Every proper subgroup of G is cyclic.
What is subgroup?In abstract algebra, a subgroup is a subset of a group that is itself a group under the same operation as the original group. More precisely, a subgroup H of a group G is a subset of G that satisfies the following three conditions:
The identity element of G is in H.H is closed under the group operation of G.Every element in H has an inverse in H.What are cyclic subgroups?A cyclic subgroup is a subgroup of a group that is generated by a single element. In other words, a subgroup H of a group G is cyclic if there exists an element a in G such that every element of H can be written as a power of a, and H is the smallest subgroup of G containing a.
In the given question,
Let H be a proper subgroup of G. Since H is a proper subgroup, we have lHl < lGl = 77.
Now, by Lagrange's Theorem, the order of any subgroup of G must divide the order of G. That is, lHl divides lGl = 77.
Since H is a proper subgroup, we have lHl > 1. Therefore, lHl can only be 7, 11, or 77 (the divisors of 77).
Suppose lHl = 7. Then, by Lagrange's Theorem, the elements of G are partitioned into disjoint cosets of H, each of size 7. But 77 is not divisible by 7, so there cannot be a whole number of cosets. Therefore, this case is not possible.
Similarly, suppose lHl = 11. Then, the elements of G are partitioned into disjoint cosets of H, each of size 11. But 77 is not divisible by 11, so there cannot be a whole number of cosets. Therefore, this case is also not possible.
Thus, we are left with the case where lHl = 77. But this implies that H = G, which contradicts the fact that H is a proper subgroup. Therefore, this case is also not possible.
We have exhausted all possibilities for the order of H, and in each case, we have arrived at a contradiction. Therefore, the assumption that there exists a proper non-cyclic subgroup of G must be false.
Hence, every proper subgroup of G is cyclic.
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In 1980, Town A had a population of 1200 people and the population increased by 300 people each year. In
1980, Town B had a population of 500 people and the population increased by 12% each year.
Which town will eventually have the greater population?
Which town had a greater population in 2005?
To answer this question, we need to calculate the population of both towns for each year and compare them.
For Town A:
1980 population = 1200
1981 population = 1200 + 300 = 1500
1982 population = 1500 + 300 = 1800
1983 population = 1800 + 300 = 2100
and so on
For Town B:
1980 population = 500
1981 population = 500 + 0.12(500) = 560
1982 population = 560 + 0.12(560) = 627.2
1983 population = 627.2 + 0.12(627.2) = 702.464
and so on
To determine which town will eventually have the greater population, we can calculate the limit of each population as time goes to infinity. For Town A, the population limit is infinity because the population keeps increasing by 300 every year. For Town B, the population limit is also infinity because the population grows exponentially due to the 12% increase each year. However, we can still compare their growth rates.
For Town A, the annual growth rate is constant at 300 people per year, whereas for Town B, the growth rate is decreasing over time due to the compounding effect of the percentage increase.
Therefore, initially, Town A has a smaller population than Town B. However, over time, Town A's growth rate remains constant while Town B's growth rate slows down. Eventually, Town A will surpass Town B in population.
To determine which town had a greater population in 2005, we need to calculate the population of each town in 2005 using the growth rates we calculated above.
For Town A:
1980 population = 1200
2005 population = 1200 + 300(25) = 8700
For Town B:
1980 population = 500
2005 population = 500(1.12)^25 = 4544.73
Therefore, in 2005, Town A had a greater population than Town B.
Find the equation for the ellipse
The value of the equation of the ellipse is (x + 1)² + (y + 1)² = 9, the center of the ellipse is at (-1, -1), thus graph B is correct.
What makes this ellipse eccentric?E = c/a, where c is the distance from the ellipse's centre to either focus and an is the distance from the ellipse's centre to either end of the main axis, is the formula for an ellipse's eccentricity.
The distance between the foci is 2a = 6, where an is the separation between any focus and the elliptic centre.
The middle of the ellipse is at the midway of the foci since they are both on the same horizontal line, which is (-1, -1).
The main axis measures 18 inches, or 2b, where b is the distance between the ellipse's center and either end of the major axis.
Inputting the values we are familiar with yields:
b² + (c = 3)² = a²
b² + 9 = 9
b^2 = 0
So b = 0, which means the ellipse is a circle with radius a = 3 centered at (-1, -1). The equation for a circle with center (h, k) and radius r is:
(x - h)² + (y - k)² = r²
Substituting the values we know, we get:
(x + 1)² + (y + 1)² = 9
Hence, the value of the equation of the ellipse is (x + 1)² + (y + 1)² = 9, the center of the ellipse is at (-1, -1), thus graph B is correct.
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Graph y=100(1+0.06)x
Answer:
Step-by-step explanation:
Consider the following function.
r(x)=4−32x
Step 2 of 2 : Find two points on the line to graph the function.
f(x)=
Answer:
Step-by-step explanation:
To find two points on the line represented by the function r(x) = 4 - 32x, we can arbitrarily choose two values of x and then find the corresponding values of r(x) using the equation.
For example, if we choose x = 0, then r(0) = 4 - 32(0) = 4. So the point (0,4) is on the line.
Similarly, if we choose x = 1, then r(1) = 4 - 32(1) = -28. So the point (1,-28) is on the line.
Therefore, we have two points (0,4) and (1,-28) on the line represented by the function r(x) = 4 - 32x. These two points can be used to graph the function.
In a math class with 30 students, a test was given the same day that an assignment was due. There were 21 students who passed the test and 20 students who completed the assignment. There were 7 students who failed the test and also did not complete the assignment. What is the probability that a student completed the homework given that they passed the test?
"Given that they passed the test" says we're only considering the 21 students who passed the test.
There were 9 students who didn't pass the test and 7 of those didn't complete the HW. This means that 2 students who did the HW didn't pass the test.
Since 20 students completed the HW and 2 didn't pass the test, 18 students completed the HW and passed the test.
18/21 = 6/7 or 85.71%
Your probability is 85.71%.
The crust of the pizza has a circumference
37.7 inches. The pizza fits perfectly inside the box (a
circle inscribed in a square). What is the length of one
side of the box?
Round your answer to the nearest tenth.
The length of one side of the box is approximately 8.5 inches whose circumference is 37.7 inches.
What is circumference?Circumference is the distance around the edge of a circle or any circular object. It can be found by multiplying the diameter of the circle by pi (π), which is approximately 3.14159. Alternatively, it can be found by multiplying twice the radius of the circle by pi.
According to question:The diameter of the circle inscribed in the square is equal to the side length of the square. Since the circumference of the pizza crust is 37.7 inches, the diameter of the circle is:
d = circumference / pi = 37.7 / pi ≈ 12
Therefore, the radius of the circle is r = d/2 ≈ 6. To find the side length of the square, we need to find the diagonal of the square, which is equal to twice the radius of the circle. As a result, the square's length of one side is:
s = diagonal / √(2) = 2r / √(2) = 6 √(2) ≈ 8.5
Therefore, the length of one side of the box is approximately 8.5 inches.
The circumference is an important measurement in geometry and is used to calculate the perimeter of circles, as well as to determine the length of curved lines in various other contexts.
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Evaluate the function graphically.
Find f (4)
To evaluate the function graphically, we need to plot the two lines on graph and see where they intersect.Then find the value of f(x) at that point.Therefore, f(4) = -4, which matches with the graphical evaluation.
What is function?A function is a mathematical rule that maps every input from a set (the domain) to a unique output from another set (the range), where each input has only one corresponding output.
The two lines intersect at the point (2,-4). Therefore, the value of f(x) at x=4 is -4.
To find f(4), we can use the equation of line 1, which is:
y - (-8) = (2 - (-8))/(1 - (-5)) × (x - (-5))
Simplifying, we get:
y + 8 = (10/6) × (x + 5)
y + 8 = (5/3) × (x + 5)
y = (5/3) × x - 19/3
Substituting x=4, we get:
y = (5/3) × 4 - 19/3
y = -4
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The price of a pencil is 1½. What is the cost of 10 such pencils? What about 20 such
pencils? Is the price directly proportional to the number of pencils?
Tο find the cοst οf 10 such pencils, we need tο multiply the price οf οne pencil by 10. We can write 1½ as a mixed number οr as an imprοper fractiοn.
In arithmetic, what is prοpοrtiοn?A prοpοrtiοn is an equatiοn that sets twο ratiοs at the same value. Fοr instance, yοu cοuld express the ratiο as fοllοws: 1: 3 if there is 1 bοy and 3 girls (fοr every οne bοy there are 3 girls) There are 1 in 4 bοys and 3 in 4 girls. 0.25 are male (by dividing 1 by 4)
A mixed number is a whοle number and a fractiοn tοgether, such as 1½. An imprοper fractiοn is a fractiοn where the numeratοr (tοp number) is larger than the denοminatοr (bοttοm number), such as 3/2. Tο multiply a mixed number by a whοle number, we can either cοnvert the mixed number tο an imprοper fractiοn first, οr multiply the whοle number by the whοle part and the fractiοnal part separately and then add them tοgether. Fοr example:
1½ × 10 = (1 + ½) × 10 = (1 × 10) + (½ × 10) = 10 + 5 = 15
Or
1½ × 10 = (3/2) × 10 = (3 × 10) / 2 = 30 / 2 = 15
Either way, we get the same answer: The cοst οf 10 such pencils is 15.
Tο find the cοst οf 20 such pencils, we can use the same methοd but multiply by 20 instead οf 10. Fοr example:
1½ × 20 = (1 + ½) × 20 = (1 × 20) + (½ × 20) = 20 + 10 = 30
Or
1½ × 20 = (3/2) × 20 = (3 × 20) / 2 = 60 / 2 = 30
Either way, we get the same answer: The cοst οf 20 such pencils is 30.
Tο check if the price is directly prοpοrtiοnal tο the number οf pencils, we need tο see if the ratiο οf the price tο the number οf pencils is cοnstant. A ratiο is a cοmparisοn οf twο quantities using divisiοn. A cοnstant ratiο means that nο matter hοw many pencils we buy, the price per pencil will always be the same. Fοr example:
Price / Number οf Pencils = Cοnstant Ratiο 15 / 10 = Cοnstant Ratiο 30 / 20 = Cοnstant Ratiο
We can simplify these fractiοns by dividing bοth numbers by their greatest cοmmοn factοr (GCF). The GCF οf 15 and 10 is 5, sο we divide bοth numbers by 5:
(15 / 5) / (10 / 5) = Cοnstant Ratiο 3 / 2 = Cοnstant Ratiο
The GCF οf 30 and 20 is alsο 5, sο we divide bοth numbers by 5:
(30 / 5) / (20 / 5) = Cοnstant Ratiο 6 / 4 = Cοnstant Ratiο
Nοw we can cοmpare these simplified fractiοns and see if they are equal. If they are equal, then it means that the price per pencil is always the same regardless οf hοw many pencils we buy. If they are nοt equal, then it means that the price per pencil changes depending οn hοw many pencils we buy.
3 /2= ?6 /4 12 /8= ?12 /8 12 /8=12 /8
Since these fractiοns are equal, it means that:
Price / Number οf Pencils = Cοnstant Ratiο Price ∝ Number οf Pencils
The symbοl ∝ means “is prοpοrtiοnal tο”. Sο this equatiοn tells us that yes, the price is directly prοpοrtiοnal tο the number οf pencils because fοr every increase in οne unit in οne quantity (number οf pencils), there is a cοrrespοnding increase in a fixed amοunt in anοther quantity (price).
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Write the quadratic equation whose roots are 6 and -5 and who’s leading coefficient is 4
[tex]\begin{cases} x = 6 &\implies x -6=0\\ x = -5 &\implies x +5=0\\ \end{cases} \\\\[-0.35em] ~\dotfill\\\\ \stackrel{original~polynomial}{a ( x -6 )( x +5 ) = \stackrel{0}{y}}\hspace{5em}\stackrel{\textit{since the leading coefficient is 4}}{a=4} \\\\\\ 4(x-6)(x+5)=y\implies 4(x^2-x-30)=y\implies \boxed{4x^2-4x-120=y}[/tex]
x[tex]x{2} -16\\[/tex]
Answer:
Step-by-step explanation:
To solve for x in the equation x^2 - 16 = 0, we can use the difference of squares formula:x^2 - 16 = (x + 4)(x - 4) = 0
Now we can set each factor equal to zero and solve for x:x + 4 = 0 or x - 4 = 0x = -4 or x = 4
Therefore, the solutions to the equation x^2 - 16 = 0 are x = -4 and x = 4.
Write the function whose graph is the graph of y=x³, but is shifted to the left 5 units.
Y=
(Simplify your answer.)
Check the 1st picture below.
from the template below, hmm we can say that a horizontal shift of +5 simply means B = 1 and C = 5, so C/B = 5
[tex]y=x^3\hspace{4em}y=1(1x+0)^3+0\implies y=1(1x+5)^3+0\implies y=(x+5)^3[/tex]
Check the 2nd picture below.
what would a number line representing g≠7 look like help asap
Answer:
hollow circle at 7, with an arrow extending to the left
Step-by-step explanation:
Write
(6^7) ^8
in the form 6k where k is an
integer to be found.
Answer:
When we raise a power to another power, we multiply the exponents. So:
(6^7)^8 = 6^(7*8) = 6^56
Therefore, (6^7)^8 can be written in the form 6^56, where k=56.
Step-by-step explanation:
Anybody know the answer to this one ?
The function value of g[f(-1)] when f(x) = 2x² + 4x and g(x)=-3x-1 is 5.
What is function?In mathematics, a function is a rule that assigns each element from a set (called the domain) to a unique element from another set (called the range or codomain). It is a way of describing the relationship between two sets of numbers. The input values of a function are called the independent variable or argument, and the output values are called the dependent variable. The most common way to represent a function is by using an equation.
Here,
To evaluate g[f(-1)], we need to first find f(-1), then use that value as the input to the function g.
We have:
f(x) = 2x² + 4x
So, to find f(-1), we substitute -1 for x:
f(-1) = 2(-1)² + 4(-1)
= 2 - 4
= -2
Now that we have f(-1) = -2, we can evaluate g[f(-1)] by substituting -2 for x in the function g:
g[f(-1)] = g[-2]
= -3(-2) - 1
= 5
Therefore, g[f(-1)] = 5.
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63(p4-11p3+24p2)/9p(p-8)
Answer:
7p(p-3) (if you're trying to find P then p is 3)
Step-by-step explanation:
First multiply the bottom so you get 9p^2 - 72p and then you can remove 1 variable of p from the bottom and top to get 63(p^3-11p^2+24p) and take one p out for 63p(p^2-11p+24) and factor for 63p(p-8)(p-3)/9(p-8) and take p-8 out of both. Then, you're left with 63p(p-3)/9 to get 7p(p-3) and to solve for p if the equation is equal to 0 -> 7(p)(p-3) = 0
-> 7p^2-21p
->7p^2 = 21p
7p = 21
p = 3
Lorene plans to make several open-topped boxes in which to carry plants. She makes the boxes from rectangular sheets of cardboard from
which she cuts out 4 in squares from each corner. The length of the original piece of cardboard is 8 in more than the width. If the volume of
the box is 420 in", determine the dimensions of the original piece of cardboard.
The dimensions of the original piece of cardboard are __ in by _ in.
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Done and Rev
The dimensions of the original piece of cardboard are 15 inches by 23 inches.
Let's start by defining the variables we'll need for this problem:
w = width of the original piece of cardboard (in inches)
l = length of the original piece of cardboard (in inches)
h = height of the box (in inches)
We know that the length of the original piece of cardboard is 8 inches more than the width, so we can write:
l = w + 8
When Lorene cuts out 4-inch squares from each corner of the rectangular sheet, the resulting height of the box will be 4 inches. Therefore, the height of the box is:
h = 4
The base of the box will have dimensions (w - 8) inches by (l - 8) inches, since 4 inches are removed from each side of the width and length. The volume of the box is given as 420 cubic inches, so we can write:
V = (w - 8)(l - 8)(4)
Substituting the expression for l in terms of w, we get:
420 = (w - 8)(w + 8 - 8)(4)
420 = (w - 8)(w)(4)
105 = w(w - 8)
Solving for w using the quadratic formula, we get:
w = 15 or w = -7
Since w must be a positive value, we can discard the solution w = -7. Therefore, the width of the original piece of cardboard is:
w = 15 inches
Using the expression for l in terms of w, we can find the length of the original piece of cardboard:
l = w + 8 = 23 inches
Therefore, the dimensions of the original piece of cardboard are 15 inches by 23 inches.
Find the equation of a line parallel to y+2 =-1/2x that passes through the point (2,-7)
Answer: The given equation of the line is y + 2 = -1/2 x, or equivalently, y = -1/2 x - 2. This equation is in slope-intercept form, y = mx + b, where the slope m is -1/2.
A line parallel to this line will have the same slope, so its equation will also be in the form y = -1/2 x + b, where b is the y-intercept that we need to find.
To find b, we can use the fact that the line passes through the point (2, -7). Substituting x = 2 and y = -7 into the equation, we get:
-7 = -1/2 (2) + b
Simplifying and solving for b, we get:
b = -7 + 1 = -6
Therefore, the equation of the line parallel to y + 2 = -1/2 x that passes through the point (2, -7) is:
y = -1/2 x - 6
Step-by-step explanation:
Bilquis and Kayla are reading the same book. At the beginning of the month, Bilquis was on page 23 and Kayla was on page 2. Bilquis will read 7 pages per day and Kayla will read 14 pages per day. Let B represent the page of the book that Bilquis is on at the end of t days into the month, and let K represent the page of the book that Kayla is on at the end of t days into the month. Write an equation for each situation, in terms of, t, and determine whether Bilquis or Kayla is farther along in the book after 4 days.
Answer:
Kayla is farther along in the book after 4 days, since she is on page 58 while Bilquis is on page 51.
Step-by-step explanation:
Determine whether the following equation defines y as a function of x: x^2 + y^2 = 169. How was this solved? Where does the 2 (pictured) come from?
Answer:
Step-by-step explanation:
Assuming you can follow how the equation was solved for y, the 2 comes from the "±" in front of the square root. That means that there is a positive and a negative result when you plug in a number for x. For example, if we were asked to solve
[tex]x^2=4[/tex]
we would take the square root of both sides, giving us
x = ±2
This is because a positive 2 squared gives us a positive 4, and a negative 2 squared also gives us a positive 4. There will ALWAYS be 2 solutions when you take a square root of a number, both the positive (principle) and negative roots.
Does the point (-3,5) lie inside,outside or on the circle with the equasion (x+2)^2+(y-4)^2=9
complete a cash budget
Ending Balance: $14,450
What is cash?Cash typically refers to physical currency and coins that are used as a medium of exchange for goods and services. It includes banknotes and coins that are issued by a government or central bank and are accepted as legal tender within a country.
Given by the question.
Certainly, here is an example of a monthly cash budget:
Monthly Cash Budget:
Starting Balance: $10,000
Income:
Salary: $5,000Rental Income: $1,500Other Income: $500Total Income: $7,000Expenses:
Rent/Mortgage: $1,200Utilities: $300Groceries: $400Transportation: $200Insurance: $100Entertainment: $100Clothing: $50Gifts/Charity: $50Other Expenses: $150Total Expenses: $2,550Net Cash Flow: $4,450Notes:
Starting Balance refers to the amount of cash you have at the beginning of the month.
Income includes all sources of cash inflow.
Expenses includes all sources of cash outflow.
Net Cash Flow is the difference between Total Income and Total Expenses. If it is positive, you have a surplus. If it is negative, you have a deficit.
Ending Balance is the amount of cash you have at the end of the month after all income and expenses have been accounted for.
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You choose 9 CDs from your collection and place them in a rack. What is the probability that the CDs in rack ends up in alphabetical order? Give the answer as a fraction with factorial notation. The probability that the CDs are in alphabetical order =
the probability that the CDs are in alphabetical order is 1/12528
What is Probability?Probability is a branch of mathematics that deals with the study of random events or outcomes. It is the measure of the likelihood or chance of an event occurring, expressed as a number between 0 and 1. An event with a probability of 0 is impossible, while an event with a probability of 1 is certain to occur.
Given by the question.
Let's assume that the CDs are labeled as CD1, CD2, CD3, ..., CD9.
If CD1 is placed first, then there is only one way to place the remaining CDs in alphabetical order.
If CD2 is placed first, then there is only one way to place CD1 and CD3-CD9 in alphabetical order.
If CD3 is placed first, then there are two ways to place CD1-CD2 and CD4-CD9 in alphabetical order arrange (CD1-CD2-CD4-CD5-CD6-CD7-CD8-CD9 or CD2-CD1-CD4-CD5-CD6-CD7-CD8-CD9).
Similarly, for CD4-CD9, there are 3, 4, 5, 6, and 7 ways to place the remaining CDs in alphabetical order.
Therefore, the total number of ways to arrange the CDs in alphabetical order is:
[tex]1 + 1 + 2 + 3 + 4 + 5 + 6 + 7 = 29[/tex]
Hence, the probability of the CDs ending up in alphabetical order is:
[tex]1/12528[/tex]
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Question I conduct a survey, and find out that the proportion of college students who are women is 0. also learn that the proportion of psychology majors is 0.1. Assume that these two variables are indepen What is the probability of randomly selecting a student who is either female or a psychology major? Rou the hundredths place.
Answer: Since we know that the two variables are independent, we can use the addition rule of probability to find the probability of selecting a student who is either female or a psychology major:
P(female or psychology major) = P(female) + P(psychology major) - P(female and psychology major)
Since the variables are independent, we know that P(female and psychology major) = P(female) x P(psychology major) = 0 x 0.1 = 0.
Therefore:
P(female or psychology major) = 0 + 0.1 - 0 = 0.1
So the probability of randomly selecting a student who is either female or a psychology major is 0.1.
Step-by-step explanation:
find a quadratic with roots of x = -1 ± 2i
Answer:
If the roots of a quadratic equation are given as x = -1 ± 2i, then the equation can be written in the form:
(x - (-1 + 2i))(x - (-1 - 2i)) = 0
Simplifying this expression, we get:
(x + 1 - 2i)(x + 1 + 2i) = 0
Expanding the left side of this equation, we get:
x^2 + x(2 - 2i + 2 + 2i) + (1 - 2i)(1 + 2i) = 0
Simplifying this expression, we get:
x^2 + 2x + 5 = 0
Therefore, the quadratic equation with roots of x = -1 ± 2i is:
x^2 + 2x + 5 = 0
A continuación se muestra una recta numérica. 11*11 31 2 0 Z ¿Cuál es la distancia, en unidades, de 0 al punto P en la recta numérica?
The distance between zero and point P is 5 units.
¿Cuál es la distancia, en unidades, de 0 al punto P en la recta numérica?To find the distance between two points on the number line, we must take the difference between the values corresponding to those points.
The value of 0 is trivially 0.
The value of P is between -4 and -6, so we conclude that:
P = -5
Taking the difference between these values we obtain:
distance = 0 - (-5)
distance = 0 + 5 = 5
The distance between those values is 5 units.
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Find the equation of the slope 1/3 and u-intercept 2
Answer:
y = 1/3x +2
Step-by-step explanation:
The slope intercept form of the equation of a line is
y = mx+b
where m is the slope and b is the y intercept.
y = 1/3x +2
Answer:
y = 1/3x + 2
Step-by-step explanation:
The equation for slope intercept form of a line is:
y = mx + c
Here,
m = slope
c = y intercept
Given that,
m = 1/3
c = 2
So, to find equation of the line, replace m and c with the given values.
y = mx + c
y = 1/3x + 2
a person randomly grabs candies out of a bag there are 480 candles in the bag the person looked and they pulled out 20% if the candies were yell0ow and 15% of them were green how many more were yellow than green
Answer:
a person randomly grabs candies out of a bag there are 480 candles in the bag the person looked and they pulled out 20% if the candies were yell0ow and 15% of them were green how many more were yellow than green
Step-by-step explanation:
it's B
Answer: 5 more yellow candies than 15 candies, or 5
Step-by-step explanation: Think, if there are 20 yellow candies and 15 of them green;
You need to take away 20 - 15 which will be 5, 5 more yellow candies than 15 candies
Keep up the good work=)
Carly picked m mangoes
a) On Monday, she added 4 mangoes to what she picked and shared the total equally into 2 groups. Write an expression to represent the number of mangoes in each group.
b) On Tuesday, she had twice the number of mangoes picked then added 2 more mangoes. If she shared this total into 3 equal groups, write an expression to represent the number of mangoes in
each group.
c) If Carly subtracted the number of mangoes in each group on Tuesday from the number of mangoes in each group on Monday, she will have 3 mangoes left Write an equation to represent
this information.
d) Hence, calculate the number of mangoes she picked.
These expression shows Carly picked 20 mangοes
a. (m+4)/2
b. (2m+2)/3
c. (m+4)/2 - (2m+2)/3 = 3
d. Carly picked 20 mangοes.
What is Algebraic Equatiοn?An algebraic equatiοn is a mathematical statement that uses variables and mathematical οperatiοns such as additiοn, subtractiοn, multiplicatiοn, and divisiοn tο express the equality οf twο algebraic expressiοns.
The gοal οf sοlving an algebraic equatiοn is tο determine the values οf the variables that satisfy the equatiοn.
a) The number οf mangοes in each grοup that Carly shared equally οn Mοnday can be represented by the expressiοn: (m+4)/2
b) The number οf mangοes in each grοup that Carly shared equally οn Tuesday can be represented by the expressiοn: (2m+2)/3
c) If Carly subtracted the number οf mangοes in each grοup οn Tuesday frοm the number οf mangοes in each grοup οn Mοnday and gοt 3 mangοes left, the equatiοn representing this infοrmatiοn is:
(m+4)/2 - (2m+2)/3 = 3
d) Sοlving the equatiοn frοm part c) fοr m, we get:
9m + 6 - 4(2m + 2) = 18
9m + 6 - 8m - 8 = 18
m = 20
Therefοre, Carly picked 20 mangοes.
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Each side of a square classroom is 6 yards long. The school wants to replace the carpet in the classroom with new carpet that costs $53.00 per square yard. How much will the new carpet cost?
The new carpet will cοst $1,908.00.
What is an area?
The size οf a regiοn οn a surface is measured by its area. While surface area refers tο the area οf an οpen surface οr the bοundary οf a three-dimensiοnal οbject, the area οf a plane regiοn οr plane area refers tο the area οf a shape οr planar lamina.
Area can be interpreted as the quantity οf material with a particular thickness required tο create a mοdel οf the shape οr as the quantity οf paint required tο cοmpletely cοver a surface in a single cοat. It is the twο-dimensiοnal equivalent οf the vοlume οf a sοlid οr the length οf a curve (a οne-dimensiοnal cοncept) (a three-dimensiοnal cοncept).
The area οf the classrοοm is the square οf the length οf its side, which is:
6 yards × 6 yards = 36 square yardsTο find the cοst οf the new carpet, we need tο multiply the area οf the classrοοm by the cοst per square yard:
36 square yards × $53.00/square yard = $1,908.00
Therefοre, the new carpet will cοst $1,908.00.
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100 points!!!! - Centers of a Triangle
Part 1:
Define the following:
- Circumcenter
- Orthocenter
- Centroid
Part 2: Using a straightedge, draw a non-equilateral triangle at least 6 inches wide and tall. (The bigger the triangle the easier part 2 will be to complete)
Using a straightedge and compass, construct the centers (circumcenter, orthocenter, and centroid) of that triangle. Label each of these in your triangle. Each of these 3 centers should be all on that same triangle. (Hint: Use a different color for each
center to make the constructions easier to see). It is very important that you use a straightedge and a compass to construct these centers (no freehand). Provide pictures!
Part 3: Once you have constructed the 3 centers, describe what you notice about those centers. What is a hypothesis that you can make about these 3 centers?
Part 1 - Circumcenter: The point where all three sides of the triangle meet and intersect, also the center of the circle that passes through all three vertices of the triangle.
What is intersect?Intersect is an operation used in mathematics and computing to determine the common elements of two or more sets. It is usually written as a symbol (∩) and is used to return the set of all elements that are in both sets. For example, if A is the set of all even numbers and B is the set of all prime numbers, then A ∩ B will return the set of all prime even numbers.
- Orthocenter: The point where the three altitudes of the triangle intersect.
- Centroid: The point where the three medians of the triangle intersect.
Part 3: The three centers of the triangle (circumcenter, orthocenter, and centroid) can all be constructed using a straightedge and compass. The circumcenter is located at the intersection of the three sides, the orthocenter is located at the intersection of the three altitudes, and the centroid is located at the intersection of the three medians. From this, we can make the hypothesis that these three centers of a triangle are related in that they are all located at the intersection of two or more lines that form the triangle.
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