The true statements are,
A. The coefficients of xⁿ and yⁿ both equal 1.
D. For any term x and y in the expansion, a + b = n
What is an expression?
Mathematical expression is defined as the collection of the numbers variables and functions by using operations like addition, subtraction, multiplication, and division.
Given that;
The expression is,
⇒ (x + y)ⁿ
Now,
We know that;
A binomial expansion is,
⇒ [tex](x + y)^n = x^n + nx^n^-^1 y + ^{n} C_{2} x^n^-^2y^2 + ..... + nxy^n^-^1 + y^n[/tex]
Clearly, For Option A:
The coefficients of xⁿ and yⁿ both equal 1.
If we see first and last term of the expansion, This statement is true.
For Option D:
For any term xᵃyᵇ in the expansion, a + b = n.
Let we take 3rd term of expansion
Here, a = n-2 and b=2
So, a + b = n-2+2 = n
Thus, a + b = n is true statement.
For Option B:
For any term xᵃyᵇ in the expansion, a - b = n.
Let we take 3rd term of expansion
Here, a=n-2 and b=2
If we do a-b = n-2-2=n-4≠n
Thus, a-b=n is false statement.
Thus, The true statements are given below,
A. The coefficients of xⁿ and yⁿ both equal 1.
D. For any term x and y in the expansion, a + b = n
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Grace is going to invest $140 and leave it in an account for 13 years. Assuming the interest is compounded continuously, what interest rate, to the nearest hundredth of a percent, would be required in order for Grace to end up with $190?
The interest rate, to the nearest hundredth of a percent, which would be required in order for Grace to end up with $190 is 2.38%.
How to determine the interest rate?Mathematically, compound interest can be calculated by using this formula:
A(t) = P(1 + r)^{t}
Where:
A represents the future value.P represents the principal.r represents the interest rate.T represents the time measured in years.Substituting the given parameters into the compound interest formula, we have;
190 = 140(1 + r)^{13}
190/140 = (1 + r)^{13}
1.3571 = (1 + r)^{13}
Taking the 13th root of both sides of the equation, we have:
[tex]\sqrt[13]{1.3571} = \sqrt[13]{(1 + r )^{13} }[/tex]
1.0238 = 1 + r
r = 1.02378 - 1
Interest rate, r = 0.0238
Interest rate, r = 2.38%
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Given that A,O & B lie on a straight line segment, evaluate the size of the smallest angle.
The diagram is not drawn to scale.
Based on the definition of the angles of a straight line, the measure of the smallest angle is: are: 40°.
What is the Sum of Angles on a Straight Line?The sum of all the angles on a straight line is equal to 180 degrees.
To find the size of the smallest angle, find the value of a by creating the equation below:
(2a + 62) + (54 - a) + (5a - 20) = 180
Open the parentheses:
2a + 62 + 54 - a + 5a - 20 = 180
Combine like terms
6a + 96 = 180
6a = 180 - 96
6a = 84
6a/6 = 84/6
a = 14
Find the measure of each angles:
2a + 62 = 2(14) + 62 = 90°
54 - a = 54 - 14 = 40°
5a - 20 = 5(14) - 20 = 50°
The size of the smallest angle is 40°.
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Solve g(x) = 10, if g(x) = 2/x+1
Answer:
x = -0.8.
Step-by-step explanation:
To solve for x in the equation g(x) = 10, we need to find the value of x that makes g(x) equal to 10. Since we are given that g(x) = 2/x+1, we can substitute this expression for g(x) in the equation g(x) = 10 to get:
2/x+1 = 10
Next, we can multiply both sides of the equation by x+1 to get rid of the fraction on the left side:
2 = 10 * (x+1)
This simplifies to 2 = 10x + 10, which we can rearrange to get 2 - 10 = 10x. Subtracting 10 from both sides gives us -8 = 10x. Finally, dividing both sides by 10 gives us the solution: x = -0.8.
Therefore, the value of x that makes g(x) equal to 10 is x = -0.8.
10) Find the missing angle measure. Round answers to the
nearest tenth.
37
24
Answer:
Θ = 49.6°
Step-by-step explanation:
cos Θ = adj/hyp
cos Θ = 24/37
cos Θ = 0.64864
Θ = cos^-1 0.64864
Θ = 49.6°
Use the common ratio to find the next term of the geometric sequence. 120,60,30, ...
Answer:
15.
Step-by-step explanation:
The common ratio = 60/120 = 1/2
So the next term is 30 * 1/2 = 15.
what is the rate of change (slope) of the distance in feet below sea level with respect to time that the submarine traveled? place your answer in the box:
The rate of change (slope) of the distance in feet below sea level with respect to time that the submarine traveled is 11 feet .
Is the rate of change and slope the same thing?Slope and rate of change are the same thing. The difference is just semantics of the situation.
Typically we use the word “slope” when we are referring to a graph, and we use the phrase “rate of change” when we are referring to a table or a situation in real life.
But that’s not absolute. We can count the slope on a graph, but there’s also a “slope formula” for use when you have two coordinate points or two data points — even though you are not creating a graph.
What do you call "the rate of change of slope"?It is the derivative of of the slope, or the second derivative of the original function.
For example, if you plot position verses time, the slope between two points(change in position per change in time) is the velocity. This is the first derivative.
Take any two points ( 0 , 380 ) and ( 15 , 545 )
slope m = 545 - 380 / 15 - 0
= 165 / 15
= 33 / 3
= 11 / 1
= 11 feet
Hence the slope m is 11 feet .
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An object moving with a speed of 21 m/s and has a kinetic energy of 140 J, what is the mass of the object?
Answer:
0.63 kg
Step-by-step explanation:
KE = 1/2mv²
solve for m:
m = 2(KE)/v²
m = 2(140J) / (21m/s)²
m = 280J / 441 = 0.63 kg
Richard will increase the amount of time he studies each night from a period of 40 minutes to a period of 50 minutes. By what percentage will Richard increase the amount of time he studies each night?
Richard will increase the amount of time he studies each night by 25%.
What is the percentage?The percentage is defined as a ratio expressed as a fraction of 100.
To find the percentage increase, we need to first find the difference between the two amounts of time.
In this case, the difference is 50 minutes - 40 minutes = 10 minutes.
Next, we need to divide the difference by the original amount of time and multiply by 100% to express the answer as a percentage.
In this case, the percentage increase is :
⇒ (10 / 40) x 100% = 25%.
Therefore, he will increase the amount of time he studies each night by 25%.
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Noah has 5 m of rope how many pieces of rope of length 1/2 meter can he cut from it
Answer: 10 pieces
Step-by-step explanation:
Answer:
10 ropes
Step-by-step explanation:
5 divided by 1/2 is 10
Easy question, pls explain:
In a particular class, 72% of the students have black hair. Five black-haired students leave the class, so that now 65% of the students have black hair. How many students were originally in the class?
Answer:
72% are black haired - 5 students becomes 65% students are black haired.
meaning 7% are 5 students
Base = Percentage/Rate
Base = 5/0.07
Base = 71 students
CHECKING71 (students) * 0.72 (rate of black haired) = 51
There are 51 black hair students51 - 5 (leave the class) = 46 students
71 (students) * 0.65 (rest of black hair) = 46 students
Therefore, we checked our answers and its correctThere are total of 71 students in the class. 51 are black hair, but since 5 left, there will be 46 black hair students leftStep-by-step explanation:
heart and star pls <3 brainliest will be appreciated <3(っ◔◡◔)っ -{ elyna s }-HELP PLSS ILL MARK BRAINLIEST!!!
Select the correct images on the graph.
Identify which shapes on the graph are congruent to shape I by performing these sequences of transformations on shape I:
a reflection across the y-axis, followed by a 90° counterclockwise rotation about the origin, and then a translation 3 units down
. a 90° counterclockwise rotation about the origin and then a translation 2 units up and 2 units left
a 180° rotation about the origin and then a translation 1 unit right
Which of the following is equivalent to
16^3/4
Answer:
B. 8
Step-by-step explanation:
I’m glad to help you
Use what you know about multiplying integers to explain why this equation is true:
20 ÷ (−5) = (−4).
Here, in the given equation it is true because two different sign, and the result is negative.
What about multiplying integers?To multiply or divide signed integers, always multiply or divide the absolute values and use these rules to determine the sign of the answer.
When multiply two integers with the same signs, the result is always positive. Just multiply the absolute values and make the answer positive.
When multiply two integers with the different signs, the result is negative.
Here in this equation,
20 ÷ (-5)
= - 4
So, it is true.
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let x be the set of 2000 correct: your answer is correct. people (the pigeons) and y the set of all 1825 incorrect: your answer is incorrect. possible birthdays (the pigeonholes).
Yes, in a group of 2,000 people, at least 5 must have the same birthday.
This is due to the Pigeonhole Principle, which states that if there are more pigeons than pigeonholes, then at least one pigeonhole must contain more than one pigeon.
In this case, there are 2,000 people (the pigeons) and 365 possible birthdays (the pigeonholes). This means that, since 2,000 is greater than 365, at least 5 people must share the same birthday.
The Pigeonhole Principle states that if you have more items than boxes, then at least one box must contain more than one item. It can also be used to prove that certain problems have no solution.
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Complete Question:
In a group of 2,000 people, must at least 5 have the same birthday? Why? Let X be the set of 2000 people (the pigeons) and Y the set of all possible birthdays (the pigeonholes). Define a function B: X - Y by specifying th B(x) = x's birthday. Now 2,000 > 4.366 = 1,464, and so by the generalized pigeonhole principle, there must be some birthday y such that B-1(y) has --Select- 4 + 1 = 5 elements. Hence at -Select- 5 people must share the same birthday.
I’m how many ways can 25 be expressed as the sum of three prime numbers
Answer:
Below
Step-by-step explanation:
Primes under 25 2 3 5 7 11 13 17 19 23
5 7 13
7 7 11 ( can we use numbers twice?)
you can look for more....but that may be it ....
Graph TU with endpoints T(1,2) and U(4,6) and its image after the composition
TU with endpoints T(1,2) and U(4,6) and its images after the composition are T' = (-1, -1) and U' = (2,3) and T'' = (-3, 7) and U'' = (0,11).
What is the image point?Tradition holds that the reflection of point P across the line, denoted by the word P', is the "image" of point P. (pronounced "P prime").
Given, line TU with endpoints T(1,2) and U(4,6).
To find the translation:
In 1:
Translate TU 2 units left and 3 units down.
That means,
T' = (-1, -1) and U' = (2,3)
In 2:
Translate TU 4 units left and 5 units up.
That means,
T'' = (-3, 7) and U'' = (0,11)
Therefore, the image points are T' = (-1, -1) and U' = (2,3) and T'' = (-3, 7) and U'' = (0,11)
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Evan bought 12 postcards during 6 days of vacation. After 9 days of vacation,
how many total postcards will Evan have bought? Assume the relationship is
directly proportional.
Answer:
Step-by-step explanation:
Evan bought 18 postcards in 9 days
since he got 12 in 6 days that means he got 2 each day so 9x2=18
or if you add up by 2 9 times you would get 18 :)
y = 4x - 9
y=x - 3
HELP WILL GIVE BRAINLIEST
The solution to the system of equations, y = 4x - 9 and y = x - 3, is: (2, -1)
How to Solve a System of Linear Equations?The coordinates of the ordered pair that will make both equations true is the solution to the system.
Given the linear equations as:
y = 4x - 9 --> equation 1
y = x - 3 --> equation 2
Make both linear equations equal to each other and find the value of x. Thus:
4x - 9 = x - 3
Combine like terms
4x - x = 9 - 3
3x = 6
Divide both sides by 3
3x/3 = 6/3
x = 2
To find the value of y, substitute x = 2 into equation 1:
y = 4(2) - 9
y = 8 - 9
y = -1
The solution is, (2, -1).
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When solving the following equations, multiply the first and second equation by a constant in order to eliminate the y terms.
2x + 2y = 7
6x + 3y = -2
What are the constants of both equations?
Answer:
multipliers: 3, 2 for subtraction; or 3, -2 for addition.solution: (x, y) = (-4 1/6, 7 2/3)Step-by-step explanation:
You want to solve the given system of equations by eliminating the y-term. You also want to know what constants are used in the process.
EliminationThe y-term can be eliminated from these equations by combining them in such a way as to may the coefficient of y be zero. This can be done an infinite number of ways.
One way that works reasonably well is to identify the coefficients of y, and multiply each equation by the y-coefficient in the other equation. The resulting equations can be subtracted one from the other to eliminate the y-terms, which now have identical coefficients. You can choose the equation with the smaller x-coefficient to subtract from the other one.
ApplicationThe y-term coefficients are 2 and 3.
Multiply the first equation by 3, and the second by 2.
3 × (2x +2y) = 3 × 7 ⇒ 6x +6y = 21
2 × (6x +3y) = 2 × (-2) ⇒ 12x +6y = -4
The x-coefficients are 6 and 12. We choose to subtract the first equation (with x-coefficient 6) from the second equation:
(12x +6y) -(6x +6y) = (-4) -(21)
6x = -25 . . . . . . . . simplified
x = -25/6 = -4 1/6 . . . . . divide by 6
Substituting for x in the first equation, we have ...
2(-25/6) +2y = 7
2y = 7 +25/3 . . . . . add 25/3
2y = 46/3 . . . . . . . .combine terms
y = 23/3 = 7 2/3 . . . . divide by 2
The solution to the equations is (x, y) = (-4 1/6, 7 2/3).
CheckSince these numbers are not integers, we choose to check them in the original equations. We used the first equation to find y, so we'll use the second equation for checking.
6(-4 1/6) +3(7 2/3) = -25 +23 = -2 . . . . . . as required
__
Additional comment
If you solve these by addition, rather than subtraction, you want to choose multipliers that result in opposite coefficients, so they add to zero. Essentially, you use the opposite coefficients, as described above, but you negate one of them.
Above, we chose the subtraction in a way that resulted in the x-coefficient being positive. If you arbitrarily choose the signs of the multipliers (always negating the second one, for example), the combined equations may make the x-coefficient negative. That works, too, but may require more mental effort to deal with the minus sign.
Of course, you can use the x-coefficients and eliminate the x-terms.
Or, you can use a single multiplier that corresponds to the ratio of the coefficients. For example the ratio of the 2nd y-coefficient to the first one is 3/2. Multiplying the first equation by this will give the y-term a coefficient of 3, matching the y-coefficient in the 2nd equation. Then the subtraction can proceed as already described.
(6x +3y) -3/2(2x +2y) = (-2) -3/2(7)
3x = -25/2 . . . . . simplified; y is gone.
K
Determine whether the following statement is true or false, and explain why.
If a function is concave upward on an interval, it must be increasing on the interval.
Is the statement true or false?
A. The statement is true. Functions that are concave upward are always increasing.
B. The statement is false. Functions that are concave upward are always decreasing.
C. The statement is false. Functions that are concave upward are constant and do not increase or decrease.
D. The statement is false. Functions that are concave upward can be either increasing or decreasing.
11
The statement is true. Functions that are concave upward are always increasing.
What is Concave Upward Function?
Where the derivative f′ is growing, a function f is concave up (or upwards). This is equivalent to the derivative of f′, which is represented by the notation f′′f, start superscript, prime, end superscript, positive.
Explanation:
If a function is concave upward on an interval, it must be increasing on the interval. Given statement is true. Functions that are concave upward are always increasing.
A function is said to be concave upward on an interval if f″(x) > 0 at each point in the interval and concave downward on an interval if f″(x) < 0 at each point in the interval.
At interval if function is increasing then functions that are concave upward are always increasing.
Hence, The statement is true. Functions that are concave upward are always increasing.
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Question 2(Multiple Choice Worth 2 points)
(Multiplying Integers MC)
What is the product of (-25)(15)(-3)?
O 1,125
60
O-200
O-1,125
The product of the given values would be = 1,125. That is option A.
What is multiplication?Multiplication is defined as one of the major arithmetic operation along with others such as addition, subtraction and division which is used to to determine the product of two or more values.
The rule of multiplication is that
when a positive figure is multiplied by a positive figure, a positive value is obtained. When a negative figure multiplies a negative figure a positive value is obtained.Therefore;
= (-25)× (15) × (-3)
= -375 × -3
= 1,125.
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What is 1 1/2 + 5/7?
A. 2 1/14
B. 2 2/7
C.11/14
D 2 3/14
The fraction 1¹/₂ + ⁵/₇ = 2³/₁₄
What are fractions?Fractions are numbers written in the form a/b where a and b are integers and b ≠ 0.
How to find what 1¹/₂ + ⁵/₇?Since we have the fractions 1¹/₂ and ⁵/₇ and we want to compute the sum
1¹/₂ + ⁵/₇, we re-write the first fraction as an improper fraction.
So, 1¹/₂ = ³/₂
Now, re-writting the equation, we have
³/₂ + ⁵/₇
To find the sum, wetake the L.C.M of both numbers which 2 × 7 = 14.
So, we make both denominators equal to the L.C.M.
So,
³/₂ × ⁷/₇ + ⁵/₇ × ²/₂ = ²¹/₁₄ + ¹⁰/₁₄
= (21 + 10)/14
= 31/14
= (28 + 3)/14
= 28/14 + 3/14
= 2 + 3/14
= 2³/₁₄
So, 1¹/₂ + ⁵/₇ = 2³/₁₄
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Suppose you have the following three student loans: $12,000 with an APR of 7.5% for 17 years, $18,000 with an APR
of 8% for 22 years, and $13,500 with an APR of 9% for 12 years.
a. Calculate the monthly payment for each loan individually.
b. Calculate the total you'll pay in payments during the life of all three loans.
c. A bank offers to consolidate your three loans into a single loan with an APR of 8% and a loan term of 22 years.
What will your monthly payments be in that case? What will your total payments be over the 22 years?
a) The monthly payment for each loan individually is as follows:
Loan A Loan B Loan C Total
Monthly payments $133.82 $188.18 $195.00 $517.00
b) The total payments during the life of all three loans is $105,060, classified as follows:
Loan A Loan B Loan C Total
Total amount $27,300 $49,680 $28,080 $105,060
c) The monthly payments after consolidating the three loans are $350.69.
d) The total payments over the 22 years period will be $92,581.47.
How are the monthly payments determined?Note that student loans attract simple interest and not compound interest.
After the bank consolidates it, the interest system changes to compounding.
The total interest is computed by applying the annual percentage rate (APR) to the principal and the loan term. The result is added to the principal to obtain the total payment for each loan.
The total payment is divided by the loan period to get the monthly payments.
But for the consolidated loan by the bank, we can use an online finance calculator to determine the monthly payments and total payment over the loan term as follows.
Loan A Loan B Loan C Total
Student loan $12,000 $18,000 $13,500 $43,500
APR 7.5% 8% 9%
Loan term 17 years 22 years 12 years
Period in months 204 264 144
Total interests $15,300 $31,680 $14,580 $61,560
Total amount $27,300 $49,680 $28,080 $105,060
Monthly payments $133.82 $188.18 $195.00 $517.00
Consolidated Loans:Total amount of loans = $43,500 ($12,000 + $18,000 + $13,500)
Bank APR = 8%
Loan term = 22 years
N (# of periods) = 264 months (22 years x 12)
I/Y (Interest per year) = 8%
PV (Present Value) = $43,500
FV (Future Value) = $0
Results:
PMT = $350.69
Sum of all periodic payments = $92,581.47
Total Interest = $49,081.47
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Find the exact area (in units2) of the region bounded by the given equations if possible. If you are unable to determine the intersection points analytically, use a calculator to approximate the intersection points with three decimal places and determine the approximate area (in units2) of the region, then round your answer to three decimal places.x^2 = y^3 and x = 2y
The region bounded by the equations has a precise area of 21.33 units.
A boundary or some set of constraints are imposed on a bounded region. To put it another way, the size of a bounded shape cannot be infinite. Anything that is bound must be able to be contained within certain parameters.
x2 = 4y2 from equation (1), y3 = 4y2 y3 - 4y2 = 0 y2(y - 4) = 0 y2 = 0 or y-4 = 0 or y = 4 These will be our integration bounds. We have given that x2 = y3 --------- (1) and x = 2y --------- (2).
Therefore, we will express area as A = [- 4]04 = [- 4 - 0] = 64 - 85.33 A = -21.33, or A = 21.33. This indicates that the area of the bounded region is 21.33 units.
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Let f be the function given by f(x) = x+4/(x-1)(x+3) on the closed interval [-5, 5]. On which of the following closed intervals is the function f guaranteed by the extreme value theorem to have an absolute maximum and an absolute minimum?a. [-5, 5]b. [-3, 1]c. [-2, 0]d. [0, 5]
The function f is guaranteed by the extreme value theorem to have an absolute maximum and an absolute minimum on the closed interval a. [-5, 5].
The extreme value theorem states that if a function f is continuous on a closed interval [a, b], then it must have both an absolute maximum and an absolute minimum on that interval. Since the function f is given as being defined on the closed interval [-5, 5], it must have both an absolute maximum and an absolute minimum on this interval.
We can check that the function f is continuous on the interval [-5, 5] as follows:
First, we need to check that the function is defined at every point in the interval. The function f is defined as f(x) = x+4/(x-1)(x+3). The only points at which this function is not defined are x = 1 and x = -3, which are not within the interval [-5, 5]. Therefore, the function is defined at every point in the interval.
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Find the interest rate for a principal of $5305 and charged $39534 in interest for 20 years
Answer:
37.237%
Step-by-step explanation:
Use the Formula I = P x R x T
P = Principal amount (the beginning balance).
R = Interest rate (usually per year, expressed as a decimal).
T = Number of time periods (generally one-year time periods).
Plug in the values
I = 5305 x R x 20
Solve for R by dividing 20 by 5305
I=5305 x R x 20
20 / 5305 = 0.0037
Answer: 10.56 or 10.6 %
Step-by-step explanation:
5305 x x^20 = 39534
39534/5305 = x^20
20√39534/5305 = x
x = 1.1056
1.1056 x 100 = 110.56
110.56 - 100 = 10.56 or 10.6
Anna has 7,209 cans of soup that need to be boxed. If she puts 9 cans of soup in 1 box, how many boxes will she need.
Answer:
801 boxes
Step-by-step explanation:
[tex]\frac{7209}{9}[/tex] = 801
What is the equation for the hyperbola shown?
The hyperbola's standard equation is [(x2/a2) - (y2/b2)] = 1, where X denotes the transverse axis and Y denotes the conjugate axis.
Define hyperbola.A hyperbola is a particular kind of smooth curve that lies in a plane and is classified by its geometric characteristics or by equations for which it is the solution set. A hyperbola is made up of two mirror images of one another that resemble two infinite bows. These two sections are known as connected components or branches. A hyperbola is a geometric shape in mathematics where the difference between the lengths from any point on the figure to two fixed locations is a constant (the Greek letter o literally means "overshooting" or "excess"). We refer to the two fixed spots as foci (plural of focus).
Given
The equation for the hyperbola
The hyperbola's standard equation is [(x2/a2) - (y2/b2)] = 1, where X denotes the transverse axis and Y denotes the conjugate axis.
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Last year, Reagan grew 9/10 of an inch and her brother grew 2/5 of an inch. How much more did Reagan grow than her brother?
Reagan will grow 1/2 inch than her brother if Reagan grew 9/10 of an inch and her brother grew 2/5 of an inch by using ratio and proportion concept,
What is ratio and proportion?When b does not equal 0, an ordered pair of numbers a and b, represented as a / b, is said to be a ratio. Two ratios are set to be equal in an equation called a proportion. For instance, if there is 1 boy and 3 girls, the ratio would be written as 1: 3 (there are 3 girls for every boy), meaning that there are 1 in 4 boys and 3 in 4 girls.
A: b a/b is the ratio formula, which may be used to calculate any two values. A:b::c:da:b::c:da, on the other hand, is how the percentage formula is written.
Given,
The height grown by Reagan=9/10 inch
The height grown by Reagan brother=2/5 inch
(9/10)-(2/5)
=1/2 inch
Therefore, Reagan will grow 1/2 inch than her brother.
Therefore,
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Which can be proven? Select all that apply.
A. ∠ C E D ≅ ∠ A H J
B. A B ≅C B
C. C B ≅D B
D. ∠ D A G ≅ ∠ J C F
The Option that can be proven is:
∠ C E D ≅ ∠ A H J (Option A) The principle at play here is that ∠CDJ and ∠AJD are both right-angled triangles, (given)LInes CF and AG are congruent (given)Distance between Line AH and LInes CE are congruent (given), hence, ∠ C E D ≅ ∠ A H J.What is mathematical proof?Definitions, assertions, and methods are linked in a proper fashion to get the desired outcome in mathematical proof.
This procedure helps pupils understand the rationale behind the statement. This is also true of counterexamples and their important significance in mathematics.
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