X and Z are not independent random variable as P(X = x and Z = z) and P(X = x) * P(Z = z) are not equal for most values of x and z by using their joint probability
To prove that two random variables are independent, to show that their joint probability distribution is equal to the product of their individual probability distributions. Start with part (a) and then move on to part (b).
(a) Let X be the first digit and Y be the second digit of the chosen number from the set {10, 11, 12, ..., 99}.
Step 1: Calculate the probabilities of X and Y individually.
To find the probability of X taking any particular value,
there are 9 numbers (10 to 99) that start with each digit from 1 to 9. Since we choose a number uniformly at random, each first digit has an equal chance of being selected.The probability of X taking any value x (where x is a digit from 1 to 9) is given by:
P(X = x) = (number of numbers starting with digit x) / (total numbers)
= 9 / 90
= 1 / 10
Similarly, for Y, since there are 10 digits (0 to 9) and choose the second digit uniformly at random, the probability of Y taking any value y is:
P(Y = y) = (number of numbers with digit y as the second digit) / (total numbers)
= 10 / 90
= 1 / 9
Step 2: Calculate the joint probability P(X, Y).
The joint probability is the probability that X takes a particular value x and Y takes a particular value y simultaneously.
P(X = x and Y = y) = (number of numbers with digit x as the first digit and digit y as the second digit) / (total numbers)
= 1 / 90
Step 3: Check if X and Y are independent.
Two random variables X and Y are independent if and only if their joint probability equals the product of their individual probabilities.
P(X = x and Y = y) = P(X = x) * P(Y = y)
1 / 90 = (1 / 10) * (1 / 9)
Since the equation holds for all values of x and y, conclude that X and Y are independent random variables.
(b) Let X be the first digit of the chosen number and Z be the sum of the two digits.
Step 1: Calculate the probabilities of X and Z individually.
Calculated the probability distribution for X in part (a), which is P(X = x) = 1 / 10 for x = 1 to 9.
To calculate the probability distribution for Z, the sum of the two digits:
For Z = 0: There is only one number, 10, with a sum of digits equal to 0.
P(Z = 0) = 1 / 90
For Z = 1: There are two numbers, 10 and 01, with a sum of digits equal to 1.
P(Z = 1) = 2 / 90
= 1 / 45
For Z = 2: There are three numbers, 11, 20, and 02, with a sum of digits equal to 2.
P(Z = 2) = 3 / 90
= 1 / 30
For Z = 3: There are four numbers, 12, 21, 03, and 30, with a sum of digits equal to 3.
P(Z = 3) = 4 / 90
= 2 / 45
For Z = 4: There are five numbers, 13, 31, 04, 40, and 22, with a sum of digits equal to 4.
P(Z = 4) = 5 / 90
= 1 / 18
For Z = 5: There are four numbers, 14, 41, 23, and 32, with a sum of digits equal to 5.
P(Z = 5) = 4 / 90
= 2 / 45
For Z = 6: There are three numbers, 15, 51, and 24, with a sum of digits equal to 6.
P(Z = 6) = 3 / 90
= 1 / 30
For Z = 7: There are two numbers, 25 and 52, with a sum of digits equal to 7.
P(Z = 7) = 2 / 90
= 1 / 45
For Z = 8: There are two numbers, 26 and 62, with a sum of digits equal to 8.
P(Z = 8) = 2 / 90
= 1 / 45
For Z = 9: There is only one number, 36, with a sum of digits equal to 9.
P(Z = 9) = 1 / 90
Step 2: Calculate the joint probability P(X, Z).
The joint probability is the probability that X takes a particular value x and Z takes a particular value z simultaneously.
Formula:
P(X = x and Z = z) = (number of numbers with digit x as the first digit and the sum of digits equal to z) / (total numbers)
Check the case where X = 1 and Z = 2 as an example:
Numbers with digit 1 as the first digit and a sum of digits equal to 2: 12 and 21 (two numbers)
P(X = 1 and Z = 2) = 2 / 90
= 1 / 45
Step 3: Check if X and Z are independent.
To determine if X and Z are independent, we need to compare their joint probability to the product of their individual probabilities for all values of x and z.
However, if compare the joint probabilities P(X = x and Z = z) to P(X = x) * P(Z = z), find that they are not equal for most values of x and z.
Therefore, conclude that X and Z are not independent random variables.
as P(X = x and Z = z) and P(X = x) * P(Z = z) are not equal for most values of x and z.
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1(x-3)^2-2/(x^2-9)+1/(x+3)^2
Answer:
Here is your answer. Hope this helps.
Find a general solution to the differential equation -6y = 1+x+y+xy by solving the equation and then applying the initial condition y(-1) = C.
The general solution with the initial condition y(-1)=C is y=-(1+x)/(6+x)=-2/7.
The general solution to the differential equation -6y = 1+x+y+xy is y = -(1+x)/(6+x). To apply the initial condition, we can substitute x=-1 and y=C into the above equation and solve for C. We get C=-(2/7). Therefore, the general solution to the differential equation with the initial condition y(-1)=C is y=-(1+x)/(6+x)=-2/7.
Substituting x=-1 and y=C into the equation:
-6C = 1-1+C+(-1)C
-7C = 1
C = -1/7
Therefore, the general solution with the initial condition y(-1)=C is y=-(1+x)/(6+x)=-2/7.
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Which interval can be used to construct the inverse of f(x)?
Answer:
pie/8, 3pie/8
Or
B
PLS HELP Select the correct answer.
You work delivering food for a local farmer. You sell onions from the farm at these prices: 2 for $1.04, 4 for $2.52, 8 for $4.32, and 16 for $9.12. What is the unit cost per onion for the worst deal?
A.
$0.52
B.
$0.57
C.
$0.63
D.
$0.67
Answer:
C
Step-by-step explanation:
2.52/4 =0.63/1
A car dealership pays a wholesale price of $14,000 to purchase a vehicle. They sell the car for $16,100. If the dealership decided to use a 20% markup instead, how much MORE would they make?
If the dealership decided to use a 20% markup instead, then they earn $700 more.
What is Percentage?percentage, a relative value indicating hundredth parts of any quantity.
Given that A car dealership pays a wholesale price of $14,000 to purchase a vehicle.
They sell the car for $16,100
If the dealership decided to use a 20% markup instead, we have to find the hu=ow much more would they make.
Let us find 20% of 14000
20/100×14000
0.2×14000
2800
So with 20% the price will become $16800.
As they sell the car for $16100.
The difference between $16800-$16100 is $700
Hence, If the dealership decided to use a 20% markup instead, then they earn $700 more.
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How to answer this one?
Answer:
this is a little typical
Find the class width for this histogram.
The class width of the histogram is 5
What is histogram?A histogram is a visual representation of statistical data that makes use of rectangles to illustrate the frequency of data points over a range of successive numerical intervals of equal width called the class width.
The independent variable and dependent variable are plotted along the horizontal axis and the vertical axis, respectively, in the most common type of histogram.
The class width is plotted in the x - direction. this is the difference between the two consecutive values in the boundary.
class width = 100.5 - 95.5 = 5
class width = 85.5 - 80.5 = 5
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The radioactive substance cesium-137 has a half-life of 30 years. The amount A (t) (in grams) of a sample of cesium-137 remaining after t years is given by the
following exponential function.
- 647 (12)
A (t) = 64
30
Find the initial amount in the sample and the amount remaining after 100 years.
Round your answers to the nearest gram
?
To get the initial mass, set t = 0.
A(0) = 621 (1/2)0/30 g = 621 (1) g = 621 g
To get the mass after 100 years, just plug t = 100 into A(t) and evaluate.
A(100) = 621 (1/2)100/30 g = ? g
What is mass?Mass, in physics, quantitative measure of inertia, a fundamental property of all matter. It is, in effect, the resistance that a body of matter offers to a change in its speed or position upon the application of a force. The greater the mass of a body, the smaller the change produced by an applied force. The unit of mass in the International System of Units (SI) is the kilogram, which is defined in terms of Planck’s constant, which is defined as equal to 6.62607015 × 10−34 joule second. One joule is equal to one kilogram times metre squared per second squared. With the second and the metre already defined in terms of other physical constants, the kilogram is determined by accurate measurements of Planck’s constant.To learn more about International System of Units (SI) refer to:
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A right triangle A B C. Angle A C B is a right angle. Angle B A C is unknown. Side A C is six units. Side B C is three units.
The picture explains it all. We use tangent rule to find out the measure of angle A.
The measure of angle unknown angle is 30°.
Use the concept of the triangle defined as:
A triangle is a three-sided polygon, which has three vertices and three angles which has a sum of 180 degrees.
Given that,
Side AC = 6 units
Side BC = 3 units
Let ∠BAC = θ
Since we know that,
Sin θ = opposite/Hypotenuse
Sin θ = BC/AC
Put the values,
Sin θ = 3/6
Sin θ = 1/2
Since we know that,
sin(π/6) = 1/2
Therefore,
Sin θ = sin(π/6)
Take the inverse of sin on both sides we get,
θ = π/6 or 30°
Hence,
The measure of angle unknown angle is 30°.
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a company prudces 3 types of cables a b c in-house production costs per foot of cables abc are 6 8 10
The in-house production amount costs per foot of cables A, B, and C are $6, $8, and $10, respectively.
1. The question states that a company produces three types of cables, A, B, and C.
2. The in-house production amount costs for each type of cable is given in the question. The cost per foot of cable A is $6, the cost per foot of cable B is $8, and the cost per foot of cable C is $10.
The company produces three types of cables, A, B, and C, and each type of cable has a different in-house production cost. Cable A has a cost per foot of $6, cable B has a cost per foot of $8, and cable C has a cost per foot of $10. This means that if the company produces 100 feet of cable A, the in-house production cost would be $600, whereas if they produce 100 feet of cable B, the in-house production cost would be $800, and if they produce 100 feet of cable C, the in-house production cost would be $1000.
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The first three terms of a sequence are given. Round to the nearest thousandth (if necessary).
2, 6, 10,..
Find the 41st term.
The given sequence appears to be an arithmetic sequence, as the common difference between consecutive terms is 4.
The nth term of an arithmetic sequence is given by the formula: a_n = a_1 + (n-1)d, where a_1 is the first term, d is the common difference and n is the nth term.
Given that the first term of the sequence is 2, the common difference is 4.
To find the 41st term of the sequence, we can use the formula:
a_41 = 2 + (41-1) * 4 = 2 + 40 * 4 = 2 + 160 = 162
Therefore, the 41st term of the sequence is 162.
-25 POINTS-
Y=[?]x+[?]
Don’t need to explain. Guessers will be reported.
The linear equation in two variables is -5X + 10 = Y
What is linear equation?A linear equation in mathematics is an equation that may be written as follows: Y = aX + b where a and b are the coefficients, which are frequently real numbers, and X and Y are the variables (or unknowns). As long as they don't contain any of the variables, the coefficients can be thought of as equation parameters and can be arbitrary expressions. The coefficients must not all be 0 in order to produce a meaningful linear equation.
A linear polynomial over a field, from which the coefficients are taken, can also be equalised to zero to produce a linear equation.
Such an equation's solutions are the values that, when used as placeholders for the unknowns, result in the equality being true.
Let us assume that given equation as linear equation in two variables
Y = aX + b
Also, from the given table, the value which satisfy the equations are
X =3, Y = -5 and X = 4, Y = -10
using these values we have
-5 = 3a +b and -10 = 4a +b
subtracting both the equations we get,
-5 = a
Now putting this value in an equation we get
b = 10
so the general equation is -5X + 10 = Y
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what is the volume of the figure below, which is composed of two cubes with side lengths of 5 units?
Answer:
250u2
Step-by-step explanation:
The volume of a cube is found by multiplying LxWxH. 5x5x5-125, 125x2-250
Hope this helped :)
Sam has 3 1/4 pounds of blueberries. Ben also has some blueberries. Together, they have 4 2/3 pounds of blueberries. Create an equation to represent the number of pounds of blueberries, b, Ben has.
Answer:
2 1/2 + b = 4 2/3
What is the positive solution to this equation?
2x2+13x=57
your solution
4+13x=57
x=53÷4 = 13.25
What’s the difference between these two
at a party with 10 people each person shakes the hand of every other person once how many handshakes occur
The number of handshakes altogether exists 45.
What is meant by combinations?In mathematics, there are two alternative methods for dividing up a collection of components into subsets: combination and permutation. The subset's components can be arranged in any sequence when used together. The components of the subset are arranged in a particular order in a permutation.
A combination in mathematics is a choice of elements from a set with distinct members, where the order of the choices is irrelevant.
Combinations are a mathematical method for calculating the number of alternative arrangements in a collection of objects where the order of the selection is irrelevant. You are free to choose any combination of the things.
Given:
Number of people in the conference = 10
Each person shakes hands with others.
The total number of handshakes [tex]$={ }^n \mathrm{C}_2$[/tex]
Total number of handshakes [tex]$={ }^{10} C_2=(10 \times 9) / 2=45$[/tex] handshakes.
The complete question is:
At the end of a meeting all the ten people present shake hands with each other once. The number of handshakes altogether is
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dividend in arrears means that
Answer: Since the company is running short of cash they have failed or late to pay the shareholders' share of the profit which is the dividend.
find a set r of representatives such that every power of x is equal to exactly one element of r (and prove it.)
Consequently, x∈R denotes that x belongs to the set of Real numbers. This means that x is a real number.
What is meant by Real numbers?Any number that can be used to quantify a continuous, one-dimensional quantity, such as time, temperature, or distance, is referred to as a real number. Continuous in this context suggests that values may vary by arbitrary little amounts.An endless decimal expansion allows for a nearly universal representation of every real number. Rational numbers like positive and negative integers, fractions, and irrational numbers are all examples of real numbers. In other words, every number we can come up with aside from complicated numbers is a real number. Examples of real numbers include 3, 0, 1.5, 3/2, √5, and so forth.Real numbers are all entire numbers. Irrational numbers are integers. Natural numbers are numbers that make sense.Hence, As a result, the symbol x∈R indicates that x is a member of the Real number set. Thus, x must be a real number.
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What is the degree of the polynomial
Answer:
A polynomial’s degree is the highest or the greatest power of a variable in a polynomial equation. The degree indicates the highest exponential power in the polynomial (ignoring the coefficients).
For example: 6x4 + 2x3+ 3 is a polynomial. Here 6x4, 2x3, 3 are the terms where 6x4 is a leading term and 3 is a constant term. The coefficients of the polynomial are 6 and 2.
The degree of the polynomial 6x4 + 2x3+ 3 is 4.
tracy noticed that the serving size for a boxed meal is 1.5 cups and each contains 148 milligrams stacy used the entire box to make 6 cups of food. How many milligrams are in the meal tracy prepared
Answer:
5.92
Step-by-step explanation:
Set up a proportion
cups/mg = cups/mg
[tex]\frac{1.5}{148}[/tex] = [tex]\frac{6}{x}[/tex]
1.5 x 4 = 6,
so 1.48 x 4 = 5.92
9x+7y=4 solve for y.
(c) use the answers to parts (a) and (b) to estimate by how much the average monthly cell phone bill changed between 1990 and 2008. (round your answer to two decimal places.)
The area above x axis is 12.6 sq units while the area under the y-axis is 42.443 sq units. The change in the average phone bill is 29.843 units.
a) Here we see that the graph is in the shape of a right-angled triangle above the x-axis
The area of a triangle = 1/2 X Base X Height
Here the base will be the measure of the distance of co-ordinates on the x-axis
The height is the distance between the coordinates on the axis for that triangle
= 1/2 X (16 - 8) X (3.15 - 0)
= 1/2 X 8 X 3.15
= 12.60 sq units
b)
Below the x-axis, the graph is a rectangle and a triangle
The area of the rectangle = length X width
Here length is the distance of the co-ordinates of the y-axis and width is the distance between the coordinates for the x-axis
Hence we get
(0 - (-5.14))(8 - 0)
= 5.14 X 8
= 41.12 sq units for the rectangle.
The area of the triangle will be
1/2 X (18 - 15.48) X (0 - (-1.05))
1/2 X 2.52 X 1.05
= 1.323 sq unit
Hence, the total area is
41.12 + 1.323
= 42.443 sq unit
c)
Since the area under the graph represents change,
Here we see that the change in the average cell phone bill will be
42.443 + 12.6
= 55.043
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Complete Question
(Image Attached)
Solve the homogeneous differential equation in terms of x and y. A homogeneous differential equation is an equation of the form M(x, y) dx + N(x, y) dy = 0, where M and N are homogeneous functions of the same degree. To solve an equation of this form by the method of separation of variables, use the substitutions y = vx and dy = x dv + v dx.
(x^3 + y^3) dx − xy^2 dy = 0
The solution to the differential equation (x^3 + y^3) dx − xy^2 dy = 0 is y/x^3 = Ce^(−(y/x)^3), where C is an arbitrary constant.
Substituting y = vx and dy = x dv + v dx, we get:
(x^3 + (vx)^3)dx − x(vx)^2(x dv + v dx) = 0
Simplifying and rearranging, we get:
x^2v dx + (x^3 + 3xv^2 − v^3x^2)dv = 0
Dividing both sides by x^2v and rearranging, we get:
dx/x + (1/v)dv + (3/xv - v^2)dv = 0
Integrating both sides, we get:
ln|x| + ln|v| + 3ln|v/x| − v^3 = C
Where C is an arbitrary constant. Simplifying, we get:
ln|vx/x^3| = C − v^3
Rearranging, we get:
vx/x^3 = Ce^(−v^3)
Finally, we can express this in terms of x and y as:
y/x^3 = Ce^(−(y/x)^3)
The solution to the differential equation (x^3 + y^3) dx − xy^2 dy = 0 is y/x^3 = Ce^(−(y/x)^3), where C is an arbitrary constant.
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Suppose that the matrix A has the following eigenvalues and eigenvectors: λ1 = 3 + I with v1 = [ -1 1 -4]Andλ2 = 3 + i with v2 = [ -1 1 + 4i] Write the solution to the linear system r = Ar in the following forms. A. In eigenvalue/eigenvector form: B. In fundamental matrix form: C. As two equations: (write c1 and c2 for c1 and c2)
A) r(t) =[tex]c_{1}[/tex]×e^(∧×t)[tex]v_{1}[/tex]+ [tex]c_{2}[/tex] × e^(∧²×t)[tex]v_2[/tex] ; B) r(t) = X×e^(∧×t)c ; C) [tex]r_1[/tex](t) = [tex]c_{1}[/tex](-1×[tex]e^{3t}[/tex])×cos(t) -1×[tex]e^{3t}[/tex]×sin(t) -4×[tex]e^{3t}[/tex], [tex]r_2[/tex](t) = [tex]c_2[/tex](-1×[tex]e^{3t}[/tex]×cos(t) -1×[tex]e^{3t}[/tex]×sin(t) + 4i×[tex]e^{3t}[/tex].
A) The solution to the linear equation in eigenvalue/eigenvector form:
r(t) = [tex]c_{1}[/tex]×e^(∧×t)[tex]v_{1}[/tex]+ [tex]c_{2}[/tex] × e^(∧²×t)[tex]v_2[/tex]
B) The solution to the linear equation in fundamental matrix form:
r(t) = X×e^(∧×t)c
C) As two equations: The solution to the linear system can also be written as two equations as follows:
[tex]r_1[/tex](t) = [tex]c_1[/tex] ×e^(∧²×t)[tex]v_1[/tex] = [tex]c_{1}[/tex](-1×[tex]e^{3t}[/tex])×cos(t) -1×[tex]e^{3t}[/tex]×sin(t) -4×[tex]e^{3t}[/tex]
[tex]r_2[/tex](t) = [tex]c_2[/tex]×e^(∧² ×t)[tex]v_2[/tex] = [tex]c_2[/tex](-1×[tex]e^{3t}[/tex]×cos(t) -1×[tex]e^{3t}[/tex]×sin(t) + 4i×[tex]e^{3t}[/tex]
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through: (-2, 3), slope = -1
Slope is y is -1x + 1.
Find slope?To enter data and find the equation's solution, we can utilize the equation's point-slope form.
use the point ( 2, 3) and the slope of 1.
m \s= \s− \s1
x \s1 \s= \s− \s2
y \s1 \s= \s3
In the point-slope formula,
y \s− \sy \s1 \s= \sm \s( \sx \s− \sx \s1 \s)
the values into the plug
y \s− \s3 \s= \s− \s1 \s( \sx \s− \s( \s− \s2 \s) \s)
symbol reduction
Distribute the 1 according to y 3 = 1 (x + 2).
Use additive inverse to find the y in the equation y = 3 = 1 x 2
y \s− \s3 \s+ \s3 \s= \s− \s1 \sx \s− \s2 \s+ \s3
simplify
y \s= \s− \s1 \sx \s+ \s1.
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a line y=mx+5 passes through the point 1,6 and is parallel to y=4x+6 what is the value of b
A y= -1/4x+2
B y=1/4x+5
C y=4x+2
D y=4x+5
The equation of the straight line would be -
y = 4x + 5.
What are algebraic expressions?In mathematics, an expression or mathematical expression is a finite combination of symbols that is well-formed according to rules that depend on the context.Mathematical symbols can designate numbers (constants), variables, operations, functions, brackets, punctuation, and grouping to help determine order of operations and other aspects of logical syntax.Given is that a line y = mx + 5 that passes through the point (1, 6) and is parallel to y = 4x + 6.
The general equation of a straight line is given by -y = mx + c
Here - {m} is slope and {c} is y - intercept.Equation of the line is -
y = mx + 5
It is parallel to -
y = 4x + 6
Then -
m = 4 {slopes of || lines are equal)
So, we get the equation as -
y = 4x + 5
Therefore, the equation of the straight line would be -
y = 4x + 5.
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Pls help me with questions, I will give brainliest
Cotangent is the inverse of tangent:
cotΘ = 1/tanΘ
cotΘ = [tex]1/\frac{\sqrt{6} }{2}[/tex]
=[tex]\frac{2}{\sqrt{6}}[/tex]
=[tex]\frac{2}{\sqrt{6}}*\frac{\sqrt{6}}{\sqrt{6}}=\frac{2\sqrt{6}}{6}[/tex] Rationalize the denominator
cotΘ = [tex]\frac{\sqrt{6}}{3}[/tex]
Plug and solve.
[tex]\frac{\frac{\sqrt{6}}{2}+\frac{\sqrt{6}}{3} }{\frac{\sqrt{6}}{2}-\frac{\sqrt{6}}{3}}[/tex]
We'll break the problem into 2, simplify the denominator, then simplify the numerator.
Simplify the denominator with the common denominator 6, of 2 and 3:
[tex]{\frac{\sqrt{6}}{2}-\frac{\sqrt{6}}{3} = {\frac{3\sqrt{6}}{6}-\frac{2\sqrt{6}}{6} = {\frac{3\sqrt{6}-2\sqrt{6}}{6} = {\frac{\sqrt{6}}{6}[/tex]
Simplify the Numerator with the common denominator 6, of 2 and 3:
[tex]{\frac{\sqrt{6}}{2}+\frac{\sqrt{6}}{3} = {\frac{3\sqrt{6}}{6}+\frac{2\sqrt{6}}{6} = {\frac{3\sqrt{6}+2\sqrt{6}}{6} = {\frac{5\sqrt{6}}{6}[/tex]
Back in original equation with simplified numerator and denominator:
[tex]\frac{\frac{5\sqrt{6}}{6}}{\frac{\sqrt{6}}{6}}[/tex]
Multiply by reciprocal and simplify:
[tex]\frac{5\sqrt{6}}{6}*\frac{6}{\sqrt{6}} = \frac{5\sqrt{6} * 6}{6\sqrt{6}} = 5*\frac{6\sqrt{6}}{6\sqrt{6}} = 5*1 = 5[/tex]
a computer that normally cost 500 dolars was on sale for 200 dollars. what is the percent decrease. please show work for brainlist
Answer:
(100/500) *200 =40%
Step-by-step explanation:
A radio station plays tree commercials between two songs. The commercials play for two minutes altogether. The first commercial is 1/2 minute and the second commercial is 1 1/4 minutes how long is the third commercial
