Answer:
710
Step-by-step explanation:
Numeration of pages goes from 1.
9 pages with 1-digit numbers, from 1 to 9.
90 pages with 2-digit numbers, from 10 to 99.
So, 2*90 + 9 = 189 digits are just used to numerate 9 + 90 = 99 pages.
The rest of numbers are 3-digit.
[tex]\frac{(2022-189)}{3}[/tex] = 611 pages with 3-digit numbers.
Total pages = 9 + 90 + 611 = 710.
A nutritionist wants to determine how much time nationally people spend eating and drinking. Suppose for a random sample of 972 people age 15 or older, the mean amount of time spent eating or drinking per day is 1.54 hours with a standard deviation of 0.65 hour. Complete parts (a) through (d) below.
The correct answers to the question are:
A. b
B. c.
C. (1.486, 1.594)
d. b
A. why a large sample size is needed to calculate a confidence intervalThis is because it would help to make the mean normal. This is given that the distribution for eating and drinking is not properly skewed.
B. The random sample of 972 is less than the population by 5 percent so it can be said to have satisfied the requirement. option c
C. sample size n = 972
mean = 1.54
sd = 0.65
1 - 0.99 = 0.01
Z0.01/2 = 2.58
The confidence interval would be gotten as
1.54 ± 2.58 ( 0.65/√972)
= 1.54 ± 2.58*0.021
= 1.54 - 0.054 , 1.54 + 0.054
= (1.486, 1.594)
d. The correct answer is b. The mean may differ given that the interval is about those that are 15 or more.
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Complete question5. A nutritionist wants to determine how much time nationally people spend eating and drinking. Suppose for a random sample of 969 people age 15 or older, the mean amount of time spend eating or drinking per day is 1.85 hours with a standard deviation of 0.67 hours. Complete parts (a) through (d) below.
A). A histogram of time spends eating and drinking each day is skewed right. Use this result to explain why a large sample size is needed to construct a confidence interval for the mean time spent eating and drinking each day.
a. The distribution of the sample mean will always be approximately normal.
b. Since the distribution of time spent eating and drinking each day is not normally distributed (skewed right), the sample must be larger so that the distribution of the sample mean will be approximately normal.
c. Since the distribution of time spent eating and drinking each day is normally distributed, the sample must be larger so that the distribution of the sample mean will be approximately normal.
d.The distribution of the sample mean will never be approximately normal.
B). In 2010, there were over 200 million people nationally age 15 or older. Explain why this, along with the fact that the data were obtained using a random sample, satisfies the requirements for constructing a confidence interval.
a. The sample size is less than 10% of the population.
b. The sample size is greater than 5% of the population.
c. The sample size is less than 5% of the population.
d. The sample size is greater than 10% of the population.
C). Determine and interpret a 99% confidence interval for the mean amount of the time Americans age 15 or older spend eating and drinking each day. Select the correct choice and fill in the blanks.
a. There is a 99% probability that the mean amount of time spent eating and drinking per day is between _____ and _____ hours.
b. The nutritionist is 99% confident that the amount of time spent eating and drinking per day for any individual is between _____ and _____ hours.
c. The nutritionist is 99% confident that the mean amount of time spent eating and drinking per day is between _____ and _____ hours.
d. The requirements for conducting a confidence interval are not satisfied.
D). Could the interval be used to estimate the mean amount of time a 9 year old spends eating and drinking each day?
a. Yes; the interval is about the mean amount of time spend eating and drinking per day for people age 15 or older and can be used to find the mean amount of time spent eating and drinking per day for a 9 year olds.
b. No; the interval is about people age 15 or older. The mean amount of time spent eating and drinking per day for 9 years old may differ.
c. No; the interval is about individual time spent eating and drinking per day and cannot be use to fin the mean time spent eating and drinking per day for specific age.
d. Yes; the interval is about individual time spent eating and drinking per day and can be used to find the mean amount of time a 9 year old spends eating and drinking each day.
e. A confidence interval could not be constructed in part (c)
Are there any other types of numbers other then real and imaginary numbers
Answer:
Yes
Step-by-step explanation:
In mathematics, there are real numbers, imaginary numbers, and complex numbers. Is there any other category? We have infinite Cardinal and Ordinal numbers, not covered above.
Step-by-step explanation:
that is a tricky question.
basically all numbers we know are either a real or an imaginary number.
but there are many sub-types and one super-type that are defined in mathematics.
the superset of real and imaginary numbers is called complex numbers. that means it contains all real and all imaginary numbers.
the real numbers have then many sub-types :
e.g.
natural numbers
whole numbers
integer numbers
rational numbers
irrational numbers
so, you see, if the question means if there are any additional forms of numbers than real and imaginary numbers, then no.
if the question means if there are more number type definitions than just real and imaginary numbers, then yes.
Dominick borrowed $6,000 from a credit union at 9% simple interest for 30 months. What is the total amount of interest Dominick will pay throughout the life of the loan?
The total amount of interest Dominick will pay throughout the life of the loan is $1350
What is Simple Interest ?Simple Interest is the interest applied only on the Principal amount invested or borrowed .
It is calculated by the formula
I = ( P * R * T ) /100
P is the Principal
R is the Rate of Interest
T is the Time Period
Here
P = $ 6000
R= 9 %
T = 30 months = 30/12 year = 2.5 years
I = ( 6000 * 9 * 2.5 ) / 100
Interest = $1350
Therefore , the total amount of interest Dominick will pay throughout the life of the loan is $1350
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The weights in ounces of a sample of running shoes for men and women are shown. Test the claim that the means are different. Use the P-value method with α = 0.05.
Men
9.6 12.6 11.2 15.0 11.6 13.3 12.5 12.6 12.7 13.6
Women
10.2 9.2 9.5 9.9 8.8 10.4 10.5 10.6 10.4 9.9
8.9 9.8 9.9 10.1 12.0
Group of answer choices
Since the P-value is greater than α = 0.05, do not reject the null hypothesis. There is not enough evidence to support the claim that there is a difference in the weights of running shoes.
Since the P-value is less than α = 0.05, reject the null hypothesis. There is enough evidence to support the claim that there is a difference in the weights of running shoes.
Since the P-value is less than α = 0.05, do not reject the null hypothesis. There is not enough evidence to support the claim that there is a difference in the weights of running shoes.
Since the P-value is greater than α = 0.05, reject the null hypothesis. There is enough evidence to support the claim that there is a difference in the weights of running shoes.
The conclusion will be; Since the P-value is less than α = 0.05, reject the null hypothesis. There is enough evidence to support the claim that there is a difference in the weights of running shoes. Option B.
What is the conclusion?Parameters
Group 1
n1=10
x=18.47
[tex]s^2_1[/tex]=2.12
Group 1
n2=15
x2=10.01
[tex]s^2_2[/tex]=0.62
Generally, the equation for the hypothesis is mathematically given as
[tex]H_0=\mu_1 =\mu=2 \\\\H_a= \mu_1 \neq \mu=2[/tex]
Therefore
[tex]t=\frac{12.47-10.01}{\sqrt{\frac{2.12}{10}+\frac{0.61}{15}}}[/tex]
t=4.8966
In conclusion, The pvalue from table is
p-value (4.8966, [tex]\alpha[/tex] =0.06)
[tex]p-value < \alpha=0.05[/tex]
since [tex]p-value < \alpha=0.05[/tex] we reject the null hypothesis and state that there is no difference b/w the two means.
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Question Each answer choice below represents a relation by a set of ordered pairs. In which of the answer choices is the relation a function? Select all correct answers. Select all that apply: {(-3, 10), (-2, 13), (3, 13), (8, 12)} {(-2, 13), (-5, 9), (-5,-5), (-3,-1)} □ {(4,-4), (4,2), (-3, 4), (-2,-5)} □ {(-4,11), (5, 10), (-3, 3), (-1, 10)} □ {(1,7), (1,-5), (0, 13), (2, 1)}
The ordered pairs that represent functions are:
{(-3, 10), (-2, 13), (3, 13), (8, 12)} {(-4,11), (5, 10), (-3, 3), (-1, 10)}How to determine the ordered pairs?For a set of ordered pairs to represent a function, the following must be true:
The x value must point to only one y value
In {(-3, 10), (-2, 13), (3, 13), (8, 12)} and {(-4,11), (5, 10), (-3, 3), (-1, 10)}, the x value point to only one y value
Hence, {(-3, 10), (-2, 13), (3, 13), (8, 12)} and {(-4,11), (5, 10), (-3, 3), (-1, 10)} are functions
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Find the sum of the first 5,000 odd numbers
Step-by-step explanation:
this is an arithmetic sequence :
a1 = 1
an = an-1 + 2 = a1 + (n-1)×2
the sum of n terms of an arithmetic sequence is
S = n×(a1 + an)/2
n = 5000
an = a1 + 4999×2 = 9999
S = 5000×(1 + 9999)/2 = 5000×5000 = 25,000,000
The tile edging for a bathroom is in the shape of a triangle. The area of a triangle with base b and height h is A=1/2 bh. Find the area of the tile if the base measures 3.5 cm and the height measures 8 cm.
Final answer:
Area of tile is 14 cm².
Step-by-step explanation:
Area of triangle is:
[tex]\frac{1}{2} base.height[/tex]
Putting [tex]b=3.5[/tex] and [tex]h=8[/tex] in [tex]\frac{1}{2} base.height[/tex]
[tex]\frac{1}{2}bxh=\frac{1}{2}3.5x8[/tex]
[tex]=3.5x4[/tex]
[tex]=14[/tex]
What is 42% of 17?
round up
Hello
42. ?
100 17
We use cross product.
[tex] \frac{42 \times 17}{100} = 7.14[/tex]
42% of 17 => 7.14
Have a nice day ;)
[tex]\huge\text{Hey there!}[/tex]
[tex]\huge\textbf{Question reads....}[/tex]
[tex]\large\textbf{What is 42\% of 17?}[/tex]
[tex]\huge\textbf{Answering your question}[/tex]
[tex]\mathbf{42\% \ of \ 17}[/tex]
[tex]\huge\textbf{Translating it to easier to terms}[/tex]
[tex]\mathbf{42\% \ of \ 17}[/tex]
[tex]\mathbf{= 42\%\times 17}\\[/tex]
[tex]\mathbf{= \dfrac{42}{100}\times 17}[/tex]
[tex]\mathbf{= \dfrac{42\div2}{100\div2}\times17}[/tex]
[tex]\mathbf{= \dfrac{21}{50}\times17}[/tex]
[tex]\mathbf{= \dfrac{21}{50} \times \dfrac{17}{1}}[/tex]
[tex]\mathbf{= \dfrac{21\times17}{50\times1}}[/tex]
[tex]\mathbf{= \dfrac{357}{50}}[/tex]
[tex]\mathbf{= 357 \div 17}[/tex]
[tex]\mathbf{= 7.14}[/tex]
[tex]\huge\textbf{Answer }\bf \huge\boxed{\downarrow}[/tex]
[tex]\huge\text{Therefore, your answer should be: \boxed{\mathsf{7.14}}}\huge\checkmark[/tex]
[tex]\huge\text{Good luck on your assignment \& enjoy your day!}[/tex]
~[tex]\frak{Amphitrite1040:)}[/tex]
A square room is 14m long. Find the area of its floor
Answer: 196m squared
Step-by-step explanation: A square has equal sides, so if one side were to be equal to 14m, the other would too. 14 x 14 = 196.
140⁰
D 20⁰
Z 180°
O True
1350
O False
CA
90°
60°
45°
m₂ COX= mzAOX+ m2 COA.
40⁰
The given statement m<COX = m<COA + <AOX is TRUE
What is a protractor?A protractor is a mathematical instrument for measuring angle. From the given protractor, we have that;
The measure of angle COA and angle AOX is equivalent to angle COA. Mathematically;
m<COX = m<COA + <AOX
This shows that the given statement is TRUE
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DUE SOON!! URGENT!!!
Which table represents a linear function?
Answer:
C
Step-by-step explanation:
please mark brainliest if correct
The following subjects reference the same area on a polynomial graph: zeros,
solutions, roots, x-intercepts, and factors.
True or false?
Zeros, solutions, roots, x-intercepts, and factors are subjects that reference the same area on a polynomial graph.
What is a polynomial graph?A polynomial graph can be defined as a type of graph that can be used to represent only positive (non-negative) integer exponents of a variable in an equation.
This ultimately implies that, a polynomial graph touches the x-axis at zeros, solutions, roots, x-intercepts, and factors with even multiplicities.
In conclusion, we can logically deduce that zeros, solutions, roots, x-intercepts, and factors are subjects that reference the same area on a polynomial graph.
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fin the domain of the function
f(x)=x^2+7
Answer:
All real numbers
Step-by-step explanation:
there arent any domain restrictions
Given z1 = startroot 2 endroot (cosine (startfraction 3 pi over 4 endfraction) i sine (startfraction 3 pi over 4 endfraction) ) and z 2 = (cosine (startfraction pi over 2 endfraction) i sine (startfraction pi over 2 endfraction) ) , what is the value of z1 – z2?
The subtraction of complex numbers [tex]z_{1} -z_{2}[/tex] is cos(π)+i sin(π).
Given [tex]z_{1} =\sqrt{2}[/tex][cos(3π/4+i sin(3π/4) and [tex]z_{2}[/tex]=cos (π/2) +i sin(π/2)
We have to find the value of [tex]z_{1} -z_{2}[/tex].
A complex number is a number that includes real number as well as a imaginary unit in which [tex]i^{2} =1[/tex]. It looks like a+ bi.
We have to first solve [tex]z_{1} and z_{2}[/tex] and then we will be able to find the difference.
[tex]z_{1} =[/tex][tex]\sqrt{2}[/tex][ cos (3π/4)+i sin (3π/4)]
[tex]=\sqrt{2}[/tex][cos(π-π/4)+ i sin (π-π/4)]
=[tex]\sqrt{2}[/tex] [-cos(π/4)+sin (π/4)]
=[tex]\sqrt{2}[/tex](-1/[tex]\sqrt{2}[/tex]+1/[tex]\sqrt{2}[/tex])
=[tex]\sqrt{2} *0[/tex]
=0
[tex]z_{2} =[/tex]cos(π/2)+i sin (π/2)
=0+i*1
=1
Now putting the values of [tex]z_{1} and z_{2}[/tex],
[tex]z_{1} -z_{2} =0-1[/tex]
=-1
=-1+i*0
=cos (π)+i sin(π)
Hence the value of difference between [tex]z_{1} and z_{2}[/tex] is cos(π)+i sin(π).
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(07.01 MC)
The picture shows a triangular island:
Island
Z
O x sin 50°
O
Which expression shows the value of z? (1 point)
cos 50°
O x cos 50°
O
sin 50°
y
50
Answer:
Option (2)
Step-by-step explanation:
cos 50° = y/z
z = y/cos 50°
15. Ms. Goldman is thinking of a number that is less than 20 and has three factor pairs. She says
that if she adds the factors in each factor pair the sums are 19, 11 and 9. What is her number?
Number 15.
Answer:
18
Step-by-step explanation:
Let x be the unknown number.
Given information:
x < 20x has 3 factor pairs with sums of 19, 11 and 9Factor Pair:
A pair of numbers that multiply together to give the original number.
We can immediately discount prime numbers since prime numbers only have one factor pair: itself and one.
Remaining numbers: 1, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18.
Of the remaining numbers, only 12, 16 and 18 have more than 2 pairs of factors:
Factors of 12: 1 and 12, 2 and 6, 3 and 4
Factors of 16: 1 and 16, 2 and 8, 4 and 4
Factors of 18: 1 and 18, 2 and 9, 3 and 6
If the pairs of factors sum to 19, 11 and 9, then the number must be 18.
Given that
10 - V18 = a + bV2 , where a and b are integers.
V2
Find the values of a and b.
The values of a and b if they are integers are -10 and 3
Equations and surdSurd functions are square root of prime numbers. Given the following equation;
10 - √18 = a +b√2
Equate the real and surd part to form the equation
b√2 = √18
2b² = 18
b² = 9
b = 3
Similarly
a = -10
Hence the values of a and b if they are integers are -10 and 3
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-3x+4y=2 & -6x+6y=-3 solve by elimination
Answer:(4,7/2)
Step-by-step explanation:
Given functions f and g, find f-g.
f(x) = 3x -8, g(x) = 8x - 3
OA. 5x+5
OB. 11x-11
O C. -5x -5
D.-5x-11
Answer:
[tex]-5x-5[/tex]
Step-by-step explanation:
original functions:
[tex]f(x)=3x-8\\g(x)=8x-3[/tex]
Subtract the polynomials:
[tex]f(x)-g(x) = (3x-8) - (8x-3)[/tex]
Distribute the negative sign:
[tex]3x-8-8x+3[/tex]
Combine like terms:
[tex](3x-8x)+(-8+3)[/tex]
Add like terms:
[tex]-5x-5[/tex]
What is the factored form of 8x² + 12x?
O 4(4x² + 8x).
O 4x(2x + 3)
O8x(x + 4)
O 8x(x² + 4)
Answer:
Step-by-step explanation:
Solution
The factored form is going to be the HCF (highest common factor)
The fest way to do this is to turn this into primes.
8x^2 + 12x
8x^2 = 2*2*2*x*x
12x = 2 * 2 * x
Now bold the HCF.
8x^2 = 2*2*2*x*x
12x = 2 * 2 * x
The HCF = 2*2*x
Divide that into the original question to determine the second factor.
8x^2 / 4x = 2x
12x / 4x = 3 Notice how the x-s cancel
Answer
4x (2x + 3)
If the sum of all other interior angles is 620∘, what is the angle measure of x. do not include x= or the degree sign in your answer.
The measure of angle x from the diagram is 100 degrees
Sum of interior angle of an hexagonThe formula for calculating the sum of interior angle of a polygon is given as:
s = (n-2)180
Since hexagon has 6 interior angles, the sum of the interior will be 720 degrees. Hence
x + 620 = 720
x = 720 - 620
x= 100
Hence the measure of angle x from the diagram is 100 degrees
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(a) Find the equation of the normal to the hyperbola x = 4t, y = 4/t at the point (8,2).
The slope of the tangent line to the curve at (8, 2) is given by the derivative [tex]\frac{dy}{dx}[/tex] at that point. By the chain rule,
[tex]\dfrac{dy}{dx} = \dfrac{dy}{dt} \times \dfrac{dt}{dx} = \dfrac{\frac{dy}{dt}}{\frac{dx}{dt}}[/tex]
Differentiate the given parametric equations with respect to [tex]t[/tex] :
[tex]x = 4t \implies \dfrac{dx}{dt} = 4[/tex]
[tex]y = \dfrac4t \implies \dfrac{dy}{dt} = -\dfrac4{t^2}[/tex]
Then
[tex]\dfrac{dy}{dx} = \dfrac{-\frac4{t^2}}4 = -\dfrac1{t^2}[/tex]
We have [tex]x=8[/tex] and [tex]y=2[/tex] when [tex]t=2[/tex], so the slope at the given point is [tex]\frac{dy}{dx} = -\frac14[/tex].
The normal line to the same point is perpendicular to the tangent line, so its slope is +4. Then using the point-slope formula for a line, the normal line has equation
[tex]y - 2 = 4 (x - 8) \implies \boxed{y = 4x - 30}[/tex]
Alternatively, we can eliminate the parameter and express [tex]y[/tex] explicitly in terms of [tex]x[/tex] :
[tex]x = 4t \implies t = \dfrac x4 \implies y = \dfrac4t = \dfrac4{\frac x4} = \dfrac{16}x[/tex]
Then the slope of the tangent line is
[tex]\dfrac{dy}{dx} = -\dfrac{16}{x^2}[/tex]
At [tex]x = 8[/tex], the slope is again [tex]-\frac{16}{64}=-\frac14[/tex], so the normal has slope +4, and so on.
When you start your career, you decide to set aside $250 every month to deposit into an investment account. The investment firm claims that historically their accounts have earned an annual interest rate of 9.0% compounded monthly. Assuming this to be true, how much money will your account be worth after 20 years of depositing and investing? Round your answer to the nearest cent. Do not include labels or units. Just enter the numerical value.
After 20 years of depositing and investing there will be 168474.01 dollars in your account.
What is compound interest?It is defined as the interest on the principal value or deposit and the interest which is gained on the principal value in the previous year.
We can calculate the compound interest using the below formula:
[tex]\rm A = P(1+\dfrac{r}{n})^{nt}[/tex]
Where A = Final amount
P = Principal amount
r = annual rate of interest
n = how many times interest is compounded per year
t = How long the money is deposited or borrowed (in years)
The formula for calculating how much money will your account be worth after 20 years of depositing and investing is:
P = $250, r = 9% = 0.09, t = 20 years, n = 12 and PMD = $250(per month deposit)
[tex]\rm A = P(1+\dfrac{r}{n})^{nt}+\dfrac{PMD((1+\dfrac{r}{n})^{nt}-1)}{\dfrac{r}{n}}[/tex]
[tex]\rm A = 250(1+\dfrac{0.09}{12})^{12\times20}+\dfrac{250((1+\dfrac{0.09}{12})^{12\times12}-1)}{\dfrac{0.09}{12}}[/tex]
A = 1502.287 + 166971.717
A = $168474.01
Thus, after 20 years of depositing and investing there will be 168474.01 dollars in your account.
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What is the range for y=x^2-7x-8 ?
Answer:
[-20.25;+∞).
Step-by-step explanation:
1) according to the properties of the given parabola the minimum is:
[tex]x_0=-\frac{-7}{2}=3.5; \ y_0=-20.25;[/tex]
2) y₀=-20.25 means, the value -20.25 is minimum of the range, the maximum →+∞;
3) finally, the range y≥-20.25.
a company promises to release model every month. each model's battery life will be 6% longer than the previous model's. if the current model's battery is 671.0 minutes, in how many months will the latest model's battery life reach 1,008.9 minutes?
The number of months that it will take the latest model's battery life to reach 1,008.9 minutes is; 8 months
How to solve geometric progression?
Each month, there is an increase by a factor of 0.06 of the previous months model.
From geometric sequence formula of aₙ = ar^(n - 1),
where;
a is first term
r is common ratio
aₙ is nth term
we have;
1,008.9 = 671 * 1.06^(n - 1)
1008.9/671 = 1.06^(n - 1)
In 1.504 = (n - 1) In 1.06
0.408 = (n - 1) * 0.058
n - 1 = 0.408/0.058
n = 7.03 + 1
n ≈ 8 months
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For a population with μ = 60 , z-score =3.15, and σ = 10. Find random variable / data value X . Round your answer to the nearest hundredths place. Round your answer to 2 decimal places.
Answer:
X = 91.50
Step-by-step explanation:
The relation that defines a Z value can be used to find an X value when the appropriate parameters are given.
Z-scoreThe equation for finding the Z-score associated with an X-value is ...
Z = (X -μ)/σ
Solving this for X using the values given, we have ...
3.15 = (X -60)/10
31.5 = X -60 . . . . . . . multiply by 10
91.5 = X . . . . . . . . add 60
The random variable value is ...
X = 91.50
Find the error in the calculations below:
Line (1): -3(2x + 5) <21
Line (2): -6x
15 < 21
Line (3): -6x < 36
Line (4): x <- 6
Line (5):
-
O The error occurred from line (1) to line (2).
O The error occurred from line (4) to line (5).
O The error occurred from line (3) to line (4).
O The error occurred from line (2) to line (3).
Answer: Line 3 to line 4
Step-by-step explanation:
When dividing by a negative, you need to flip the inequality sign.
Identify a horizontal or vertical stretch or compression of the function f(x) = x by observing the equation of the function g(x) = 5x.
The vertical stretch of f(x) to g(x) is a factor of 5
How to determine the dilation?The functions are given as:
f(x) = x
g(x) = 5x
Substitute f(x) = x in g(x) = 5x
g(x) = 5f(x)
In th above equation, the factor is 5.
5 is greater than 1 (this means vertical stretch)
Hence, the vertical stretch of f(x) to g(x) is a factor of 5
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Choose what the expressions below best represent within the context of the word problem.
The tens digit of a number is twice the ones digit. The sum of the digits in the number is 12. What is the number?
x represents
the ones
digit
2x represents
the tens
digit
The number which represent tens digit of a number is twice the ones digit is;
x represents the ones digit2x represents the tens digitWord problemComplete question:
The tens digit of a number is twice the ones digit. The sum of the digits in the number is
12. What is the number?
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In your own words, explain why every natural number is also a rational number but not every rational number is a natural number.
Answer:
Every natural number is also a rational number but not every rational number is a natural number because the set of natural numbers is a subset of the set of rational numbers.
Explanation:
Since all natural numbers, n can be written as n/1. Thus all natural numbers are rational. Since 3/4 is a rational number but not natural thus all rational numbers are not natural.
Hint:
Natural numbers : All positive numbers except zero.
Rational numbers: All numbers in the form of p/q, where q ≠ 0 (p,q are integers)