On pet day 18 children brought a pet to school ⅔ of the pets were dogs 1\9 of the pets were cats how many dogs were there how many cats were there and how mant animals were neither dogs nor cats

Answer:

Step-by-step explanation:

Let x represent the total number of red and blue beads

Angie has some red and blue beads. 40% of her beads were red. This means that the total number of red beads is

40/100 × x = 0.4x

The number of blue beads would be

x - 0.4x = 0.6x

When she lost 50 blue beads, the number of blue beads was reduced by 1/3 its original number of beads. This means that

0.6x - 50 = 0.6x - 0.6x/3

0.6x - 50 = 0.6x - 0.2x = 0.4x

0.6x - 0.4x = 50

0.2x = 50

x = 50/0.2 = 250

The number of blue beads that Angie had initially is

0.6 × 250 = 150

The number of blue beads that Angie has left is

150 - 50 = 100 beads

The number of beads that Angie has in the end is

250 - 50 = 200 beads

Answer:

a = 50 houses

b = 45 houses

Step-by-step explanation:

Given

Number of houses called Model A = a

Number of houses called Model B = b

Total of single windows = 1800

Total of double windows = 375

then we have the system of equations

18a + 20b = 1800  (I)

3a + 5b = 375     (II)

Solving the system by whatever method we prefer, we obtain

(I)     a = (1800 - 20b)/18

then (II)

3((1800 - 20b)/18) + 5b = 375

⇒ 300 - (10/3)*b + 5b = 375

⇒ (5/3)*b  = 75

⇒ b = 45 houses

then

a = (1800 - 20*45)/18

⇒ a = 50 houses

50 model A homes and 45 model B homes will be built.

To solve the problem, let's define our variables:

We then create the following system of equations based on the given information:

1. For single windows:

18A + 20B = 1800

2. For double windows:

3A + 5B = 375

We can solve this system using the substitution or elimination method.

Step-by-Step Solution:

The solution to the system is A = 50 and B = 45, meaning that the construction company plans to build 50 model A homes and 45 model B homes.

Answer:

output = n+3.

Step-by-step explanation:

input : 1  5   6    n

output: 4 8   9     -

Here   , 4= 1+3 , 8= 5+3  ,9 = 6+3

So,

output = input +3

When input =n

Then output = n+3.

Answer:

(C )  angle E = angle X; DF = XY

Step-by-step explanation:

a triangle is said to be congruent if SAS(two sides and included angle)

SSS (all the sides are equal), ASA (two angles and an included sides).

therefore,  if angle E and angle X are equal

line DF and line XY are also equal.

Answer:

The standard deviation of given data is 12.36  

Step-by-step explanation:

We are given the following data in the question:

40, 12, 17, 25, 9, 46, 13, 22, 16,7

Formula:

[tex]\text{Standard Deviation} = \sqrt{\displaystyle\frac{\sum (x_i -\bar{x})^2}{n}}[/tex]  

where [tex]x_i[/tex] are data points, [tex]\bar{x}[/tex] is the mean and n is the number of observations.  

[tex]Mean = \displaystyle\frac{\text{Sum of all observations}}{\text{Total number of observation}}[/tex]

[tex]Mean =\displaystyle\frac{207}{10} = 20.7[/tex]

Sum of squares of differences =

372.49 + 75.69 + 13.69 + 18.49 + 136.89 + 640.09 + 59.29 + 1.69 + 22.09 + 187.69 = 1528.1

[tex]\sigma = \sqrt{\dfrac{1528.1}{10}} = 12.36[/tex]

Thus, the standard deviation of given data is 12.36

Final answer:

The standard deviation of the provided data set (number of merchandise returns on Mondays) is approximately 14.8.

Explanation:

To calculate the standard deviation of the provided data set (40, 12, 17, 25, 9, 46, 13, 22, 16, 7), we will follow these steps:

Let's apply these steps:

Therefore, the standard deviation is about 14.8.

Answer and Step-by-step explanation:

For general measurements, different people or organizations normally make slightly different measurements. Measurements are never a hundred percent accurate.

The published values are usually updated because in the modern world of discoveries, change is the only constant thing. As new discoveries roll in or not, it becomes necessary to update the current standards; no change in the updated value means the old standards hold, and any change is also updated in the published update.

For health standards/ranges, Different countries have different standards of health

And this requires regular updating because standards of health changes frim time to time.

Answer:

The second choice:

          [tex]\large\boxed{\large\boxed{1/x\cdot x=1}}[/tex]

Explanation:

The closure property on an operation means that the operation between two elements of a set produce one element of the same set.

In this case, the operation is multiplication and the set is the polynomials.

Then, the closrue property is that the multiplication of two polynomials will always produce a polynomial.

Since, [tex]1/x[/tex] is not a polynomial, the equation [tex]1/x\cdot x=1[/tex] does not support the fact that polynomials are closed under multiplication.

Answer:

80 buses will be required for 1120 soldiers.

Step-by-step explanation:

Given:

Number of seats on One side = 12 seats.

Now Given:

five seats are reserved for equipment.

So we can say that;

Number of seats used by soldiers = [tex]12-5=7\ seats[/tex]

Number of soldiers on Each seat =2

So we will now find number of soldiers on each bus.

number of soldiers on each bus is equal to Number of seats used by soldiers multiplied by Number of soldiers on Each seat.

framing in equation form we get;

number of soldiers on each bus = [tex]7\times2 = 14\ soldiers[/tex]

Now we know that;

For 14 soldiers = 1 bus

So 1120 soldiers = Number of buses required for 1120 soldiers.

By Using Unitary method we get;

Number of buses required for 1120 soldiers = [tex]\frac{1120}{14} =80[/tex]

Hence 80 buses will be required for 1120 soldiers.

80 students gained more than 48 marks.

In the given box plot, we have the following information:

- The lowest test score is **10**, and the highest is **100**.

- The 25th percentile (Q1) is **30**, the median (Q2) is **48**, and the 75th percentile (Q3) is **70**.

- No student scored exactly **30**, **48**, or **70** marks.

- **120 students** scored less than **70** marks.

Let's analyze this:

1. The interquartile range (IQR) contains the middle **50%** of the data. Since the median (Q2) is **48**, we know that **50%** of the students scored more than **48** marks.

2. We are interested in how many students scored above **48**. Since **50%** scored more than **48**, the remaining **50%** scored less than or equal to **48**.

3. Given that **120 students** scored less than **70**, we can infer that **75%** of the students scored below **70** (since each region contains **25%** of the data).

4. Therefore, **25%** of the students scored between **48** and **70** (the region between Q2 and Q3).

5. To find out how many students scored more than **48**, we look at the region above Q2. Since there are **2 regions** above Q2, each containing **40 students** (since 120 students = 75%), the total number of students who scored more than **48** is:

[tex]\(2 \times 40 = 80\)[/tex]

Therefore, 80 students gained more than 48 marks.

The number of students who gained more than 48 marks is : 60

Using the information in the boxplot ;

The median represents 50% of the data .

The number above the median value can be calculated thus ;

Now we have :

50% × 120 = 60

Hence, the number of students who gained more than 48 marks is : 60

Slope of this line is -2.5

Step-by-step explanation:

Here, from the graph, x1 = -2, x2 = -4, y1 = 3, y2 = 8

⇒ m  = (8 - 3)/(-4 - -2) = 5/-2 = -2.5

Final answer:

The depth of the gutter that will maximize its cross-sectional area is 4 inches, and the maximum cross-sectional area that allows the greatest amount of water to flow is 32 square inches.

Explanation:

To determine the depth of the gutter that will maximize its cross-sectional area, we first need to assume that turning up the edges of the aluminum sheet at right angles will form a rectangular cross-section. If the width of the aluminum is 16 inches and 'x' represents the depth of the gutter (the height of the sides when bent), the width of the base of the gutter will be 16 - 2x (since both sides are turned up).

This means the cross-sectional area 'A' in square inches will be A = x(16 - 2x). This is a quadratic equation and can be expanded as A = -2x^2 + 16x. To find the maximum area, we need to find the vertex of this parabola, which occurs at x = -b/(2a), where 'a' is the coefficient of x^2 and 'b' is the coefficient of 'x'.

In our case, a = -2 and b = 16, so the depth that maximizes the area is x = -16/(2*(-2)) = 4 inches. Therefore, the maximum cross-sectional area is A = 4(16 - 2*4) = 4(8) = 32 square inches.

The depth of the gutter that will maximize its cross-sectional area is 16 inches, and the maximum cross-sectional area is[tex]\( 768 \)[/tex] square inches.

To solve this problem, we will use calculus to find the depth of the gutter that maximizes its cross-sectional area. We will start by defining the dimensions of the gutter and then use the derivative of the area function to find the critical points. Finally, we will determine which of these critical points gives the maximum area.

Let's denote the depth of the gutter as [tex]\( x \)[/tex]inches. Since the width of the aluminum sheets is 16 inches, the base of the gutter will also be 16 inches. When the edges are turned up to form right angles, the gutter will have a rectangular base and two rectangular sides.

 The area of the base of the gutter is [tex]\( 16x \)[/tex]. The area of each side is [tex]\( x^2 \),[/tex] and there are two sides, so the total area of the sides is[tex]\( 2x^2 \).[/tex] Therefore, the total cross-sectional area [tex]\( A \)[/tex]of the gutter is the sum of the area of the base and the areas of the two sides:

[tex]\[ A(x) = 16x + 2x^2 \][/tex]

To find the depth that maximizes the area, we need to take the derivative of [tex]\( A(x) \)[/tex] with respect to[tex]\( x \)[/tex]and set it equal to zero:

[tex]\[ A'(x) = \frac{d}{dx}(16x + 2x^2) = 16 + 4x \][/tex]

 Setting [tex]\( A'(x) \)[/tex] equal to zero gives us the critical points:

[tex]\[ 16 + 4x = 0 \][/tex]

[tex]\[ 4x = -16 \][/tex]

[tex]\[ x = -4 \][/tex]

Since the depth of the gutter cannot be negative, we discard[tex]\( x = -4 \)[/tex]and realize that we need to consider the physical constraints of the problem. The actual critical point occurs at the endpoint of the domain of [tex]\( x \),[/tex]which is[tex]\( x = 0 \)[/tex](no gutter) or[tex]\( x = 16 \)[/tex] (the gutter's width). Since[tex]\( x = 0 \)[/tex]gives a minimum area (no gutter at all), the maximum area must occur at [tex]( x = 16 \).[/tex]

 Now, we calculate the cross-sectional area at [tex]\( x = 16 \)[/tex]

[tex]\[ A(16) = 16(16) + 2(16)^2 \][/tex]

[tex]\[ A(16) = 256 + 2(256) \][/tex]

[tex]\[ A(16) = 256 + 512 \][/tex]

[tex]\[ A(16) = 768 \][/tex]

 Therefore, the maximum cross-sectional area of the gutter is[tex]\( 768 \)[/tex]square inches when the depth is equal to the width, which is 16 inches.

Answer:

Part A) The amplitude = 24

Part B) The period = 24

Part C) Vertical shift = 36

Step-by-step explanation:

The general equation of the sine function:

y = A sin (Bx) + C

Where A is the amplitude and B = 360°/Period  and C is the vertical shift

See the attached figure which represents the graph of m and f(m)

So,

Part A:

The function has minimum at 12 and maximum at 60

The difference is = 60 - 12 = 48

So, The amplitude = 48/2 = 24

Part B:

Period: The period of a periodic function is the interval on which the cycle of the graph that's repeated in both directions lies.

We can deduce that the function completes one cycle within 24 months

So, the period = 24

Part C:

Vertical shift is obtained at m = 0

So, f(m) = 36

36 = A sin (0) + C

C = 36 ⇒ Vertical shift

So, The amplitude = 24

The period = 24

Vertical shift = 36

Answer:

Answer:

Part A) The amplitude = 24

Part B) The period = 24

Part C) Vertical shift = 36

Step-by-step explanation:

The general equation of the sine function:

y = A sin (Bx) + C

Where A is the amplitude and B = 360°/Period  and C is the vertical shift

See the attached figure which represents the graph of m and f(m)

So,

Part A:

The function has minimum at 12 and maximum at 60

The difference is = 60 - 12 = 48

So, The amplitude = 48/2 = 24

Part B:

Period: The period of a periodic function is the interval on which the cycle of the graph that's repeated in both directions lies.

We can deduce that the function completes one cycle within 24 months

So, the period = 24

Part C:

Vertical shift is obtained at m = 0

So, f(m) = 36

36 = A sin (0) + C

C = 36 ⇒ Vertical shift

So, The amplitude = 24

The period = 24

Vertical shift = 36

Step-by-step explanation:

Answer:

  52.2 ft

Step-by-step explanation:

Triangle JSV is similar to triangle HTV so you have the proportion ...

  JS/SV = HT/TV

  JS/(36 ft) = (5.8 ft)/(4 ft) . . . . . . . fill in the given values

  JS = (36 ft)(5.8/4) = 52.2 ft . . . . multiply by 36 ft

The height of the wall is 52.2 ft.

We know that m<HVT = m<JVS because the mirror projects equal angles. We can claim this about the angle theta.

tan(θ) = 5.8/4

θ = [tex]tan^{-1}(5.8/4)=55.4[/tex] degrees approx.

So, we want sin theta in the other triangle. Luckily, we also know that...

cos(55.4°) x hypotenuse = 36

hypotenuse = 63.4 ft approx.

So we can find the height by evaluating...

sin(55.4°) x 63.4 = 52.2 ft

answer: 52.2 ft

Answer:

[tex] a_{6} = - 11[/tex]

Final answer:

The 6th term in the recursive sequence with the first term 9 and each subsequent term decreasing by 4 from the previous term is -11.

Explanation:

To find the 6th term in the sequence described by the recursive formula given:

a1=9

an=an-1−4

We will apply the formula recursively to determine each term up to the 6th term.

First term (a1): 9

Second term (a2): a1 − 4 = 9 − 4 = 5

Third term (a3): a2 − 4 = 5 − 4 = 1

Fourth term (a4): a3 − 4 = 1 − 4 = -3

Fifth term (a5): a4 − 4 = -3 − 4 = -7

Sixth term (a6): a5 − 4 = -7 − 4 = -11

Therefore, the 6th term in the sequence is -11.

Answer:

Blue M&M's

Step-by-step explanation:

Let the total M&M's bought be 100%.

If 1/3 of the M&M's is blue this means we have 1/3 of 100 of blue M&M's i.e 33.3% are blue.

If 1/4 were red, then we have 1/4 of 100 of red M&M's i.e 25% of reds

If 1/8 of the M&M's were yellow, this means there are 1/8 of 100% M&M's i.e 12.5% of yellow.

According to the results, Jordy have more of blue M&M's

Answer:

Step-by-step explanation:

surface area=6*2.2²=6×4.84=29.04 m²

Answer:

29.04

Step-by-step explanation:

just took the test

Subtract b from both sides: ax=c-b

Divide both sides by a: x=(c-b)/a

Hope this helped

Explanation:

To solve any equation that is linear in the variable of interest, you first look at what is done to the variable, using the Order of Operations as your guide.

Here, the variable is ...

You find the value of the variable by reversing these steps, starting from the last one on the list and working up the list.

To "undo" the addition of "b", we add its opposite (-b) to the equation. The rules of equality tell you that anything you do to one side of the equation must also be done to the other side, so we add -b to both sides:

  ax +b -b = c -b

  ax = c -b . . . . . . . . simplify

To "undo" the multiplication by "a", we multiply by its reciprocal. That is, we multiply by 1/a, or, equivalently, divide by "a". Again, we must do this to both sides of the equation:

  (1/a)ax = (1/a)(c -b)

  x = (c -b)/a . . . . . . . simplify

__

In short, we subtract the added constant (b) and divide by the multiplier (a).

Final answer:

Tesha's savings account balance was decreased by $91.00 after withdrawing $22.75 each week for four weeks to pay for piano lessons.

Explanation:

To calculate the change in Tesha's savings account balance due to payment for her piano lessons, we need to multiply the weekly withdrawal amount by the number of weeks. Tesha withdrew $22.75 each week for four weeks.

Answer: Anna eats 1/8 of pie

Step-by-step explanation: Let total pie =1

Mollie eats 1/4 of pie

Brandon eats half the amount of pie that Mollie eats i.e 1/2 of (1/4)

⇒1/8

Yuki eats 4 times as much as pie as Brandon i.e 4*(1/8)

⇒1/2

Total pie eaten by Mollie +Brandon+Yuki = 1/4+1/8+1/2

⇒7/8

Therefore Anna eats (1-7/8)

⇒ 1/8 of pie

Answer:

Therefore the value of x are = -1 and -5 for which f(x)=g(x).

Step-by-step explanation:

Given,

f(x)=x²+6x+6    and    g(x)=1

f(x)=g(x)

⇒x²+6x+6=1

⇒x²+6x+6-1=0

⇒x²+6x+5=0

⇒x²+5x+x+5=0

⇒x(x+5)+1(x+5)=0

⇒(x+5)(x+1)=0

⇒x+5=0     or       x+1=0

⇒x=-5                     x=-1

Therefore the value of x are = -1 and -5.

To find the values of x for which f(x) equals g(x), we set the equations x² + 6x + 6 equal to 1 and solve. The quadratic equation factors to yield x = -1 and x = -5 as solutions.

To find the values of x for which f(x) equals g(x), we need to set the functions equal to each other and solve for x:

Given:
f(x) = x² + 6x + 6
g(x) = 1

We need to solve for x when:

x² + 6x + 6 = 1

Subtract 1 from both sides to set the equation to zero:

x² + 6x + 5 = 0

Next, we factor the quadratic equation:

(x + 1)(x + 5) = 0

Set each factor to zero and solve for x:

x + 1 = 0
x = -1

x + 5 = 0
x = -5

Thus, the values of x for which f(x) = g(x) are x = -1 and x = -5.

Answer:

11x5=55

Step-by-step explanation:

The equation that best shows that 55 is a multiple of 11 is 11 × 5 = 55.

The equation that best shows that 55 is a multiple of 11 is Choice B: 11 × 5 = 55.

To determine if a number is a multiple of another number, we need to check if the first number can be divided evenly by the second number without any remainder. In this case, 55 can be divided evenly by 11 because 11 × 5 equals 55.

The other choices, A, C, and D, do not represent the relationship of 55 being a multiple of 11.

Stratified sampling method - where population were divided into different groups called strata and the sample is then drawn from EACH group.

Systematic sampling method - sampling method using probability where elements are chosen from a target population.

Random sampling - each sample chosen has equal probability to be chosen.

Convenience sampling method - Not a random sampling method where the sample is being chosen as per ease of access.

Cluster sampling method - population were divided into different groups as known as clusters. The clusters were then chosen randomly as the samples. Each individuals in the clusters chosen are used as the samples.

Answer: E. Cluster sampling where selected towns refer to the clusters and were chosen randomly.

The type of sampling used by the researcher is cluster sampling.

Cluster sampling is a technique used when it becomes difficult to study the target population spread across a wide area and simple random sampling cannot be applied.

The researcher divides the entire population into sections or clusters. Then the researcher randomly selects a few clusters from the total clusters for the research.

In this case, the clusters are the five randomly selected towns. Everyone in these selected towns is included in the sample. This makes it a cluster sample.

Find more exercises on cluster sampling;

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Answer:

24.66

Step-by-step explanation:

You use the Pythagorean Theorem and do 27^2 -11^2= x            

Answer:

The total amount of commission on these vehicles is 1686.

Step-by-step explanation:

Given:

Miles earns a 6% commission on each vehicles he sells.

Today he sold a truck for 18500 and a car for 9600.

Now, to get the total amount of his commision on these vehicles.

Percent of commission Miles earns on each vehicles he sells = 6%.

He sold a truck for = 18500.

He sold a car for = 9600.

So, the total amount of vehicles:

[tex]18500+9600=28100.[/tex]

Now, to get the total amount of commission on vehicles:

[tex]6\%\ of\ 28100[/tex]

[tex]=\frac{6}{100} \times 28100[/tex]

[tex]=0.06\times 28100[/tex]

[tex]=1686.[/tex]

Therefore, the total amount of commission on these vehicles is 1686.

Answer:

[tex]6x+7y+3=0[/tex]

Step-by-step explanation:

We are asked to find the equation of the line in general form, which has a of -6/7 and containing the point (10,-9).

We know that genera equation of a line is in form [tex]Ax+By+C=0[/tex], where, A, B and C are real numbers.

First of all, we will write our equation in point-slope form as:

[tex]y-y_1=m(x-x_1)[/tex], where,

m = Slope of line,

[tex](x_1,y_1)[/tex] = Given point on line.

[tex]y-(-9)=-\frac{6}{7}(x-10)[/tex]

[tex]y+9=-\frac{6}{7}x+\frac{60}{7}[/tex]

[tex]y*7+9*7=-\frac{6}{7}x*7+\frac{60}{7}*7[/tex]

[tex]7y+63=-6x+60[/tex]

[tex]6x+7y+63-60=-6x+6x+60-60[/tex]

[tex]6x+7y+3=0[/tex]

Therefore, our required equation would be [tex]6x+7y+3=0[/tex].

Answer: her regular hourly rate is $12.5

Step-by-step explanation:

Let x represent the regular payment that Sonya receives per hour for the first 40 hours of work. This means that her total pay for the first 40 hours of work is

40 × x = $40x

Sonya is paid time and a half for hours worked in excess of 40 hours. This means that her hourly pay for working more than 40 hours would be x + 0.5x = $1.5x

She worked for 52 hours. This means that the extra hours that she worked is

52 - 40 = 12 hours

Total pay for 12 extra hours is

12 × 1.5x = 18x

If she had gross weekly wages of $725, then

40x + 18x = 725

58x = 725

x = 725/58 = $12.5

To find Sonya's regular hourly rate, we need to calculate her overtime pay and then subtract it from her total weekly wages.

To find Sonya's regular hourly rate, we need to calculate her overtime pay and then subtract it from her total weekly wages. Sonya earned $725 for 52 hours worked. Since she is paid time and a half for hours worked more than 40, she worked 12 overtime hours (52 - 40 = 12).

Her overtime pay is calculated by multiplying her regular hourly rate by 1.5, so her overtime pay is 1.5 * regular hourly rate * 12 hours. Subtracting her overtime pay from her total wages gives us her regular wages, so we have the equation: $725 - (1.5 * regular hourly rate * 12) = regular wages. Solving for  regular hourly rate will give us the answer.

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Answer:

87.5%

Step-by-step explanation:

64:64-8

64:56

56/64 *100 = 87.5%

Answer:

Step-by-step explanation:

Let us record the station number 1, 2 or 3 for each family member A, B or C.

I am attaching a table containing total outcomes. Outcomes are presented along rows while the assigned station to each member is written along columns. For ease of understanding, 1 3 2 in the table should be interpreted as family member A being assigned to station 1, member B to station 3 and member C to station number 2, respectively.

From table it is clear that the total outcomes possible are 27.

We know that, probability can be defined as,

[tex]PROBABITILY = \frac{NUMBER\;OF\;DESIRED\;OUTCOMES}{TOTAL\;NUMBER\;OF\;OUTCOMES}[/tex]

a) All Members Assigned to the Same Station.

Cases for all members being assigned to same station are as follows:

[1 1 1], [2 2 2], [3 3 3] (outcome number 1, 14 and 27 in the table).

Therefore,

[tex]PROBABILITY\;(Case\;a) = \frac{3}{27}\\\\PROBABILITY\;(Case\;a) = 0.111[/tex]

b) At Most Two Members Assigned to the Same Station.

It means that maximum of 2 members can have the same station. Cases for this situation are as follows:

[1 1 2], [1 1 3], [1 2 1], [1 2 2], [1 3 1], [1 3 3], [2 1 1], [2 1 2], [2 2 1], [2 2 3], [2 3 2],

[2 3 3], [3 1 1], [3 1 3], [3 2 2], [3 2 3], [3 3 1], [3 3 2]

(outcome number 2, 3, 4, 5, 7, 9, 10, 11, 13, 15, 17, 18, 19, 21, 23, 24, 25 and 26 in the table).

Therefore,

[tex]PROBABILITY\;(Case\;b) = \frac{18}{27}\\\\PROBABILITY\;(Case\;b) = 0.666[/tex]

c) All Members Assigned to a Different Station.

For this scenario, we have the following results:

[1 2 3], [1 3 2], [2 1 3], [2 3 1], [3 1 2], [3 2 1] (outcome number 6, 8, 12, 16, 20 and 22 in the table).

Therefore,

[tex]PROBABILITY\;(Case\;c) = \frac{6}{27}\\\\PROBABILITY\;(Case\;c) = 0.222[/tex]

Answer:

The quadratic mean of 2 real positive numbers is greater than or equal to the arithmetic mean.

Step-by-step explanation:

x and y      Quadratic Mean       Arithmetic mean

3 and 3                    3                                    3

2 and 3                   2.55                               2.5

3 and  6                  4.74                                4.5

2 and 5                   3.8                                  3.5

2 and 17                 12.1                                   9.5

18 and 28              23.5                                  23

10 and  48             34.7                                  29

The quadratic mean is always greater than the arithmetic mean except when  x and y are the same.

When the difference between the pairs is  small the difference in the means is also small. As that difference increases the difference in the means also increases.

So we conjecture that the quadratic mean is always greater than or equal to the arithmetic mean.

Proof.

Suppose it is true then:

√(x^2 + y^2) / 2) ≥ (x + y)/2       Squaring  both sides:

(x ^2 + y^2) / 2 ≥ (x + y)^2 / 4    Multiply through by 4:

2x^2 +2y^2  ≥  (x + y)^2

2x^2 +2y^2 >=  x^2 + 2xy + y^2

x^2 + y^2 >= 2xy.

x^2 - 2xy + y^2 ≥ 0

(x - y)^2 ≥ 0

This is true  because the square of any real number is positive so the original inequality must also be true.

The quadratic mean of two real numbers, x and y, is given by the formula sqrt((x^2 + y^2)/2). A conjecture can be made that the quadratic mean is greater than or equal to the arithmetic mean for positive real numbers. This conjecture can be proved using the AM-QM inequality and algebraic manipulations.

The quadratic mean of two real numbers, x and y, is given by the formula:

Q(x, y) = sqrt((x^2 + y^2)/2)

To formulate a conjecture about the relative sizes of the arithmetic and quadratic means of different pairs of positive real numbers, we can compare the two means for various pairs of numbers. Based on observations, it can be conjectured that the quadratic mean is always greater than or equal to the arithmetic mean for positive real numbers.

To prove the conjecture, we can use the AM-QM inequality, which states that the quadratic mean is greater than or equal to the arithmetic mean:

Q(x, y) >= A(x, y)

Where Q(x, y) is the quadratic mean and A(x, y) is the arithmetic mean.

Let's consider two positive real numbers, a and b:

Q(a, b) = sqrt((a^2 + b^2)/2)

A(a, b) = (a + b)/2

Now, we need to prove that Q(a, b) >= A(a, b):

(a^2 + b^2)/2 >= (a + b)/2

a^2 - a + b^2 - b >= 0

(a^2 - a) + (b^2 - b) >= 0

a(a - 1) + b(b - 1) >= 0

a(a - 1) >= 0

Therefore, Q(a, b) >= A(a, b), which confirms the conjecture that the quadratic mean is always greater than or equal to the arithmetic mean for positive real numbers.

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