Pythagorean theorem. Then, we'll demonstrate that if ABC is a triangle with sides a, b, and c where a2 + b2 = c2, it is congruent to DEF and is thus a right triangle because a2 + b2 = c2.

By the Pythagorean theorem, because △DEF is a right triangle, a² + b² = x².

When a2 + b2 = c2 and a2 + b2 = x2, c2 equals x2.

If △ABC is congruent to △DEF, then it must also be a right triangle.Thus, the two triangles have congruent sides and are congruent.

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If △ABC is congruent to △DEF, then it must also be a right triangle. Thus, the two triangles have congruent sides and are congruent.

A key idea in geometry known as the Pythagorean theorem explains the relationship between the sides of a right triangle. The square of the hypotenuse, or side opposite the right angle, is said to be equal to the sum of the squares of the other two sides. It can be expressed mathematically as: a² + b² = c²

By defining a right triangle, DEF, with sides a, b, and x, we can demonstrate the opposite of the Pythagorean theorem. Then, we'll demonstrate that if ABC is a triangle with sides a, b, and c where [tex]a^2 + b^2 = c^2[/tex], it is congruent to DEF and is thus a right triangle because a2 + b2 = c2.

By the Pythagorean theorem, because △DEF is a right triangle, a² + b² = x².

When[tex]a^2 + b^2 = c^2[/tex] and [tex]a^2 + b^2 = x^2[/tex], [tex]c^2[/tex] equals [tex]x^2[/tex].

If △ABC is congruent to △DEF, then it must also be a right triangle. Thus, the two triangles have congruent sides and are congruent.

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Answer: A repeated decimal 0.833333333

Step-by-step explanation: Reduce the expression by canceling common factors.

The value after minimizing f(x, y) = x2 + y2 with respect to constraint - x + 2y − 10 = 0, using Lagrange multipliers, is 50.

To solve this problem using Lagrange multipliers, we first write the function to be minimized as:

f(x,y) = x² + y²

And the constraint equation as:

g(x,y) = x + 2y - 10 = 0

We then form the Lagrangian function L(x,y,λ) as follows:

L(x,y,λ) = f(x,y) - λg(x,y)

Substituting in our expressions for f(x,y) and g(x,y), we get:

L(x,y,λ) = x² + y² - λ(x + 2y - 10)

Now, we take partial derivatives of L with respect to x, y and λ and set them equal to zero:

∂L/∂x = 2x - λ = 0 ∂L/∂y = 2y - 2λ = 0 ∂L/∂λ = x + 2y - 10 = 0

Solving these equations simultaneously gives us:

x = λ y = λ/2 x + 2y - 10 = 0

Substituting these values back into our original function f(x,y), we get:

f(5,5) = (5)² + (5)² = 50

Therefore, the minimum value of f(x,y) subject to the given constraint is 50.

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Answer:

Step-by-step explanation:

To solve each of these equations, we need to isolate the variable (x) on one side of the equation. Here are the steps to solve each equation:

9x + 6 = 24

Subtract 6 from both sides:

9x = 18

Divide both sides by 9:

x = 2

Therefore, the solution to the equation is x = 2.

8x - 4 = 28

Add 4 to both sides:

8x = 32

Divide both sides by 8:

x = 4

Therefore, the solution to the equation is x = 4.

-18 - x = 57

Add 18 to both sides:

-x = 75

Multiply both sides by -1:

x = -75

Therefore, the solution to the equation is x = -75.

-4 - 8x = 8

Add 4 to both sides:

-8x = 12

Divide both sides by -8:

x = -1.5

Therefore, the solution to the equation is x = -1.5.

3x + 0.7 = 4

Subtract 0.7 from both sides:

3x = 3.3

Divide both sides by 3:

x = 1.1

Therefore, the solution to the equation is x = 1.1.

Answer:

The value could be any length between 3 and 15

Step-by-step explanation:

9 - 6 = 3

and

9 + 6 = 15

Important points to conduct a survey are; to gather information, make informed decisions, evaluate programs or services, identify trends, assess needs.

Surveys are conducted for a variety of reasons, including gathering information, making informed decisions, evaluating programs or services, identifying trends, and assessing needs. By using surveys, organizations can collect valuable data that can be used to inform decisions, improve programs or services, and better understand their target audience.

Surveys, also known as questionnaires, are used to gather information from a targeted group of individuals or a population. Surveys are an important tool for collecting data in a structured manner and can be used for a variety of reasons. Here are some of the reasons why surveys are conducted:

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Answer:

Step-by-step explanation:

[tex]p(x)=2x^2-ax+1\\\\p(1)=3\times 1^2-a\times1+1=4-a\\\\\text{but } p(1)=2 \text{ So,}\\\\4-a=2 \rightarrow a=2[/tex]

Answer:2

Step-by-step explanation:

p(x)--->p(1)

means you should write 1 instead of every x and then make whole equation equal ro 2:

3*1^2-a*1+1=3-a+1=2

-a+4=2

-a=-2

a=2

Answer:

a=18 b=24

Step-by-step explanation:

We know that BC=25 and QR=30, the key term is that they are similar triangles. Therefore, BC: QR=25:30=5:6. Then BA:A=5:6=15:X

x=a=18

20:b=5:6

b=24

the sides from shortest to longest in the triangle is DE,DF and EF.

A triangle is defined as a two-dimensional geometric shape that has three straight sides and three angles. The three sides of a triangle are referred edges or legs, and the three angles are formed where the edges or legs meet.

∠DEF=58°

∠DFE=180°-147°=33°

∠DEF+∠DFE+∠EDF=180°

∠EDF=180°-33°-58°

=89°

The greatest side and the largest angle of a triangle are opposite one another, as are the shortest side and the smallest angle.

Increasing order of angle

∠DFE,∠DEF,∠EDF

Arranging the sides from shortest to longest DE,DF and EF.

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Answer:

So the lengths of the legs are approximately 8.6 cm and 15.6 cm.

Step-by-step explanation:

Let's call the smaller leg "x" and the larger leg "x + 7". According to the Pythagorean theorem, we know that:

x^2 + (x + 7)^2 = 17^2

Expanding the square on the left side and simplifying, we get:

2x^2 + 14x - 210 = 0

Dividing both sides by 2, we get:

x^2 + 7x - 105 = 0

Now we can solve for x using the quadratic formula:

x = (-7 ± sqrt(7^2 - 4(1)(-105))) / 2(1)

x = (-7 ± sqrt(649)) / 2

x ≈ -15.6 or x ≈ 8.6

Since we're dealing with lengths of sides in a triangle, we can't have a negative value for x. So we discard the negative solution and conclude that the smaller leg is approximately 8.6 cm.

To find the larger leg, we add 7 to x:

x + 7 ≈ 15.6 cm

The z-score for Alexandria's test grade is 0.95 standard deviations.

What is standard deviation ?

Standard deviation is a measure of the amount of variation or dispersion in a set of data. It tells us how spread out the data is from the mean or average value. A low standard deviation indicates that the data points are close to the mean, while a high standard deviation indicates that the data points are spread out over a wider range of values.

To calculate the standard deviation, we first find the mean of the data. Then, for each data point, we subtract the mean and square the difference. We take the average of these squared differences, and then take the square root of that average. This gives us the standard deviation.

a) The z-score for Alexandria's test grade can be calculated using the formula:

z = (x - μ) / σ

where x is the test score, μ is the mean of the Math test scores, and σ is the standard deviation of the Math test scores.

Plugging in the values, we get:

z = (82 - 71.5) / 11.1 = 0.95

So Alexandria's test score is 0.95 standard deviations above the mean of the Math test scores.

b) The z-score for Christina's test grade can be calculated in the same way:

z = (x - μ) / σ

where x is the test score, μ is the mean of the Science test scores, and σ is the standard deviation of the Science test scores.

Plugging in the values, we get:

z = (61.2 - 62.2) / 8.2 = -0.12

So Christina's test score is 0.12 standard deviations below the mean of the Science test scores.

c) To determine who did relatively better, we need to compare the z-scores for Alexandria and Christina. Alexandria's z-score of 0.95 indicates that her test score is above average compared to the other Math test scores. Christina's z-score of -0.12 indicates that her test score is slightly below average compared to the other Science test scores. Therefore, Alexandria did relatively better than Christina.

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Answer:

Yes

Step-by-step explanation:

Yes because 1+14=15 hours and that is more than two

Once we have this data, we can substitute the values into the formula to find the empirical probability that a person prefers apple pie given that they prefer whipped cream. Therefore, the missing probability is 1/12, and we know this because it is the value that makes the sum of all probabilities equal to 1.

Probability is a measure of the likelihood of an event occurring. It is expressed as a number between 0 and 1, where 0 represents an event that is impossible, and 1 represents an event that is certain to occur. For example, if the probability of winning a coin toss is 1/2, this means that there is an equal chance of the coin landing heads or tails. Probability can be calculated by dividing the number of favorable outcomes by the total number of possible outcomes. This is known as the classical probability approach. Another approach is empirical probability, where probabilities are calculated based on observed data or experiments. Lastly, subjective probability involves making an informed guess or estimate about the likelihood of an event occurring based on subjective factors such as experience, intuition, or expert opinion. Probability is a fundamental concept in statistics and is used in many application

Here,

1. The formula needed to calculate the empirical probability that a person prefers apple pie given that they prefer whipped cream is:

P(Apple Pie | Whipped Cream) = P(Apple Pie and Whipped Cream) / P(Whipped Cream)

where P(Apple Pie and Whipped Cream) is the probability that a person prefers both apple pie and whipped cream, and P(Whipped Cream) is the probability that a person prefers whipped cream.

This formula is used because it is a conditional probability, which is a measure of the probability of an event occurring given that another event has occurred. In this case, we want to find the probability that a person prefers apple pie given that they already prefer whipped cream.

To calculate P(Apple Pie and Whipped Cream), we would need to gather data on the number of people who prefer both apple pie and whipped cream. Similarly, to calculate P(Whipped Cream), we would need to gather data on the number of people who prefer whipped cream.

2. To find the missing probability, we need to use the fact that the sum of all probabilities in a probability distribution must be equal to 1. Therefore, we can set up an equation to solve for the missing probability:

1/6 + 1/3 + x + 5/12 = 1

Simplifying the equation by finding a common denominator gives:

2/12 + 4/12 + x + 5/12 = 1

Combining like terms gives:

11/12 + x = 1

Subtracting 11/12 from both sides gives:

x = 1 - 11/12

x = 1/12

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