Answer:
Problem 1) [tex] m = \dfrac{1}{4} [/tex] [tex] slope_{perpendicular} = -4 [/tex]
Problem 2) [tex] m = \dfrac{1}{3} [/tex] [tex] slope_{perpendicular} = -3 [/tex]
Step-by-step explanation:
[tex] slope = m = \dfrac{y_2 - y_1}{x_2 - x_1} [/tex]
[tex] slope_{perpendicular} = \dfrac{-1}{m} [/tex]
Problem 1) M(9,6), N(1,4)
[tex] slope = m = \dfrac{6 - 4}{9 - 1} = \dfrac{2}{8} = \dfrac{1}{4} [/tex]
[tex] slope_{perpendicular} = \dfrac{-1}{\frac{1}{4}} = -4 [/tex]
Problem 2) M(-2,2), N(4,-4)
[tex] slope = m = \dfrac{4 - 2}{4 - (-2)} = \dfrac{2}{6} = \dfrac{1}{3} [/tex]
[tex] slope_{perpendicular} = \dfrac{-1}{\frac{1}{3}} = -3 [/tex]
The first common multiple of two number is 6. What is their fourth common multiple?
Answer:
4th multiple = 24
Step-by-step explanation:
Given
Let the two numbers be represented by m and n
Required
Find the 4th common multiple of the numbers.
From the question, we understand that the first common multiple of m and n is 6.
This can be represented as:
m * n * 1 = 6
mn = 6
Their fourth common multiple can be represented as: m * n * 4
4th multiple = m * n * 4
4th multiple = 4 * mn
Substitute 6 for mn
4th multiple = 4 * 6
4th multiple = 24
Hence, the 4th multiple of both numbers is 24.
Evaluate −x^2−5 y^3 when x = 4 and y = 1
Answer:
Simplify:
[tex]-4^2-5(1^3)[/tex]
So you get:
[tex]-21\\[/tex]
Answer:
[tex]\huge\boxed{-21}[/tex]
Step-by-step explanation:
-x²-5y³
Given that x = 4, y = 1
[tex]-(4)^2-5(1)^3[/tex]
[tex]-16-5(1)\\-16-5\\-21[/tex]
When conducting a residual analysis, which plot would you look at to determine if the equal variance assumption is satisfied?
a. Scatter plot of Yhat vs. QN X
b. Scatter Plot of Residuals vs QN X
c. Scatter Plot of Residuals vs Yhat
d. Stem-and-Leaf Plot of the Zresiduals
Answer:
C.Scatter Plot of Residuals vs Yhat
Step-by-step explanation:
A major traffic problem in the Greater Cincinnati area involves traffic attempting to cross the Ohio River from Cincinnati to Kentucky using Interstate 75. Let us assume that the probability of no traffic delay in one period, given no traffic delay in the preceding period, is 0.9 and that the probability of finding a traffic delay in one period, given a delay in the preceding period, is 0.6. Traffic is classified as having either a delay or a no-delay state, and the period considered is 30 minutes.Required:a. Assume that you are a motorist entering the traffic system and receive a radio report of a traffic delay. What is the probability that for the next 60 minutes (two time periods) the system will be in the delay state?b. What is the probability that in the long run the traffic will not be in the delay state?c. An important assumption of the Markov process model presented here has been the constant or stationary transition probabilities as the system operates in the future. Do you believe this assumption should be questioned for this traffic problem? Explain.
Answer:
a) 0.36
b) 0.3
c) Yes
Step-by-step explanation:
Given:
Probability of no traffic delay in one period, given no traffic delay in the preceding period = P(No_Delay) = 0.9
Probability of finding a traffic delay in one period, given a delay in the preceding period = P(Delay) = 0.6
Period considered = 30 minutes
a)
Let A be the probability that for the next 60 minutes (two time periods) the system will be in the delay state:
As the Probability of finding a traffic delay in one period, given a delay in the preceding period is 0.6 and one period is considered as 30 minutes.
So probability that for the next two time periods i.e. 30*2 = 60 minutes, the system in Delay is
P(A) = P(Delay) * P(Delay) = 0.6 * 0.6 = 0.36
b)
Let B be the probability that in the long run the traffic will not be in the delay state.
This statement means that the traffic will not be in Delay state but be in No_Delay state in long run.
Let C be the probability of one period in Delay state given that preceding period in No-delay state :
P(C) = 1 - P(No_Delay)
= 1 - 0.9
P(C) = 0.1
Now using P(C) and P(Delay) we can compute P(B) as:
P(B) = 1 - (P(Delay) + P(C))
= 1 - ( 0.6 + 0.10 )
= 1 - 0.7
P(B) = 0.3
c)
Yes this assumption should be questioned for this traffic problem because it implies that traffic will be in Delay state for the 30 minutes and just after 30 minutes, it will be in No_Delay state. However, traffic does not work like this in general and it makes this scenario unrealistic. Markov process model can be improved if probabilities are modeled as a function of time instead of being presented as constant (for 30 mins).
Equivalent means having ___ ___
I need two answer omg pls helppp ;-;
Answer:
equal value and function
Step-by-step explanation:
20.) The area of a circle is given by the equation A = nr2. If the radius of a circle is equal to 6 centimeters,
which of the following is closes to the area of the circle? (Use it = 3.14.)
113.04
18.84
36
28.26
please provide explanation
Answer:
113.04 cm^2
Step-by-step explanation:
The area of a circle is
A = pi r^2
We know the radius is 6 cm
A = 3.14 * 6^2
A = 3.14 * 36
A =113.04
The following shows the monthly sales in units of six salespersons before and after a bonus plan was introduced. At 95% confidence, determine whether the bonus plan has increased sales significantly. (For the following matched samples, let the difference "d" be: d = after - before.)
Salesperson After Before
1 94 90
2 82 84
3 90 84
4 76 70
5 79 80
6 85 80
Answer:
Since the calculated value of t= 2.249 does not falls in the rejection region we therefore accept the null hypothesis at 5 % significance level . On the basis of this we conclude that the bonus plan has not increased sales significantly.
Step-by-step explanation:
The null and alternative hypotheses as
H0: μd=0 Ha: μd≠0
Significance level is set at ∝= 0.05
n= 6
degrees of freedom = df = 6-1 = 5
The critical region is t ( base alpha by 2 with df=5) ≥ ± 2.571
The test statistic under H0 is
t = d/ sd/ √n
Which has t distribution with n-1 degrees of freedom
Sales Difference
Person After Before d = after - before d²
1 94 90 4 16
2 82 84 -2 4
3 90 84 6 36
4 76 70 6 36
5 79 80 -1 1
6 85 80 5 25
∑ 18 118
d`= ∑d/n= 18/6= 3
sd²= 1/6( 118- 18²/6) = 1/6 ( 118 - 54) = 10.67
sd= 3.266
t= 3/ 3.266/ √6
t= 2.249
Since the calculated value of t= 2.249 does not falls in the rejection region we therefore accept the null hypothesis at 5 % significance level . On the basis of this we conclude that the bonus plan has not increased sales significantly.
Find the general solution of the following differential equation. Primes denote derivatives with respect to x.(x+2y)y'=2x-yleft parenthesis x plus 2 y right parenthesis y prime equals 2 x minus y
Answer:
[tex]-[ln(x^2-yx-y^2)] = K\\[/tex]
Step-by-step explanation:
Given the differential equation [tex](x+2y)y'=2x-y[/tex], this can also be written as;
[tex](x+2y)\frac{dy}{dx} =2x-y[/tex]
On simplification
[tex](x+2y)\frac{dy}{dx} =2x-y\\\\\frac{dy}{dx} = \frac{2x-y}{x+2y} \\\\let \ y = vx\\\frac{dy}{dx} = v+x\frac{dv}{dx}[/tex]
The differential equation becomes;
[tex]v+x\frac{dv}{dx} =\frac{ 2x-vx}{x+2vx}\\\\v+x\frac{dv}{dx} = \frac{ x(2-v)}{x(1+2v)}\\\\v+x\frac{dv}{dx} = \frac{2-v}{1+2v}\\\\x\frac{dv}{dx} = \frac{2-v}{1+2v} - v\\\\x\frac{dv}{dx} = \frac{(2-v)-v(1+2v)}{1+2v}\\\\x\frac{dv}{dx} = \frac{2-v-v-2v^2}{1+2v}\\\\x\frac{dv}{dx} = \frac{2-2v-2v^2}{1+2v}[/tex]
[tex]\frac{dx}{x} = \frac{1+2v}{2-2v-2v^2}dv\\\\integrating\ both \ sides\\\\[/tex]
[tex]\int\limits \frac{dx}{x} = \int\limits \frac{1+2v}{2-2v-2v^2}dv\\\\lnx = \frac{1}{2} \int\limits \frac{1+2v}{1-v-v^2}dv\\\\lnx + C = -\frac{1}{2}ln(1-v-v^2)[/tex]
[tex]C = -\frac{1}{2}ln(1-v-v^2) - lnx \\\\ -ln(1-v-v^2) - 2lnx = 2C\\\\-[ln(1-v-v^2) + lnx^2] = 2C\\\\-[ln(1-v-v^2)x^2] = 2C\\since\ v = y/x\\\\- [ln(1-y/x-y^2/x^2)x^2] = K\\\\-[ln(x^2-yx-y^2)] = K\\[/tex]
Hence the solution to the differential equation is [tex]-[ln(x^2-yx-y^2)] = K\\[/tex]
Simplify the following expression. (4x − 8)(4x + 8)
Answer:
16x^2 - 64
Step-by-step explanation:
(4x − 8)(4x + 8)
We recognize that this is the difference of squares
(a-b) (a+b) = a^2 - b^2
=(4x)^2 - 8^2
=16x^2 - 64
The length of a rectangle is a inches. Its width is 5 inches less than the length. Find the area and the perimeter of the rectangle.
Answer:
[tex]Area = 5a - a^2[/tex] [tex]inches\²[/tex]
[tex]Perimeter = 10\ inches[/tex]
Step-by-step explanation:
Given
Length = a
Width = 5 - a
Required
Determine the Area and Perimeter;
Calculating Area
Area is calculated as thus;
[tex]Area = Length * Width[/tex]
Substitute values for Length and Width
[tex]Area = a * 5 - a[/tex]
[tex]Area = a * (5 - a)[/tex]
Open Bracket
[tex]Area = 5a - a^2[/tex]
Calculating Perimeter
Perimeter is calculated as thus;
[tex]Perimeter = 2 (Length + Width)[/tex]
Substitute values for Length and Width
[tex]Perimeter = 2 (a + 5 - a)[/tex]
Collect Like Terms
[tex]Perimeter = 2 (5 + a- a)[/tex]
[tex]Perimeter = 2 (5)[/tex]
Open Bracket
[tex]Perimeter = 10\ inches[/tex]
Find y. A. √22 B. 8 C. √42 D. 4
Answer:
[tex]\Large \boxed{\mathrm{D. \ 4}}[/tex]
Step-by-step explanation:
The triangle is a right triangle.
We can use trigonometric functions to solve the problem.
tan θ = opp/adj
tan 30 = y/(4√3)
y = 4√3 tan 30
y = 4
what does the inverse of f(x)=2x-3 looks like on a graph
Step-by-step explanation:
To find this inverse of a graph you have to multiply the current equation by -1.
-1(2x-3)
f(x)=-2x+3.
This graph will start at the point (0,3). Then according to the slope the rest of the points will go down two than right one. So the next two point will be (1,1) and (2,-1).
The length of a rectangle is twice the width. If the length is increased by 4 inches and the width is decreased by 1 inch, a new rectangle is formed whose perimeter is 198 inches. Find the dimensions of the original rectangle.
Answer:
Width: 32 inches.
Length: 64 inches.
Step-by-step explanation:
Let's say that the width of the rectangle is x inches, so the length is 2x inches.
If 2x + 4 and x - 1, the perimeter is 198 inches. That means that two times the width plus two times the length is 198 inches.
2(2x + 4) + 2(x - 1) = 198
4x + 8 + 2x - 2 = 198
6x + 6 = 198
x + 1 = 33
x = 32.
That means that the width of the original rectangle is 32 inches, and the length is 32 * 2 = 64 inches.
Hope this helps!
A polling company reported that 53% of 1018 surveyed adults said that secondhand smoke issecondhand smoke is "very harmful.""very harmful." Complete parts (a) through (d) below.
a. What is the exact value that is 53% of 1018?
b. Could the result from part (a) be the actual number of adults who said that secondhand smoke issecondhand smoke is "very harmful" question mark "very harmful"? Why or why not?
c. What could be the actual number of adults who said that secondhand smoke issecondhand smoke is "very harmful" question mark "very harmful"?
d. Among the 10181018 respondents, 260260 said that secondhand smoke issecondhand smoke is "not at all harmful.""not at all harmful." What percentage of respondents said that secondhand smoke issecondhand smoke is "not at all harmful" question mark "not at all harmful"?
Answer:
a. 539.54
b. No, the result from part (a) could not be the actual number of adult who said that secondhand smoke are very harmful because a count of people must result into a whole number.
c. 540
d. 25.54%
Step-by-step explanation:
Given that:
A polling company reported that 53% of 1018 surveyed adults said that secondhand smoke is "very harmful."
Complete parts (a) through (d) below.
a. What is the exact value that is 53% of 1018?
The 53% of 1018 is :
=[tex]\dfrac{53}{100} \times 1018[/tex]
= 0.53 × 1018
= 539.54
b. Could the result from part (a) be the actual number of adults who said that secondhand smoke is ''very harmful"? Why or why not?
No, the result from part (a) could not be the actual number of adult who said that secondhand smoke are very harmful because a count of people must result into a whole number.
c. What could be the actual number of adults who said that secondhand smoke is secondhand smoke "very harmful"?
Since, a count of people must result into a whole number, the actual number of adults who said that secondhand smoke is secondhand smoke "very harmful" can be determined from the approximation of the exact value into whole number which is 539.54 [tex]\approx[/tex] 540.
d. Among the 1018 respondents, 260 said that secondhand smoke is is "not at all harmful.'' What percentage of respondents said that secondhand smoke is "not at all harmful"?
Since 260 respondents out of 1018 respondents said that the second hand smoke is not harmful, then the percentage of the 260 respondents is :
= [tex]\dfrac{260}{1018} \times 100 \%[/tex]
= 25.54%
Assume that women's heights are normally distributed with a mean given by mu = 64.3 inches, and a standard deviation given by sigma= 2.2 inches.
A) If a woman is randomly selected, find the probability that her height is less than 65 inches.
B) If 34 women are randomly selected, find the probability that they have a mean height less than 65 inches.
Answer:69
Step-by-step explanation:
Helppp needed ASAP!!!!
Answer:
The two missing sides are : 79.54m and 58.62m
Step-by-step explanation:
first we look for the missing angle in the triangle
sum of angle in a triangle = 180°let make the missing angle be A 180 = A + 25 + 35180 = A + 60A = 180 - 60A = 120°so now we use sine rules:
a/(sin A) = b/(sin B) = c/(sin C)Let us make:
The side facing 35° be called bThe side facing 25° be called cThe side facing 120° be called aso :
120/(sin 120°) = b/(sin 35°)120/(0.866) = b/0.574138.57 = b/0.574b = 138.57 x 0.57479.54mso the side facing 35° = 79.54m
b/(sin B) = c/(sin C)79.54/(sin 35) = c/(sin 25)79.54/ (0.574) = c/(0.423)138.57 = c/(0.423)c = 138.57 x 0.423c = 58.62mTo find the area of this triangle
Use De Moivre's theorem to find the indicated power of the complex number. Write the answer in rectangular form.[2(cos10∘ + i sin10∘)]^3.
Answer:
[tex]\bold{4\sqrt3 + i4}[/tex]
Step-by-step explanation:
Given complex number is:
[tex][2(cos10^\circ + i sin10^\circ)]^3[/tex]
To find:
Answer in rectangular form after using De Moivre's theorem = ?
i.e. the form [tex]a+ib[/tex] (not in forms of angles)
Solution:
De Moivre's theorem provides us a way of solving the powers of complex numbers written in polar form.
As per De Moivre's theorem:
[tex](cos\theta+isin\theta)^n = cos(n\theta)+i(sin(n\theta))[/tex]
So, the given complex number can be written as:
[tex][2(cos10^\circ + i sin10^\circ)]^3\\\Rightarrow 2^3 \times (cos10^\circ + i sin10^\circ)^3[/tex]
Now, using De Moivre's theorem:
[tex]\Rightarrow 2^3 \times (cos10^\circ + i sin10^\circ)^3\\\Rightarrow 8 \times [cos(3 \times10)^\circ + i sin(3 \times10^\circ)]\\\Rightarrow 8 \times (cos30^\circ + i sin30^\circ)\\\Rightarrow 8 \times (\dfrac{\sqrt3}2 + i \dfrac{1}{2})\\\Rightarrow \dfrac{\sqrt3}2\times 8 + i \dfrac{1}{2}\times 8\\\Rightarrow \bold{4\sqrt3 + i4}[/tex]
So, the answer in rectangular form is:
[tex]\bold{4\sqrt3 + i4}[/tex]
How do you write 30,8608
Answer:
it should be 308,608. the comma is after every three in this scenario.
Step-by-step explanation:
Ana drinks chocolate milk out of glasses that each holds 1/8 fraction of a liter. She has 7/10 fraction of a liter of chocolate milk in her refrigerator. How many glasses of chocolate milk can she pour?
Answer:
6 glasses
I hope this helps!
Answer:
28/5
Step-by-step explanation:
looked up on khan
Please answer this correctly without making mistakes
Answer:
7/10 mi
Step-by-step explanation:
The total distance is 3 miles = 30/10 miles.
The other distances added gives 7/10+7/10+9/10 = 23/10
Therefore the last hop from Kingwood to Silvergrove is 30/10 - 23/10 = 7/10
Find out the Time Zone for UAE and its neighboring countries. Express them as positive or negative rational numbers with reference to Greenwich Mean Time. Note down the time of few of your daily activities such as breakfast, school time, lunch time, etc. Compare the same time with GMT.anyone please answer this.
Answer:
UAE is in the Gulf Standard Time zone.
It is GMT + 4
Breakfast: 7 am; GMT 3 am
School time 8 am: GMT 4 am
Lunch time: 12:30 pm; GMT 8:30 am
Step-by-step explanation:
UAE is in the Gulf Standard Time zone.
It is GMT + 4
Breakfast: 7 am; GMT 3 am
School time 8 am: GMT 4 am
Lunch time: 12:30 pm; GMT 8:30 am
a ball is dropped from a height of 512 inches onto a level floor. after the fourth bounce it is still 2 inches off the ground. presuming that the height the ball bounces is always the same fraction of the height reached on the previous bounce. what is that fraction? A) 1/4 B) 3/7 C) 5/9 D)4/7 E)3/5
Answer:
A. 1/4
Step-by-step explanation:
We know that before the 1st bounce, the height of the ball is 512 inches.
Say the fraction is x.
Then, after the first bounce, the height of the ball is 512 * x = 512x.
After the second bounce, the height is now x * 512x = 512x².
By similar reasoning, the height after the third bounce is 512x³ and after the fourth bounce, it is [tex]512x^4[/tex].
We also know that after the fourth bounce, the height is 2 inches. So, set 2 equal to [tex]512x^4[/tex]:
2 = [tex]512x^4[/tex]
Divide both sides by 512:
[tex]x^4=2/512[/tex][tex]x^4=2/512=1/256[/tex]
Take the fourth root of both sides:
[tex]x=\sqrt[4]{1/256} =1/4[/tex]
Hence, the answer is A.
~ an aesthetics lover
Answer:
A. 1/4.
Step-by-step explanation:
This is exponential decay so we have , where x = the fraction:
512(x)^4 = 2
x^4 = 1/256
x= 1 / (256)^0.25
= 1/4
what is 12x^3-9x^2-4x+3 in factored form?
Answer: (3x^2-1)(4x-3)
============================
Work Shown:
Use the factor by grouping method
12x^3-9x^2-4x+3
(12x^3-9x^2)+(-4x+3)
3x^2(4x-3)-1(4x-3)
(3x^2-1)(4x-3)
Write the expression
using only positive exponents.
-3/x-4
someone please help me
Answer:
3 mL
Step-by-step explanation:
The fluid level is called the concave meniscus. The adhesive force causes it to crawl up on the sides, but you should ignore that while reading the level.
here is my question hope this works now
Answer:
[tex]\boxed{x=1}[/tex] and [tex]\boxed{x=7}[/tex]
Step-by-step explanation:
This quadratic is already factored down to its factors (x - 1) and (x - 7).
Set these equal to zero and solve for x by adding 1 or 7 to both sides of the equation.
[tex]x-1=0\\\\\boxed{x=1}[/tex]
[tex]x-7=0\\\\\boxed{x=7}[/tex]
Six people attend the theater together and sit in a row with exactly six seats.
a. How many ways can they be seated together in the row?
b. Suppose one of the six is a doctor who must sit on the aisle in case she is paged. How many ways can the people be seated together in the row with the doctor in an aisle seat?
c. Suppose the six people consist of three married couples and each couple wants to sit together with the husband on the left. How many ways can the six be seated together in the row?
Answer
A. 720 ways
B. 240 ways
C. 6 ways
There are 720 ways they can be seated in a row together, 120 ways can the people be seated together in a row with the doctor in an aisle seat, and 6 ways can the six be seated together in a row.
What is a permutation?A permutation is defined as a mathematical process that determines the number of different arrangements in a set of objects when the order of the sequential arrangements.
It is assumed that six people will attend the theater together and sit in a row of six.
The following are the various ways they can be seated in a row together:
⇒ 6!
⇒ 6 × 5 × 4 × 3 × 2 × 1
⇒ 720
If the doctor sits in the aisle seat, the remaining 5 persons can sit in the remaining 5 seats 5! ways
The total possibilities are as follows:
⇒ 5 × 4 × 3 × 2 × 1
= 120
Consider that the six persons are made up of three married couples.
In addition to the aforementioned, divide the six chairs into three groups of two seats each.
There is one choice in each block for the husband to be placed on the left and the wife to be positioned on the right and the 3 couples can be seated in the 3 blocks in 3! ways.
⇒ 3 × 2 × 1
The required answer is 6.
Thus, there are 720 ways they can be seated in a row together, 120 ways can the people be seated together in a row with the doctor in an aisle seat, and 6 ways can the six be seated together in a row.
Learn more about permutation here:
brainly.com/question/1216161
#SPJ2
Can I have help with 43 and 44 I need to see how to do them thanks.
Answer:
see explanation
Step-by-step explanation:
(43)
3[tex]x^{5}[/tex] - 75x³ ← factor out 3x³ from each term
= 3x³(x² - 25) ← this is a difference of squares and factors in general as
a² - b² = (a - b)(a + b) , thus
x² - 25 = x² - 5² = (x - 5)(x + 5)
Thus
3[tex]x^{5}[/tex] - 75x³ = 3x³(x - 5)(x + 5)
(44)
81c² + 72c + 16 ← is a perfect square of the form
(ac + b)² = a²c² + 2abc + b²
Compare coefficients of like terms
a² = 81 ⇒ a = [tex]\sqrt{81}[/tex] = 9
b² = 16 ⇒ b = [tex]\sqrt{16}[/tex] = 4
and 2ab = 2 × 9 × 4 = 72
Thus
81c² + 72c + 16 = (9c + 4)²
1. 3x^5 -75x³
=3x³(x²-25)
=3x³(x²-5²)
=3x³(x-5)(x+5)
2. 81c²+72c+16
=81c²+36c+36c+16
=9c(9c+4)+4(9c+4)
=(9c+4)(9c+4)
=(9c+4)²
“Type ‘equal, supplementary, complementary, or vertical in the space provided’”
Answer:
Supplementary
Step-by-step explanation:
When the sum of 2 angles equal 180°, they are called supplementary angles. And they also form a straight line together.
<AOB (40°) and <BOC (140°) are not equal angles.
<AOB (40°) and <BOC (140°) are not complementary angles. Complementary angles add up to equal 90°.
<AOB (40°) and <BOC (140°) are not vertical angles. Vertical angles are opposite angles formed when two lines intersect.
<AOB (40°) and <BOC (140°) are supplementary angles. They add up to equal 180°.
Is 55/22 a rational
Answer:
The fraction [tex]\displaystyle \frac{55}{22}[/tex] is indeed a rational number.
Step-by-step explanation:
A number [tex]x[/tex] is rational if and only if there exist two integers [tex]p[/tex] and [tex]q[/tex] (where [tex]q \ne 0[/tex]) such that [tex]x = \displaystyle \frac{p}{q}[/tex].
[tex]\displaystyle \frac{55}{22}[/tex], the number in question here is already written in the form of a fraction. The two integers [tex]p = 55[/tex] and [tex]q = 22[/tex] ([tex]q \ne 0[/tex]) meet the requirement that [tex]\displaystyle \frac{55}{22} = \frac{p}{q}[/tex]. Therefore, [tex]\displaystyle \frac{55}{22}\![/tex] is indeed a rational number.
Side note: the [tex]p[/tex] and [tex]q[/tex] here ([tex]q \ne 0[/tex]) don't have to be unique. For example:
because [tex]\displaystyle \frac{55}{22} = \frac{5 \times 11 }{2 \times 11} = \fraac{5}{2}[/tex], both of the following pairs could satisfy [tex]\displaystyle \frac{55}{22} = \frac{p}{q}[/tex]:
[tex]p = 55[/tex] and [tex]q = 22[/tex];[tex]p = 5[/tex] and [tex]q = 2[/tex].