A clothing company produces denim jeans. The jeans are made and sold with either a regular cut or a boot-cut. To estimate the proportion of all customers in Tacoma, WA, who prefer boot-cut jeans, a marketing researcher examined sales receipts for a random sample of 178 customers who purchased jeans from the firm’s Tacoma store. 56 of the customers in the sample purchased boot-cut jeans. Construct the 99% confidence interval to estimate the proportion of all customers in Tacoma, Washington, who prefer boot-cut jeans and interpret the confidence interval (please write the interval boundaries to THREE decimal places
Answers
Answer:
99% confidence interval for the proportion of all customers in Tacoma, Washington, who prefer boot-cut jeans is [0.225 , 0.405].
Step-by-step explanation:
We are given that a marketing researcher examined sales receipts for a random sample of 178 customers who purchased jeans from the firm’s Tacoma store. 56 of the customers in the sample purchased boot-cut jeans.
Firstly, the Pivotal quantity for 99% confidence interval for the population proportion is given by;
P.Q. = \(\frac{\hat p-p}{\sqrt{\frac{\hat p(1-\hat p)}{n} } }\) ~ N(0,1)
where, \(\hat p\) = sample proportion of customers who purchased boot-cut jeans = \(\frac{56}{178}\) = 0.315
n = sample of customers = 178
p = population proportion of customers who prefer boot-cut jeans
Here for constructing 99% confidence interval we have used One-sample z test for proportions.
So, 99% confidence interval for the population proportion, p is ;
P(-2.58 < N(0,1) < 2.58) = 0.99 {As the critical value of z at 0.5% level
of significance are -2.58 & 2.58}
P(-2.58 < \(\frac{\hat p-p}{\sqrt{\frac{\hat p(1-\hat p)}{n} } }\) < 2.58) = 0.99
P( \(-2.58 \times {\sqrt{\frac{\hat p(1-\hat p)}{n} } }\) < \({\hat p-p}\) < \(2.58 \times {\sqrt{\frac{\hat p(1-\hat p)}{n} } }\) ) = 0.99
P( \(\hat p-2.58 \times {\sqrt{\frac{\hat p(1-\hat p)}{n} } }\) < p < \(\hat p+2.58 \times {\sqrt{\frac{\hat p(1-\hat p)}{n} } }\) ) = 0.99
99% confidence interval for p = [ \(\hat p-2.58 \times {\sqrt{\frac{\hat p(1-\hat p)}{n} } }\), \(\hat p+2.58 \times {\sqrt{\frac{\hat p(1-\hat p)}{n} } }\)]
= [ \(0.315-2.58 \times {\sqrt{\frac{0.315(1-0.315)}{178} } }\) , \(0.315+2.58 \times {\sqrt{\frac{0.315(1-0.315)}{178} } }\) ]
= [0.225 , 0.405]
Therefore, 99% confidence interval for the proportion of all customers in Tacoma, Washington, who prefer boot-cut jeans is [0.225 , 0.405].
The interpretation of the above confidence interval is that we are 99% confident that the proportion of all customers in Tacoma, Washington, who prefer boot-cut jeans will lie between 0.225 and 0.405.
Related Questions
Hi! ❤️ , im looking for some help here. ill give brainliest if able to.
Answers
Answer:
A. 2^11
Step-by-step explanation:
(They are basically asking what's 2^4 × 2^7, but with more words.)
I usually do each exponent individually:
2^4 is the same as 2 × 2 × 2 × 2 = 16 (or you could have read the text to figure that out)
2^7 is the same as 2 × 2 × 2 × 2 × 2 × 2 × 2 = 128
Then just multiply 128 and 16 to get 2,048, and see which option also gives you 2,048.
BUT, you can also:
(Combine the exponents together to get your answer. Just remember that if it's multiplication you add them, and if it's division you subtract them.)
2^4 × 2^7
4 + 7 = 11
2^11 (This equals 2,048 btw. You don't even have to check all the options to get the answer).
Hope this helps friend :)
The last part I learned from another user, while answering one of your other questions. I personally find this mind blowing, lol.
A baby has its first doctor's visit when it is 5 months old and it weighs 19 pounds. The doctor tells the mother to expect the baby to gain 2 pounds each month. Find an equation in the form y = m x + b that models the baby's weight, where x is the age of the baby in months and y is its weight in pounds.
Answers
A formula in which the baby's weight is represented by the equation y = m x + b, where x is the baby's age in months and y is its weight in pounds is: y = 2x + 19.
Define about the slope-intercept form?A method of formulating a line's equation that makes it simple to identify the slope and y-intercept is known as the slope-intercept form. The point where the line crosses its y-axis is known as the y-intercept, and the slope refers to how steep the line is.
For the given question:
baby's weight is give by equation: y = m x + b.
In which, x = age of the baby in months.
y = its weight in pounds.
When, x = 5, y = 19
2 pounds each month; m = 2
Find y-intercept .
19 = 2*5 + c
c = 19 - 10
c = 9
formula for baby's weight: y = 2x + 9.
Thus, a formula in which the baby's weight is represented by the equation y = m x + b, where x is the baby's age in months and y is its weight in pounds is: y = 2x + 19.
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40x +16y = ( + )
plz help me
Answers
Answer:
Write in Slope-Intercept Form
Graph
Find the x and y Intercepts
Find the Slope
Find the Slope and y-intercept
Write in y=mx+b Form
Write in Standard Form
Simplify each side of 4a+ 5a-2=5+3-1
Answers
Answer:
9a-2=7
Step-by-step explanation:
4a+ 5a-2=5+3-1
To simplify each side, combine like terms.
4a+5a = 9a
5+3-1 = 7
The equation becomes:
9a-2=7
If (3m-5) is a factor of 9m^2 -9m -10, then which is the other factor
Answers
Answer:
(3m+2)
Step-by-step explanation: trust me I gotchu
write 2.9302 as a fraction and a mixed number
Answers
rationalise 29302/100000
=14651/50000
what fraction subtracted from 3/4 will yield an asnwer that is equal to the difference between 2/3 and 1/2?
Answers
Answer:
\(\displaystyle x=\frac{5}{6}\)
Step-by-step explanation:
\(\displaystyle x-\frac{3}{4}=\frac{2}{3}-\frac{1}{2}\\\\x-\frac{3}{4}=\frac{4}{6}-\frac{3}{6}\\\\x-\frac{3}{4}=\frac{1}{6}\\\\x=\frac{1}{6}+\frac{3}{4}\\\\x=\frac{4}{24}+\frac{18}{24}\\\\x=\frac{20}{24}\\\\x=\frac{5}{6}\)
In the diagram below, side PQ has a length of 26.86 cm and side PR has a length of 40.00 cm.
Determine the measure of angle Q in degrees to one decimal place.
Goodness gracious! The diagram cannot be rendered!
Answers
Step-by-step explanation:
what is the answer to 100001/9
Help me please will give brainlist
Answers
Answer: 691
Step-by-step explanation:
(17 + 6 + 9) × (19 + 16) - 19 × 7 - 6 × 4 - 16 × 17 = 691
If a certain stock has a historical 10-year CAGR (compounded annual growth rate) of -2.3% and
currently sells for $83.80 a share, how much did the stock sell for 10 years ago?
Answers
Stock used to sell for $105.81 ten years ago.
\(CAGR = ((\frac{EV}{BV} )^{\frac{1}{n}}-1)*100\)
Here, CAGR = Compounded Annual Growth Rate = -2.3%
EV = Ending Value = x
BV = Beginning Value = $83.80
n = No. of years = 10
Putting the values in the equation,
\(-2.3 = ((\frac{83.80}{x} )^{\frac{1}{10}}-1)*100\)
\(\frac{-2.3}{100} = ((\frac{83.80}{x} )^{\frac{1}{10}}-1)\)
\(\frac{-2.3}{100}+1 = (\frac{83.80}{x} )^{\frac{1}{10}}\)
\(\frac{100-2.3}{100}= (\frac{83.80}{x} )^{\frac{1}{10}}\)
\(\frac{97.7}{100}= (\frac{83.80}{x} )^{\frac{1}{10}}\)
\(0.977 = (\frac{83.80}{x} )^{\frac{1}{10}}\)
\((0.9777)^{10} = \frac{83.80}{x}\)
\(\frac{83.80}{x}=0.792\)
\(x = \frac{83.80}{0.792}\)
x = 105.81
Hence, the beginning value is $105.81.
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Suppose that v varies directly with x, and v = 20 when x = 32.
A) Write a direct variation equation that relates x and y .
B) Find y when x = 24
Answers
\(\qquad \qquad \textit{direct proportional variation} \\\\ \textit{\underline{y} varies directly with \underline{x}}\qquad \qquad \stackrel{\textit{constant of variation}}{y=\stackrel{\downarrow }{k}x~\hfill } \\\\ \textit{\underline{x} varies directly with }\underline{z^5}\qquad \qquad \stackrel{\textit{constant of variation}}{x=\stackrel{\downarrow }{k}z^5~\hfill } \\\\[-0.35em] ~\dotfill\)
\(\stackrel{\textit{"v" varies directly with "x"}}{v = k(x)}\hspace{5em}\textit{we also know that} \begin{cases} x=32\\ v=20 \end{cases} \\\\\\ 20=k(32)\implies \cfrac{20}{32}=k\implies \cfrac{5}{8}=k\hspace{8em}\boxed{v=\cfrac{5}{8}x} \\\\\\ \textit{when x = 24, what's v?}\qquad v=\cfrac{5}{8}(24)\implies v=15\)
What is the exact value of 5 5/6-(1 1/4+3 7/8)
Answers
let's firstly convert all the mixed fractions to improper fractions.
\(\stackrel{mixed}{5\frac{5}{6}}\implies \cfrac{5\cdot 6+5}{6}\implies \stackrel{improper}{\cfrac{35}{6}} ~\hfill \stackrel{mixed}{1\frac{1}{4}} \implies \cfrac{1\cdot 4+1}{4} \implies \stackrel{improper}{\cfrac{5}{4}} \\\\\\ \stackrel{mixed}{3\frac{7}{8}}\implies \cfrac{3\cdot 8+7}{8}\implies \stackrel{improper}{\cfrac{31}{8}} \\\\[-0.35em] ~\dotfill\)
\(\cfrac{35}{6}-\left( \cfrac{5}{4}+\cfrac{31}{8} \right)\implies \cfrac{35}{6}-\cfrac{5}{4}-\cfrac{31}{8}\implies \cfrac{(8)35~~ - ~~(12)5~~ - ~~(6)31}{\underset{\textit{using this LCD}}{48}} \\\\\\ \cfrac{280~~ - ~~60~~ - ~~186}{48}\implies \cfrac{34}{48}\implies \cfrac{17}{24}\)
Which equation shows the point-slope form of the line that passes through (3, 2) and has a slope of y plus StartFraction one-half EndFraction equals 3 left-parenthesis x minus 2 right-parenthesis.?
y + 2 =y plus 2 equals StartFraction one-third EndFraction left-parenthesis x plus 3 right-parenthesis.(x + 3)
y – 2 = y minus 2 equals StartFraction one-third EndFraction left-parenthesis x minus 3 right-parenthesis.(x – 3)
y + 3 = y plus 3 equals StartFraction one-third EndFraction left-parenthesis x plus 2 right-parenthesis.(x + 2)
y – 3 = y plus StartFraction one-half EndFraction equals 2 left-parenthesis x minus 3 right-parenthesis.(x – 2)
Answers
The equation that shows the point-slope form of the line passing through (3, 2) with a slope of (1/2) is:
y - 2 = (1/2)(x - 3)
In the point-slope form of a linear equation, the formula is y - y1 = m(x - x1), where (x1, y1) represents a point on the line, and m represents the slope of the line. By substituting the given values into the formula, we can determine the correct equation.
In this case, the given point is (3, 2) and the slope is (1/2). Plugging these values into the formula, we get:
y - 2 = (1/2)(x - 3)
This equation represents the line passing through the point (3, 2) with a slope of (1/2). It is in the point-slope form, which allows us to easily determine the equation of a line based on a given point and slope.
Therefore, the correct equation is y - 2 = (1/2)(x - 3).
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Answer: B
Step-by-step explanation:
Graph the circle (x - 3)^2 + (y + 3)^2= 36.
Answers
The circle on a graph by drawing the center point and then drawing a circle around it with a radius of 6.
To graph the circle with the equation (x - 3)² + (y + 3)² = 36, we can start by finding the center and radius of the circle.
The equation of a circle in standard form is (x - h)² + (y - k)² = r² (h, k) represents the center of the circle and r represents the radius.
Comparing the given equation with the standard form, we can see that the center of the circle is (3, -3) and the radius is √36 = 6.
Using this information, we can proceed to plot the circle on a graph:
Plot the center point: (3, -3).
From the center point, move 6 units in each direction (up, down, left, and right) to determine the points on the circle.
Up: (3, -3 + 6) = (3, 3)
Down: (3, -3 - 6) = (3, -9)
Left: (3 - 6, -3) = (-3, -3)
Right: (3 + 6, -3) = (9, -3)
Connect the plotted points to form a circle.
The resulting graph should look like this:
| • (3, 3)
|
| •
| •
__|_____________________________
|
|
|
|
| • (3, -3)
The center of the circle is denoted by a solid dot (•) in the graph and the other points lie on the circumference of the circle.
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Use the following table to find the probability that a randomly chosen member of the Student Government Board is a graduate student or lives in on-campus housing. Express your answer as a fraction in lowest terms or a decimal rounded to the nearest millionth.
Students on the Student Government Board
On-Campus Housing Off-Campus Housing
Freshman 2 2
Sophomore 2 4
Junior 0 3
Senior 4 2
Graduate Student 2 0
Answers
The probability that a randomly chosen member of the Student Government Board is a graduate student or lives in on-campus housing is 8/25 or 0.32 (rounded to the nearest millionth).
1. Calculate the total number of students on the Student Government Board by summing up the numbers in the table:
Total Students = 2 + 2 + 2 + 4 + 0 + 3 + 4 + 2 = 19
2. Calculate the total number of graduate students on the Student Government Board:
Total Graduate Students = 2 + 0 = 2
3. Calculate the total number of students living in on-campus housing:
Total On-Campus Housing = 2 + 2 + 0 + 4 + 2 = 10
4. Calculate the probability of selecting a graduate student from the Student Government Board by dividing the total number of graduate students by the total number of students:
Probability of Graduate Student = Total Graduate Students / Total Students = 2 / 19
5. Calculate the probability of selecting a student living in on-campus housing by dividing the total number of students in on-campus housing by the total number of students:
Probability of On-Campus Housing = Total On-Campus Housing / Total Students = 10 / 19
6. Calculate the probability that a randomly chosen member of the Student Government Board is a graduate student or lives in on-campus housing by summing up the probabilities from steps 4 and 5:
Probability = Probability of Graduate Student + Probability of On-Campus Housing = 2 / 19 + 10 / 19
7. Simplify the fraction if necessary. In this case, the fraction cannot be simplified further, so the final probability is 2 / 19 + 10 / 19 = 12 / 19.
8. Convert the fraction to a decimal by dividing the numerator by the denominator: 12 / 19 ≈ 0.631578947, which rounds to 0.632 (rounded to the nearest thousandth).
9. Finally, express the probability as a fraction in lowest terms: 12 / 19 is already in lowest terms.
Therefore, the probability that a randomly chosen member of the Student Government Board is a graduate student or lives in on-campus housing is 12/19 or approximately 0.632 (rounded to the nearest thousandth).
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what’s the answer to this geometry question??
Answers
Step-by-step explanation:
\( \sqrt{a^{2} + a^{2} } = 10 \sqrt{2} \: in\)
\(2a ^{2} = 200 \: in^{2} \)
\( {a}^{2} = 100 \: in ^{2} \)
Which statements describe the area of the figure on the grid if each square has an area of 9 square meters? Check all that apply. O It is the sum of the areas of 7 full squares and 12 partial squares. It is greater than 162 square meters. It is less than 279 square meters. It is the sum of the areas of 18 full squares and 13 partial squares. It is greater than 63 square meters. It is less than 117 square meters.
Answers
Answer:
I dont know for sure but i think its 2,3,4,5,and 6
w2 – 2w
Complete square to make perfect square trinomial
Answers
Answer:
w² - 2(w)(1) + 1² = w² - 2w + 1 = (w - 1)²Step-by-step explanation:
(a - b)² = a² - 2ab + b²
We have
w² - 2w = w² - 2(w)(1)
We must add 1²
w² - 2(w)(1) + 1² = (w - 1)²
The answer is ,w² - 2(w)(1) + 1² = w² - 2w + 1 = (w - 1)², the perfect square.
What is perfect square?A perfect square is a number, from a given number system, that can be expressed as the square of a number from the same number system. Examples of Numbers that are Perfect Squares. 25 is a perfect square. 25 is a natural number, and since there is another natural number 5, such that 52 = 25, 25 is a perfect square.
Here, we have,
(a - b)² = a² - 2ab + b²
We have
w² - 2w = w² - 2(w)(1)
We must add 1²
w² - 2(w)(1) + 1² = (w - 1)²
Hence, The answer is ,w² - 2(w)(1) + 1² = w² - 2w + 1 = (w - 1)², the perfect square.
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Identifying the values a, b, and c is the first step in using the Quadratic Formula to find solution(s) to a quadratic equation. What are the values a, b, and c in the following quadratic equation? -3x^2 -5x +9 =0
Answers
Answer:
The quadratic equation is in the standard form: ax^2 + bx + c = 0. To use the quadratic formula to find the solutions, we need to identify the values of a, b, and c in the equation:
a is the coefficient of the x^2 term, which is -3 in this case.
b is the coefficient of the x term, which is -5 in this case.
c is the constant term, which is 9 in this case.
Therefore, the values of a, b, and c in the quadratic equation -3x^2 - 5x + 9 = 0 are:
a = -3
b = -5
c = 9
Simplify the expression:
\( ({2x + 5y})^{2} - ({3x + 4y})^{2} \)
Answers
Step-by-step explanation:
the answer is in the picture
![Simplify the expression:[tex] ({2x + 5y})^{2} - ({3x + 4y})^{2} [/tex]](https://i5t5.c14.e2-1.dev/h-images-qa/answers/attachments/nSeIe0O1s8ZhaQh5P7qY4gUCphsSO32k.jpeg)
Answer:
3x+18y
Step-by-step explanation:
2x+5y+2x+5y-3x+4y+3x+4y
2x+2x-3x+3x+5y+5y+4y+4y
4x-x+10y+8y
3x+18y
Which expression is equivalent to StartRoot 8 x Superscript 7 Baseline y Superscript 8 Baseline EndRoot? Assume x greater-than-or-equal-to 0.
x y squared StartRoot 8 x cubed EndRoot
2 x cubed y cubed StartRoot x y squared EndRoot
2 x cubed y Superscript 4 Baseline StartRoot 2 x EndRoot
4 x cubed y Superscript 4 Baseline StartRoot x EndRoot
Answers
The expression that is equivalent to StartRoot \(8 x^7 y^8\) EndRoot is (\(2 x^3 y^4\) StartRoot 2 x EndRoot)^2.
To understand why this is the case, let's break down each expression and simplify them step by step:
StartRoot \(8 x^7 y^8\) EndRoot:
We can rewrite 8 as \(2^3\), and since the square root can be split over multiplication, we have StartRoot \((2^3) x^7 y^8\) EndRoot. Applying the exponent rule for square roots, we get StartRoot \(2^3\) EndRoot StartRoot \(x^7\) EndRoot StartRoot \(y^8\) EndRoot.
Simplifying further, we have 2 StartRoot \(2 x^3 y^4\) EndRoot StartRoot \(2^2\) EndRoot StartRoot \(x^2\) EndRoot StartRoot \(y^4\) EndRoot. Finally, we obtain 2 \(x^3 y^4\) StartRoot 2 x EndRoot, which is the expression in question.
(\(2 x y^2\) StartRoot 8 x^3 EndRoot)^2:
Expanding the expression inside the parentheses, we have \(2 x y^2\)StartRoot \((2^3) x^3\) EndRoot. Applying the exponent rule for square roots, we get \(2 x y^2\) StartRoot \(2^3\) EndRoot StartRoot \(x^3\) EndRoot.
Simplifying further, we have \(2 x y^2\) StartRoot 2 x EndRoot. Squaring the entire expression, we obtain (\(2 x y^2\) StartRoot 2 x EndRoot)^2.
Therefore, the expression (\(2 x^3 y^4\) StartRoot 2 x EndRoot)^2 is equivalent to StartRoot \(8 x^7 y^8\) EndRoot.
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Find the length of side x in simplest radical form with a rational denominator
Answers
Answer:
rggsktvvg gggggggggggg yyyhhhhg
What is the complete factorization of the polynomial below?
x³-2x²+x-2
A. (x-2)(x + 1)(x-1)
B. (x + 2)(x + 1)(x-1)
C. (x + 2)(x-1)(x-1)
D. (x-2)(x-1)(x-1)
Answers
Answer:
(x-2)(x^2+1)
do not exist the answer, so check your question
Step-by-step explanation:
(x^3-2x^2)+(x-2) = x^2(x-2)+(x-2)
= (x-2)(x^2+1)
Question 12 of 22
Select the action you would use to solve = 16. Then select the property that
justifies that action.
A. Property: Multiplication property of equality.
B. Action: Add 4 to both sides.
C. Property: Addition property of equality.
D. Action: Divide both sides by 4.
E. Property: Division property of equality.
OF Action: Multiply both sides by 4.
Answers
Answer:
The correct answer is:
D. Action: Divide both sides by 4.
E. Property: Division property of equality.
I need this answer ASAP I will make first person to answer brainleist
Answers
Answer:
\(\frac{m}{3}\) + 549.33
Step-by-step explanation:
\(\frac{1}{3}\) of the total earnings m subtracted by the lunch money and added by what they earned last week.
Sam's parents gave him a map to reach his house. He is 1.5cm from his home. If each 3 cm on scale drawing equals 5 km, how far is he from the house?
Answers
Answer: Sam was 2.5 Km far from his house
Step-by-step explanation:
Sam's parents gave him a map to reach his house.
As per Map Sam was =1.5cm from his home.
To find actual distance from house
If we consider actual distance of sam house =X
We have,
If each 3 cm on scale drawing equals 5 km
We know on map 3cm = 5Km
If 1.5 cm = X Km
We need to cross multiple
3X =1.5 × 5
3X = 7.5
X= 7.5÷3
X= 2.5 Km
Sam was 2.5 Km far from his house.
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Are the equation linear or nonlinear?
Answers
Answer:
Results below
Step-by-step explanation:
Equation of the line
A straight line can be written in the form:
y = ax + b
Where a and b are constants and x is the independent variable.
The essential condition for an equation to be linear is that the x must be powered to the exponent 1, which is usually not written.
From the equations presented in the table:
\(y = x^1+2\) is linear because the exponent of the x is 1
y = 5(x+2) = 5x + 10 is linear with a=5 and b=10.
y = x is linear with a=1 and b=0
\(y = x^2+1\) is not linear because the exponent of x is 2
\(y = x(x) = x^2\) is not linear because the exponent of x is 2
The table below summarizes the results
Malcolm trains on his kayak every weekend. He paddles upstream (against current) for 3 ½ hours and then returns downstream (with current) in 2hrs 6 minutes. If the river flows at 3km/ h, find:
* The paddling speed in still water
* The distance he paddles upstream.
Answers
The probability she pulls out a purple piece of candy would be 0.22.
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 Sam's fathers collection.
We can write the equations for upstream and downstream as -
x - y = 7/2
x + y = 21/10
Solving the equations graphically -
{x} = 2.8
{y} = 0.7
In still water, the speed would be -
S = 3 - 0.7
S = 2.3 Km/h
Distance peddled upstream -
D = 2.8 x 3.5 = 9.8 Km
Therefore, the speed in still water would be 2.3 Km/h and the distance peddled upstream would be 9.8 Km.
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Use the approach in Gauss's Problem to find the following sums of arithmetic
sequences.
a. 1+2+3+4+...+99
b. 1+3+5+7+...+1001
c. 3+6+9+ 12 + ... + 300
d. 4+8+12 + 16 + ... + 400
Answers
a. Let S be the first sum,
S = 1 + 2 + 3 + … + 97 + 98 + 99
If we reverse the order of terms, the value of the sum is unchanged:
S = 99 + 98 + 97 + … + 3 + 2 + 1
If we add up the terms in both version of S in the same positions, we end up adding 99 copies of quantities that sum to 100 :
S + S = (1 + 99) + (2 + 98) + … + (98 + 2) + (99 + 1)
2S = 100 + 100 + … + 100 + 100
2S = 99 × 100
S = (99 × 100)/2
Then S has a value of
S = 99 × 50
S = 4950
Aside: Suppose we had n terms in the sum, where n is some arbitrary positive integer. Call this sum ∑(n) (capital sigma). If ∑ is a sum of n terms, and we do the same manipulation as above, we would end up with
2 ∑(n) = n × (n + 1) ⇒ ∑(n) = n (n + 1)/2
b. Let S' be the second sum. It looks a lot like S, but the even numbers are missing. Let's put them back, but also include their negatives so the value of S' is unchanged. In doing so, we have
S' = 1 + 3 + 5 + … + 1001
S' = (1 + 2 + 3 + 4 + 5 + … + 1000 + 1001) - (2 + 4 + … + 1000)
The first group of terms is exactly the sum ∑(1001). Each term in the second grouped sum has a common factor of 2, which we can pull out to get
2 (1 + 2 + … + 500)
so this other group is also a function of ∑(500), and so
S' = ∑(10001) - 2 ∑(500) = 251,001
However, we want to use Gauss' method. We have a sum of the first 501 odd integers. (How do we know there 501? Starting with k = 1, any odd integer can be written as 2k - 1. Solve for k such that 2k - 1 = 1001.)
S' = 1 + 3 + 5 + … + 997 + 999 + 1001
S' = 1001 + 999 + 997 + … + 5 + 3 + 1
2S' = 501 × 1002
S' = 251,001
c/d. I think I've demonstrated enough of Gauss' approach for you to fill in the blanks yourself. To confirm the values you find, you should have
3 + 6 + 9 + … + 300 = 3 (1 + 2 + 3 + … + 100) = 3 ∑(100) = 15,150
and
4 + 8 + 12 + … + 400 = 4 (1 + 2 + 3 + … + 100) = 4 ∑(100) = 20,200
a father was 25 years older than his son five years ago. if the product of the son and his father's ages then was 900 years. how old is the son now.
Answers
Answer:
25
Step-by-step explanation:
Let x represent the son's age now. 5 years ago, it was x-5, and the father's age was (x -5)+25 = (x+20).
The product of their ages at that time was 900, so we have ...
(x -5)(x +20) = 900
x^2 +15x -100 = 900
x^2 +15x -1000 = 0 . . . . subtract 900
(x +40)(x -25) = 0 . . . . factor
x = -40 or +25 . . . . these values make the factors zero
The son's age now is 25.