The sum of two numbers is 35. The larger number is one less than three times the smaller number. Let represent the larger number. Let represent the smaller number. Write a system of equations to represent this situation. What are the two numbers?
Answers
Explanation
To solve the question we will have to set up a simultaneous equation
If x represents the larger number
y represents the smaller number, then
The sum of the two numbers is 35
[tex]Equation\text{ 1: x+y=35}[/tex]The larger number is one less than three times the smaller number.
[tex]Equation\text{ 2: }x=3y-1[/tex]So, we will solve the equation using the substitution method
Thus, we will substitute x = 3y -1 into equation 1
[tex]\begin{gathered} 3y-1+y=35 \\ 3y+y-1=35 \\ 4y-1=35 \\ 4y=35+1 \\ 4y=36 \\ \\ y=\frac{36}{4} \\ \\ y=9 \end{gathered}[/tex]The smaller number is 9
The larger number will be
[tex]\begin{gathered} x+y=35 \\ x=35-y \\ x=35-9 \\ x=26 \end{gathered}[/tex]The larger number is 26
The answers are 9 and 26
Related Questions
Please just give me the answer straightforward I don’t need an explanation
Answers
Explanation
We are given the function
[tex]y=\frac{1}{2}(3)^{-2x}+6[/tex]First, we have to find the y-intercept
The y-intercept is the point where the graph intersects the y-axis. From the graph, the y-intercept is 6.5
To get the horizontal asymptote
We approach a horizontal asymptote by the curve of a function as x goes towards infinity.
From the graph above,
The horizontal asymptote is
[tex]y=6[/tex]For the transformation
Dilate the following points by each scale factor (k) provided.P(3, 4) by k=1/2 AndN(4, 15) by k=2
Answers
We are asked to dilate the given two points.
P(3, 4) by a scale factor of k = 1/2
Multiply the x and y coordinates by the scale factor.
[tex]P(3,4)\rightarrow P^{\prime}(\frac{1}{2}\cdot3,4\cdot\frac{1}{2})=P^{\prime}(1.5,2)[/tex]Therefore, the dilated point is P'(1.5, 2)
This is an example of reduction.
Similarly,
N(4, 15) by a scale factor of k = 2
Multiply the x and y coordinates by the scale factor.
[tex]N(4,15)\rightarrow N^{\prime}(2\cdot4,2\cdot15)=N^{\prime}(8,30)[/tex]Therefore, the dilated point is N'(8, 30)
This is an example of enlargement.
p varies directly as q. When q = 31.2, p = 20.8. Find p when q = 15.3.a.10.2b.22.95c.42.4i got B ...?
Answers
A direct relationship is an association between two variables such that they rise and fall in value together. In another terms, one of the variables is equal to the other times a constant. In our case, we have
[tex]p=kq[/tex]Where k is a constant. To find k, we can use the relation we already know the values.
[tex]\begin{gathered} p(31.2)=20.8 \\ \implies20.8=31.2k \\ k=\frac{20.8}{31.2} \\ k=\frac{2}{3} \end{gathered}[/tex]Then, the relation between our variables is
[tex]p=\frac{2}{3}q[/tex]Evaluating q = 15.3 on this expression, we have
[tex]p=\frac{2}{3}\times(15.3)=10.2[/tex]The answer is 10.2.
help me please i want to see if i did it correctly
Answers
-1/16
1) Considering that we have the following identity:
[tex]\begin{gathered} \cos (2x)=2\cos (2x)-1 \\ \end{gathered}[/tex]2) We can plug into that the cos (x).:
[tex]\begin{gathered} \cos (2x)=(\frac{\sqrt[]{15}}{4})^2-1 \\ \cos (2x)=\frac{15}{16}-\frac{16}{15} \\ \cos (2x)=-\frac{1}{16} \end{gathered}[/tex]Since for the Quadrant II cosine yields a negative value.
can someone please help me with the following?
Answers
Answer:
y^2 = -12x
Explanation:
For a parabola whose vertex is at (0,0), its standard equation is
[tex]y^2=-4ax[/tex]where the directrix is given by x = a.
Now in our case, the vertex of our parabola is at (0,0) and the directrix is x = 3. This tells us that a = 3 and so the above equation gives
[tex]y^2=-4\cdot3x[/tex][tex]\boxed{y^2=-12x\text{.}}[/tex]which is our answer!
In a sequence of numbers, a4= 98, a5= 99.2, a6= 100.4, a7= 101.6, and a8= 102.8. Based on this information,which equation can be used to find an, the nth term in the sequence?
Answers
Given:
a4 = 98
a5 = 99.2
a6 = 100.4
a7 = 101.6
a8 = 102.8
Use the arithmetic sequence formula below:
[tex]a_n=a_1+(n-1)d[/tex]Where,
an = nth term
a1 = first term
n = number of terms
d = common differnce
Let's solve for the common differnce.
d = a5 - a4 = 99.2 - 98 = 1.2
Use the 8th term a8, to find the first term:
[tex]\begin{gathered} 102.8=a_1+(8-1)1.2_{} \\ \\ 102.8=a_1+(7)1.2 \\ \\ 102.8=a_1+8.4 \\ \\ a_1=102.8-8.4\text{ = 94.4} \end{gathered}[/tex]Therefore, the first term a1 = 94.4
Thus, the equation for the nth term will be:
Input 94.4 for a1, 1.2 for d in the arithmetic formula above
[tex]\begin{gathered} a_n=94.4+(n-1)1.2 \\ \\ a_n=94.4+1.2n-1.2 \\ \\ \text{combine like terms:} \\ a_n=1.2n+94.4-1.2 \\ \\ a_n=1.2n+93.2 \end{gathered}[/tex]ANSWER:
[tex]a_n=1.2n+93.2[/tex]+0.049 where t is in hours after 6:00 AM last Sunday12The temperature in Middletown Park at 6:00 AM last Sunday was 434 degrees Fahrenheit. The temperature was changing at a rate given by r(t) = 3.27 cosROUND ALL ANSWERS TO 2 DECIMAL PLACESAt 10 00 AM last Sunday, the temperature in the park was increasing at a rate ofabout 1.68 degrees per hourFrom 6:00 AM to 1:00 PM last Sunday, the temperature in the park increasedby _degreesWhat was the temperature in the park at 1:00 PM last Sunday? _degreesWhat was the temperature in the park at 4:00 PM Last Friday (5 days later)? _degrees
Answers
1 ) According to the question, the temperature has been changing according to this function:
[tex]r(t)\text{ =}3.27\text{ }\cos (\frac{\pi t}{12})+0.049[/tex]Where t is the number of hours after 6:00 am last Sunday
b) From 6 am to 1 pm last Sunday, the temperature in the park increased by
6 am to 1 pm = 7 hours, let's plug into that:
[tex]\begin{gathered} r(t)\text{ =}3.27\text{ }\cos (\frac{\pi t}{12})+0.049 \\ r(7)\text{ =}3.27\text{ }\cos (\frac{7\pi}{12})+0.049 \\ r(7)=3.31732 \\ r(7)\approx3.32 \end{gathered}[/tex]The rate of change in 7 hours was approximately 3.32 degrees per hour
c)
The temperature in the park at 6 am was 43.4 ºF. To find the temperature we must find the value for y, 1 pm Last Sunday. Let's plug the value already found: 1.68 for r.
Considering that according to question a, the temperature increased by 1.68º per hour. 6 am to 10 pm: 4 hours
c)
Since the question wants the temperature from 6 am to 1 pm, and it has been increasing by 3.32 degrees per hour, in 7 hours
7 x 3.32 =23.24º
43.4º+23.24=110.04ºF
d) On Friday, 5 days later there was
5 x 24 at 6am + 10 hours =120+10=130 hours
[tex]\begin{gathered} r(130)\text{=}3.27\text{ }\cos (\frac{130\pi}{12})+0.049 \\ r(130)\text{ =2.7}6 \end{gathered}[/tex]Starting from 43.4º F +2.76 =46.16ºF
Identify the measure of each exterior angle of a regular dodecagon
Answers
Solution:
Given:
A dodecagon is a 12-sided polygon.
A regular dodecagon is a figure with sides of the same length and internal angles of the same size.
The sum of exterior angles of a polygon is 360°.
The formula for calculating the size of each exterior angle is;
[tex]\begin{gathered} \text{Each exterior angle = }\frac{360}{n} \\ \text{where n is the number of sides of the polygon} \end{gathered}[/tex]For a dodecagon, n = 12
Hence,
[tex]\begin{gathered} \text{Each exterior angle = }\frac{360}{12} \\ \text{Each exterior angle = }30^0 \end{gathered}[/tex]Therefore, each exterior angle of a regular dodecagon is 30 degrees.
slo also has cell phone service from Varisoon Wireless. She pays a flat fee of $39.95for 350 minutes per month. Any minutes over 350 are at an additional cost. In November,Noslo's bill for wireless service was $108.70 for 475 minutes. What is the charge perminute for the minutes over 350 per month?
Answers
hello
Noslo spent 475minutes on call for that month, but we already know the price on the first 350 minutes
if Noslo is charged $39.95 for 350 minutes, we can subtract $39.95 from $108.70 and also 475 from 350 to know the cost and number of minutes respectively.
the number of minutes is
[tex]475-350=125\text{ minutes}[/tex]the cost of the minutes is
[tex]108.70-39.95=68.75[/tex]so, what this information implies is thar Noslo was charged $68.75 for 125 minutes.
we can easily find how much he's charged per minute from here.
let's make x represent that
[tex]\begin{gathered} 125=68.75 \\ 1=x \\ \text{cross multiply both sides} \\ x\times125=68.75 \\ 125x=68.75 \\ \text{divide both sides by the coefficient of x} \\ \frac{125x}{125}=\frac{68.75}{125} \\ x=0.55 \end{gathered}[/tex]from the calculations above, Nsolo was charged $0.55 per minute
Which polynomial matches the description below?A binomial with two different variables and a degree of five.
Answers
A binomial has two monomials, that discard b and d options.
Two different variables discard a.
Therefore, the correct option is c.
Identify the like terms. 4y, (–7x), 9y, 13
Answers
Like terms are terms that have the same variables of similar exponents.
The given terms are:
4y, (–7x), 9y, 13
James makes wreaths for a living. He can make 6 wreaths in 450 minutes. How many minutes does it take him to make 2wreaths?
Answers
150 minutes to make 2 wreaths
1) Gathering the data, and setting a proportion.
Then let's cross multiply.
There is a direct proportionality, between the number
6 wreaths 450 mins
2 x
6x = 900 Dividing by 6
x=150 minutes
what's the difference between a coefficient and a variable
Answers
we have that
variables" because the numbers they represent can vary—that is, we can substitute one or more numbers for the letters in the expression. Coefficients are the number part of the terms with variables
example
5x
In the term 5x
x is the variable
5 is a coefficient
the length of cuboid is twice the breadth and thrice the height. if the volume of the cuboid is 972cm3 find the breadth of the cuboid
Answers
The breadth of the cuboid is 9 cm with the volume as 972 cm³.
Given:
length of the cuboid is twice the breadth and thrice of the height.
If the volume of the cuboid is 972 cm³ then find the breadth of the cuboid= ?
Let breadth of cuboid =x cm
(since length=2*breadth)
hence length=2x
(since given length=3*height
hence height=length/3)
hence height=(2x)/3
volume of cuboid
=length*breadth*height
=972 cm³ (given)
or x*2x*2x/3=972
or 4x³=972*3
or x³=243*3
hence x=(729)⅓
x=(9³)⅓ cm.
(breadth) x=9 cm
Hence the breadth of the cuboid is 9 cm.
Learn more about Cuboid here:
brainly.com/question/46030
#SPJ9
URGENT!! ILL GIVE
BRAINLIEST!!!! AND 100 POINTS!!!!!
Answers
Answer:
c
Step-by-step explanation:
Nelson Collins decided to retire to Canada in 10 years. What amount should he deposit so that he will be able to withdraw $80,000 at the end of each year for 25 years after he retires. Assume he can invest 7% interest compounded annually.
Answers
Answer
$1,016,699
Explanation
The amount, A that an invested sum of P, becomes over time t, at a rate of r% is given as
A = P (1 + r)ᵗ
For this question,
A = Total amount that the amount invested becomes = $80,000 × 25 = $2,000,000
P = Amount invested at the start of the 10 years before retirement = ?
r = 7% = 0.07
t = 10 years
A = P (1 + r)ᵗ
2,000,000 = P (1 + 0.07)¹⁰
2,000,000 = P (1.07)¹⁰
Note that 1.07¹⁰ = 1.967
2,000,000 = 1.967P
We can rewrite this as
1.967P = 2,000,000
Divide both sides by 1.967
(1.967P/1.967) = (2,000,000/1.967)
P = $1,016,699
Hope this Helps!!!
Among all of the pairs of numbers whose difference is 12, the smallest product is
Answers
We have two numbers x and y such that their difference is 12:
[tex]\begin{gathered} x-y=12 \\ \Rightarrow x=12+y \end{gathered}[/tex]Now, we take the product of them:
[tex]x\cdot y=(12+y)\cdot y=y^2+12y[/tex]The smallest result we can get is 0 (ignoring the negative numbers, because the meaning of "small" implies an absolute value). Looking at the expression above, it is 0 for y = 0. If y = 0 then x = 12, and the difference is:
[tex]x-y=12-0=12[/tex]And their product is:
[tex]x\cdot y=12\cdot0=0[/tex]What is the initial value of the function represented by this table?ху04192140004.0509
Answers
The initial value, or y-intercept, is the output value when the input of a linear function is zero.
From the table provided, the y-intercept is obtained when x=0
The answer
Which measurements could not represent the side lengths of a right triangle?A) 3cm, 4cm, 5cmB)3cm, 5cm, 9cmC)12cm, 16cm, 20cmD)16cm, 63cm, 65cm
Answers
The Pythagorean theorem states that for a rigth triangle, the square of the hypothenuse is equal to the sum of squares of the other two sides, symbolically:
[tex]a^2+b^2=c^2[/tex]To check if these sides lengths are of a rigth triangle you have to square them.
Remember that the hypothenuse is always the longest side.
So for the first set:
A)
3cm, 4cm and 5 cm
Lets take the side length 5cm as the hypothenuse
So a=3, b=4 and c=5
If the theorem checks then
[tex]3^2+4^2=5^2[/tex]Square all sides:
[tex]\begin{gathered} 3^2=9 \\ 4^2=16 \\ 5^2=25 \end{gathered}[/tex]Add both squared sides:
[tex]9+16=25[/tex]The result is equal to the square of the hypotenuse, this means that this side lengths corresponds to a rigth triangle.
*-*-*
B)
a=3 cm
b=5 cm
c=9 cm (hypothenuse)
Square the three sides:
[tex]\begin{gathered} a^2=3^2=9 \\ b^2=5^2=25 \\ c^2=9^2=81 \end{gathered}[/tex]If the theorem checks then 9 + 25 must be equal to 81
[tex]9+25=34[/tex]The square sum of both sides is different from the quare of the hypotenuse, these side lengths do not correspond to a rigth triangle.
C)
a=12cm
b= 16 cm
c= 20 cm (hypothenuse)
Square the sides:
[tex]\begin{gathered} a^2=12^2=144 \\ b^2=16^2=256 \\ c^2=20^2=400 \end{gathered}[/tex]If the theorem checks then 144 plus 256 must be 400
[tex]144+256=400[/tex]The sum of squares of the sides is equal to the square of the hypothenuse, this set of side lengths belong to a right triangle.
D)
a=16cm
b=63cm
c=65cm (hypothenuse)
Square the sides:
[tex]\begin{gathered} a^2=16^2=256 \\ b^2=63^2=3969 \\ c^2=65^2=4225 \end{gathered}[/tex]If the theorem checks out, then 256 + 3969 must be equal to 4225:
[tex]256+3969=4225[/tex]The sum of squares of the sides is equal to the square of the hypothenuse, this set of side lengths belong to a right triangle.
Convert: 15 meters=centimeters
Answers
EXPLANATION
The relationship between the meters and centimeters is the following:
[tex]1\text{ meter=100 centimeters}[/tex]By applying the unit method, we can get the conversion, as follows:
[tex]Number\text{ of }centimeters=15\text{ meters*}\frac{100\text{ centimeters}}{1\text{ meter}}[/tex]Multiplying terms:
[tex]Number\text{ of centimeters=1500 centimeters}[/tex]The solution is 1500 centimeters.
Use the formula P = 2l + 2w to find the length l of a rectangular lot if the width w is 55 feet and the perimeter P is 260 feet.l = ? feet
Answers
In order to determine the length of the given rectangle, Solve the equation for the perimeter of the rectangle for l and replace w=55ft and P=260ft, and simplify:
[tex]\begin{gathered} P=2l+2w \\ 2l=P-2w \\ l=\frac{P-2w}{2} \\ l=\frac{260ft-2(55ft)}{2} \\ l=\frac{260ft-110ft}{2} \\ l=\frac{150ft}{2}=75ft \end{gathered}[/tex]Hence, the length of the rectangle is 75ft
I'm having trouble with this problem "Solve the equation -8 + 6m = 1/2 (-4m +16) for m"
Answers
Let's begin by listing out the given information:
[tex]\begin{gathered} -8+6m=\frac{1}{2}(-4m+16) \\ \end{gathered}[/tex]Let's proceed to expand the bracket. We have:
[tex]\begin{gathered} -8+6m=\frac{1}{2}(-4m+16) \\ -8+6m=-2m+8 \\ \end{gathered}[/tex]We will put like terms together, we have:
[tex]\begin{gathered} 6m+2m=8+8 \\ 8m=16 \\ \text{Divide both sides by ''8'', we have:} \\ m=\frac{16}{8}=2 \\ m=2 \end{gathered}[/tex]The dot plot shows the number of wins for 16 baseball teams. Which statement about thedata is true?.Baseball Team Wins•0123 4 5 6 7 8Number of WinsThere is a data point at 8, so most teams won 8 games.The data are clustered around 2, so most teams won exactly 2games.The data are clustered from 4 to 7, so most toams lost 4 to 7gamos.The data are clustered from 1 to 3, so most teams won 1 to 3games.
Answers
We can see on the graph that the dots represent a team, and on the x-axis is the number of wins. Looking at the graph we can see that a lot of teams won around 1-3 and just one team won 8 times, therefore, the correct answer is: The data are clustered from 1 to 3, so most teams won 1 to 3 games
The following equation is a conic section written in polar coordinates.=51 + 5sin(0)Step 2 of 2: Find the equation for the directrix of the conic section.
Answers
For a conic with a focus at the origin, if the directrix is
[tex]y=\pm p[/tex]where p is a positive real number, and the eccentricity is a positive real number e, the conic has a polar equation
[tex]r=\frac{ep}{1\pm e\sin\theta}[/tex]if 0 ≤ e < 1 , the conic is an ellipse.
if e = 1 , the conic is a parabola.
if e > 1 , the conic is an hyperbola.
In our problem, our equation is
[tex]r=\frac{5}{1+5\sin\theta}[/tex]If we compare our equation with the form presented, we have
[tex]\begin{cases}e={5} \\ p={1}\end{cases}[/tex]Therefore, the directrix is
[tex]y=1[/tex]2v – 5V = -24muti step equation
Answers
2v - 5v = -24
2v - 5v = -3v, then
-3v = -24
-3 is multiplying on the left, then it will divide on the right
v = -24/(-3)
v = 8
suppose you have 5 apples and you subtract 2 of them, how many apples are left?
You are doing the next computation: 5 apples - 2 apples = 3 apples
What is the result of 2 apples - 5 apples?
if sin = -3/5 and cos >0 what is exact value of cot?5/3-4/33/4-4/5
Answers
Explanation
Given the following information:
[tex]\begin{gathered} Sin=\frac{-3}{5} \\ Cos>0 \end{gathered}[/tex]This implies that the value of sin is negative while that of cos is positive.
This occurs in the fourth quadrant. This also means that the value of tan is negative.
We know that sin uses the value of the opposite and the hypotenuse.
We need to determine the value of the adjacent.
[tex]\begin{gathered} Adjacent=\sqrt{Hyp^2-Opp^2} \\ where \\ Hyp=5 \\ Opp=3 \end{gathered}[/tex][tex]\begin{gathered} Adjacent=\sqrt{5^2-3^2}=\sqrt{25-9}=\sqrt{16} \\ Adj=4 \end{gathered}[/tex]We know that cot is the reciprocal of tan. The value of tan is given as:
[tex]\begin{gathered} Tan=\frac{Opp}{Adj}=\frac{3}{4} \\ But\text{ tan is negative in the fourth quadrant. } \\ \therefore Tan=\frac{-3}{4} \end{gathered}[/tex]We can now determine the value of cot to be:
[tex]Cot=\frac{-4}{3}(reciprocal\text{ of tan\rparen}[/tex]Hence, the answer is the second option i.e. -4/3.
Meg plans to build a fence around her yard. She draws this diagram of the yard.How many meters of fencing material does Meg need?
Answers
The required figure is,
The length of fencing material required = Perimeter of the figure.
Perimeter is calculated as,
[tex]\begin{gathered} Perimeter\text{ = sum of the length of all the sides} \\ Perimeter\text{ = 11.8 m + 18.5 m + 12.1 m +5.4 m + 6.4 m + 6.4 m } \\ Perimeter\text{ = 60.6 m} \\ \end{gathered}[/tex]Thus 60.6 m of fencing material is needed.
D. The number of people in the United States with mobile cellular phones was about 142
million in 2002 and about 255 million in 2007. If the growth in mobile cellular phones
was linear, what was the approximate rate of growth per year from 2002 to 2007?
What would the expected number of people to have phones in 2010? 2015? 2020?
Show this information on a graph (years versus the number of users).
Answers
Since it is linear, we can assume a function of the form:
[tex]y(x)=mx+b[/tex]Where:
m = Slope = rate of growth
b = y-intercept
So:
[tex]\begin{gathered} x=2002,y=142 \\ 142=2002m+b_{\text{ }}(1) \\ ----------- \\ x=2007,y=255 \\ 255=2007m+b_{\text{ }}(2) \end{gathered}[/tex]Using elimination method:
[tex]\begin{gathered} (2)-(1) \\ 255-142=2007m-2002m+b-b \\ 113=5m \\ m=\frac{113}{5}=22.6 \end{gathered}[/tex]So:
Replace m into (1):
[tex]\begin{gathered} 142=2002(22.6)+b \\ b=-45103.2 \end{gathered}[/tex]The linear equation which represents this model is:
[tex]y=22.6x-45103.2[/tex]The approximate rate of growth per year from 2002 to 2007 is 22.6 million
the expected number of people to have phones in:
[tex]\begin{gathered} x=2010 \\ y=22.6(2010)-45103.2 \\ y\approx323 \end{gathered}[/tex][tex]\begin{gathered} x=2015 \\ y=22.6(2015)-45103.2 \\ y\approx436 \end{gathered}[/tex][tex]\begin{gathered} x=2020 \\ y=22.6(2010)-45103.2 \\ y\approx549 \end{gathered}[/tex]323 million of people will have phones in 2010
436 million of people will have phones in 2015
549 million of people will have phones in 2020
In the trapezoid below, if < BAC = 124 °, what is the measure of < DCA.
Answers
ANSWER
[tex]56[/tex]Option B
EXPLANATION
Given;
[tex]\angle BAC=124[/tex]Recall;
[tex]\begin{gathered} \angle DCA+\angle BAC=180 \\ \angle DCA+124=180 \\ \angle DCA=180-124 \\ =56 \end{gathered}[/tex]This is a intro to statistics question, I was wondering if i can have the work shown soni can see the process, thanks :)Charges for advertising on a TV show are based on the number of viewers, which is measured by the rating. The rating is a percentage of the population of 110 million TV households. The CBS television show 60 Minutes recently had a rating of 7.8, indicating that 7.8% of the households were tuned to that show. An advertiser conducts an independent survey of 100 households and finds that only 10 were tuned to 60 Minutes. Assuming that the 7.8 rating is correct, find the probability of surveying 100 randomly selected households and getting 10 or fewer tuned to 60 Minutes. What does this suggest? Does the advertiser have grounds for claiming a refund on the basis that the size of the audience was exaggerated?
Answers
Step-by-step explanation:
7.8% were viewing the show.
so, when picking a single household the probability that is is watching the show is 0.078.
that also means that the probability that this household is not watching the show is 1 - 0.078 = 0.922.
"only" 10 of 100 were watching the show ? that is 10% and therefore a higher rate than measured by the rating agency.
there is no indication that the audience was exaggerated.
anyway, the probability to get 10 or fewer out of randomly picked 100 households to watch the show is
the probability of exactly 10 households watching +
the probability of exactly 9 households watching +
the probability of exactly 8 households watching +
...
the probability of exactly 1 household watching +
the probability of no (0) household watching.
to get the probabilty of any of these events :
x = number of households watching out of the overall 100.
w = probability of a household to watch = 0.078
a = probability of a household to be absent (= not watching) = 0.922
probability(x) = p(x) = w^x × a^(100-x) × c(100, x)
in other words :
the combined probability of "x" households watching and "a" households not watching (which must be 100-x) multiplied by the number of combinations of picking x households out of 100.
so, for the first one, x = 10, we get
p(10) = 0.078¹⁰ × 0.922⁹⁰ × 100! / (10! × (100 - 10)!) =
= 0.078¹⁰×0.922⁹⁰×100×99×...×91 / 10×9×8×...×2 =
= 8.335775831e-012 ×
0.00066955... ×
1.731030945644e+13 =
= 0.096612596...
the rest is then best via Excel or other spreadsheet applications.
p(9) = 0.125496...
p(8) = 0.145117...
p(7) = 0.147558...
p(6) = 0.129888...
p(5) = 0.096969...
p(4) = 0.059699...
p(3) = 0.0291
p(2) = 0.01053
p(1) = 0.002515...
p(0) = 0.000297...
and the sum of all that is the probability to have 10 or fewer households in randomly picked 100 watching the show :
0.843782...
the vast majority of samples of 100 households is expected to have 10 or fewer households watching the show.
this covers the general rating (that would suggest that 7.8 households of 100 are watching the show), some even better ranges (more than 7.8 households), but also everything that is worse than the original rating (below 7.8 households).
when we look at the individual probabilities, we see that the largest probabilities by far are in the categories of 9, 8, 7 and 6 households out of 100 are watching.
so, these are the main expected results when picking samples of 100 households.
therefore, I don't see any indication that the advertiser was "cheated".