A pizza chain was willing to pay $1 to each person interviewed about his or her likes and dis- likes of types of pizza crust. Of the people interviewed, 200 liked thin crust, 270 liked thick crust, 70 liked both, and 50 did not like either type of crust. What was the total cost of the survey?

To maximize the area of the Norman window, solve for the dimensions of the rectangle. Substitute the expression for 'h' in terms of 'w' into the area formula. Take the derivative of A with respect to 'w', set it equal to zero, and solve for 'w'.

To maximize the area of the Norman window, we need to find the dimensions of the rectangle. Let's denote the width of the rectangle as 'w' and the height as 'h'. The perimeter of the rectangle can be expressed as 2w + h + πh = 24 feet. Rearranging the equation, we have (2 + π)h + 2w = 24. Since we want to maximize the area, we can solve for 'h' in terms of 'w' using this equation.

Next, we can substitute the expression for 'h' in terms of 'w' into the area formula for the window, which is A = wh + (π/4)w^2. Simplifying this expression, we get A = (w(2 + πw))/4. To find the dimensions that maximize the area, we can take the derivative of A with respect to 'w', set it equal to zero, and solve for 'w'. This will give us the width of the rectangle. Once we have the width, we can substitute it back into the equation for 'h' to find the height.

By solving these equations, we can find the dimensions of the rectangle that will maximize the area of the Norman window, allowing in as much light as possible.

Answer:

[tex]\frac{16}{21}=\frac{8}{10.5}[/tex]

or

[tex]\frac{16}{8}=\frac{21}{10.5}[/tex]

Step-by-step explanation:

we know that

If two figures are similar then the ratio of its corresponding sides is proportional

In this problem

triangle YPQ and triangle YXZ are similar by AA Similarity Theorem

so

[tex]\frac{YP}{YX}=\frac{YQ}{YZ}[/tex]

substitute the given values

[tex]\frac{16}{21}=\frac{8}{10.5}[/tex]

Rewrite

[tex]\frac{16}{8}=\frac{21}{10.5}[/tex]

The boxes should be filled with [tex]\frac{16}{21} = \frac{8}{10.5}\\\\\frac{16}{8} = \frac{21}{10.5}[/tex]

As we know that

In the case when two figures are similar so the ratio of its corresponding sides is proportional

In this given situation  

triangle YPQ and triangle YXZ are similar by AA Similarity Theorem

So,

[tex]\frac{YP}{YX} = \frac{YQ}{YZ}[/tex]

[tex]\frac{16}{21} = \frac{8}{10.5}\\\\\frac{16}{8} = \frac{21}{10.5}[/tex]

It can be any of the both.

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Answer:

x = 0 , y = 0 , z = -3

Step-by-step explanation:

Solve the following system:

{x - 6 y + 4 z = -12 | (equation 1)

x + y - 4 z = 12 | (equation 2)

2 x + 2 y + 5 z = -15 | (equation 3)

Swap equation 1 with equation 3:

{2 x + 2 y + 5 z = -15 | (equation 1)

x + y - 4 z = 12 | (equation 2)

x - 6 y + 4 z = -12 | (equation 3)

Subtract 1/2 × (equation 1) from equation 2:

{2 x + 2 y + 5 z = -15 | (equation 1)

0 x+0 y - (13 z)/2 = 39/2 | (equation 2)

x - 6 y + 4 z = -12 | (equation 3)

Multiply equation 2 by 2/13:

{2 x + 2 y + 5 z = -15 | (equation 1)

0 x+0 y - z = 3 | (equation 2)

x - 6 y + 4 z = -12 | (equation 3)

Subtract 1/2 × (equation 1) from equation 3:

{2 x + 2 y + 5 z = -15 | (equation 1)

0 x+0 y - z = 3 | (equation 2)

0 x - 7 y + (3 z)/2 = -9/2 | (equation 3)

Multiply equation 3 by 2:

{2 x + 2 y + 5 z = -15 | (equation 1)

0 x+0 y - z = 3 | (equation 2)

0 x - 14 y + 3 z = -9 | (equation 3)

Swap equation 2 with equation 3:

{2 x + 2 y + 5 z = -15 | (equation 1)

0 x - 14 y + 3 z = -9 | (equation 2)

0 x+0 y - z = 3 | (equation 3)

Multiply equation 3 by -1:

{2 x + 2 y + 5 z = -15 | (equation 1)

0 x - 14 y + 3 z = -9 | (equation 2)

0 x+0 y+z = -3 | (equation 3)

Subtract 3 × (equation 3) from equation 2:

{2 x + 2 y + 5 z = -15 | (equation 1)

0 x - 14 y+0 z = 0 | (equation 2)

0 x+0 y+z = -3 | (equation 3)

Divide equation 2 by -14:

{2 x + 2 y + 5 z = -15 | (equation 1)

0 x+y+0 z = 0 | (equation 2)

0 x+0 y+z = -3 | (equation 3)

Subtract 2 × (equation 2) from equation 1:

{2 x + 0 y+5 z = -15 | (equation 1)

0 x+y+0 z = 0 | (equation 2)

0 x+0 y+z = -3 | (equation 3)

Subtract 5 × (equation 3) from equation 1:

{2 x+0 y+0 z = 0 | (equation 1)

0 x+y+0 z = 0 | (equation 2)

0 x+0 y+z = -3 | (equation 3)

Divide equation 1 by 2:

{x+0 y+0 z = 0 | (equation 1)

0 x+y+0 z = 0 | (equation 2)

0 x+0 y+z = -3 | (equation 3)

Collect results:

Answer: {x = 0 , y = 0 , z = -3

To solve the given system of equations, use the method of elimination to eliminate one variable at a time and solve for the remaining variables.

To solve the system of equations:

x - 6y + 4z = -12

x + y - 4z = 12

2x + 2y + 5z = -15

We can use the method of substitution or elimination. Let's use the method of elimination:

Therefore, the solution is x = -5, y = 4, and z = 1.

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Answer:

C

Step-by-step explanation:

The recommended daily intake of a nutrients supplement for a certain age group is 800 mg per day.

Actully supplements needs vary from person to person  plus or minus 50 mg.

This means the minimum value is 800 - 50 = 750 mg per day and the maximum value is 800 + 50 = 850 mg per day.

Let x mg be a possible daily amount of nutrients supplement. Then

[tex]750\le x\le 850[/tex]

Subtract 800:

[tex]750-800\le x-800\le 850-800\\ \\-50\le x-800\le 50[/tex]

This inequality can be rewritten using absolute value notation as

[tex]|x-800|\le 50[/tex]

See picture for solution and answer.

9 bottles of soda and 10 juice bottles are purchased

Solution:

Given that,

Sugar contained by each soda bottle = 45 grams

Sugar contained by each juice bottle = 35 grams

Let,

x be the number of soda bottles

y be the number of juice bottles

Given that,

Ethan purchased a total of 19 bottles of juice

Therefore, we get,

x + y = 19 ---------- eqn 1

Ethan purchased a total of 19 bottles of juice and soda which collectively contain 755 grams of sugar

Therefore, we frame a equation as:

[tex]45 \times x + 35 \times y = 755[/tex]

45x + 35y = 755 --------- eqn 2

Let us solve eqn 1 and eqn 2

From eqn 1,

x = 19 - y --------- eqn 3

Substitute eqn 3 in eqn 2

45(19 - y) + 35y = 755

855 - 45y + 35y = 755

10y = 100

y = 10

Substitute y = 10 in eqn 3

x = 19 - 10

x = 9

Thus 9 bottles of soda and 10 juice bottles are purchased

Answer:

The distance from P to origin is approximately 40.82 units.                                  

Step-by-step explanation:

We are given the following in the data:

The point P(21,35)

We have to find the distance of point P from the origin.

Coordinates of origin: (0,0)

Distance formula:

[tex](x_1,y_1),(x_2,y_2)\\\\d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}[/tex]

Putting the values, we get,

[tex](21,35), (0,0)\\\\d = \sqrt{(0-21)^2 + (0-35)^2} = \sqrt{1666} = 7\sqrt{34} \approx 40.82\text{ units}[/tex]

The distance from P to origin is approximately 40.82 units.

Answer:

C. The new tomatoes are more consistent in their weights than the old tomatoes.

Context:

Agricultural scientists are working on developing an improved variety of Roma tomatoes. Marketing research indicates that customers are likely to bypass Romas that weigh less than 70 grams. The current variety of Roma plants produces fruit that averages 74 grams, but 11% of the tomatoes are too small. It is reasonable to assume that a Normal model applies.

The question is asking to compare the consistency of weights in two types of tomatoes based on their standard deviation. Select the option that best fits the standard deviation calculated for each group.

In statistics, standard deviation is a measure of the amount of variation or dispersion in a set of values. A lower standard deviation means that the values tend to be close to the mean (average) value of the set, while a higher standard deviation implies that the values are spread out over a wider range. The question asks you to compare the consistency in weight between two types of tomatoes, determined by their standard deviations.

If the standard deviation of the old tomatoes' weight is lower than that of the new ones', option B ('The old tomatoes are more consistent in their weights than the new tomatoes') would be accurate. On the other hand, if the opposite is true, option C ('The new tomatoes are more consistent in their weights than the old tomatoes') would be correct. Option A would be right if the standard deviations are similar, indicating similar consistency, and option D would apply if along with the new tomato weights being more consistent, their weight distribution is left skewed.

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Answer:

How Pythagoras theorem relates to Trigonometry ratios

For a right angled triangle ∆ABC,

Let a = hypothenus

b = opposite

c = adjacent

Sin²θ + Cos²θ = 1

Provided that a² + b² = c²

To prove this divide through by c²

Tongive

a²/c² + b²/c² = c²/c²

Sinθ = a/c , Cosθ = b/c

So, the above equation becomes

(Sinθ)² + (Cosθ)² = 1

Sin²θ + Cos²θ = 1

Coterminal Angles

Coterminal Angles are angles who share the same initial side and terminal sides.

Finding coterminal angles is as simple as adding or subtracting 360° or 2π to each angle, depending on whether the given angle is in degrees or radians.

Final answer:

The Pythagorean Theorem and trigonometric ratios are related through the relationships between the sides of a right triangle. The theorem helps to calculate the sides involved in defining the sine, cosine, and tangent functions. Coterminal angles share the same trigonometric values despite having different measurements.

Explanation:

The Pythagorean Theorem is intimately related to trigonometric ratios, as it provides the fundamental relationship between the sides of a right-angled triangle. In trigonometry, the ratios of these sides define sine, cosine, and tangent functions. The Pythagorean theorem states that a² + b² = c², where 'a' and 'b' are the lengths of the legs and 'c' is the length of the hypotenuse of a right triangle. For example, given an angle θ within the right triangle, the cosine of θ is the adjacent side over the hypotenuse (cos(θ) = a/c), which implicitly uses the Pythagorean relationship when solving for the hypotenuse 'c'.

It's also important to remember coterminal angles in trigonometry, which are angles that share the same terminal side. Coterminal angles can be found by adding or subtracting whole multiples of 360° (or 2π radians). Although coterminal angles are different in measure, they have the same sine, cosine, and tangent values because these trigonometric functions are based on the position of the terminal side, not the actual angle measurement. Hence, coterminal angles have identical trigonometric function values.

Answer:

A: -4,3 The x coordinate of the vertex is given by x = -b/2a and the y coordinate of the vertex is given by y= f( -b/2a), we need to transform the equation to the form ax^2 + bx + c We have: 5(x+4)^2+3 = 5x^2+ 20x + 23 Then we substitute x= -20/2*5 y= f(-20/2*5) = -4 = f(-4) =3 So V(-4,3)

Step-by-step explanation:

Answer:

I still dont get what the answer is

Step-by-step explanation:

Answer:

  D. The second route, in 1.18 hours

Step-by-step explanation:

The appropriate relation is ...

  time = distance/speed

The time required on the first route is ...

  time1 = (40 mi)/(25 mi/h) = 40/25 h = 1.6 h

The time required on the second route is ...

  time2 = (65 mi)/(55 mi/h) = 65/55 h = 1 2/11 h ≈ 1.18 h

__

The second route requires a shorter time, so will get you there faster. The second route will get you there in 1.18 hours.

Answer: 6x^13-1.5w+156x+13 is the answer to the first equation and is that another equation?

The expression 13 - 0.5w + 6x evaluates to 11 when substituting w=10 and x=1/2.

The problem is to evaluate the expression 13 - 0.5w + 6x given the values w=10 and x=1/2. Following the order of operations, we first substitute the given values into the expression.

13 - 0.5(10) + 6(1/2) = 13 - 5 + 3 = 11.

The result of the evaluated expression is 11.

Continuous compound is e^rate x time

The formula would be D. 1200e^0.04t

Answer:  OPTION D A = 1200.e(0.04)(t)

Step-by-step explanation:

If the interest is compounded continuously for t years at a rate of r per year, then the compounded amount is given by:

A = P. e rt

P =$1200, r = 4% , t = t years

A = 1200.e(0.04)(t)

Answer:

Step-by-step explanation:

Given

Plane travels 240 miles [tex]10^{\circ}[/tex] East of North

Position vector [tex]\vec{r_1}=240(\cos(10)\hat{j}+\sin (10)\hat{i})[/tex]

Then the plane travels 180 miles [tex]67^{\circ}[/tex] East of North

[tex]\vec{r_{21}}=180(\cos(67)\hat{j}+\sin (67)\hat{i})[/tex]

[tex]\vec{r_2}=\vec{r_{21}}+\vec{r_1}[/tex]

[tex]\vec{r_2}=\left ( 240\sin (10)+180\sin (67)\right )\hat{i}+\left ( 240\cos (10)+180\cos (67)\right )\hat{j}[/tex]

[tex]\vec{r_2}=\left ( 207.36\right )\hat{i}+\left ( 306.68\right )\hat{j}[/tex]

Total distance traveled in North direction is given by coefficient of \hat{j}

i.e. North[tex]=306.68\ miles\approx 307\ miles[/tex]

Total distance traveled in East direction is given by coefficient of \hat{i}

East [tex]=207.36\ miles\approx 207\ miles[/tex]

Answer:

  -2

Step-by-step explanation:

We assume you want to find the number. Let it be represented by x.

  (6+2x)·5 = -3x +4 . . . . . the meaning of the problem statement

  30 +10x = -3x +4 . . . . . eliminate parentheses

  26 +13x = 0 . . . . . . . . . add 3x-4 to both sides

  2 + x = 0 . . . . . . . . . . . . divide by 13

  x = -2 . . . . . . . . . . . . . . subtract 2

The number is -2.

Answer:

a.) 23

b.) y=14

c.) 23

d.) -23

e.) T=8

f.) f=1/8

Step-by-step explanation:

a.) general equation is Asin((2π/T))

A is the amplitude. It's A value is 23

b.)  Midline = vertical_shift = 14

c.) max = positive amplitude value = 23

d.) min = negative amplitude = -23

e.) Factor out 2π from your angular frequency to get the period.

ω = π/4 = (2π)/8 = (2π)/T

Period = 8

f.) Frequency is just the inverse of the period.

f = 1/T = 1/8

Answer:

x=42

Step-by-step explanation:

x-12=30

x=30+12

x=42

To solve for x in this equation, we want to get x by itself on the left side of the equation. Since 12 is being subtracted from x, to get x by itself, we need to add 12 to the left side of the equation. If we add 12 to the left side, we must also add 12 to the right side.

On the left side, -12 and +12 cancel each other out so we are simply left with x. On the right side, 30 + 12 is 42 so we have x = 42.

It's important to understand that we can check our answer by substituting 42 back into the original equation.

So we have (42) - 12 = 30.

42 - 12 is 30 so we have 30 = 30 which is a true statement so our answer, x = 42, is correct.

Answer:

The number of small mowers are 19 and the large mowers are 11.

Step-by-step explanation:

Given:

A garden supply store sells two types of lawn mowers. Total sales of mowers for the year were $8379.70. The total number of mowers sold the 30. The small mower cost $249.99 and the large mower costs $329.99.

Now, to find the number of each type of mower sold.

Let the number of small mower be [tex]x.[/tex]

And the number of large mower be  [tex]y.[/tex]

So, total number of mowers are:

[tex]x+y=30[/tex]

[tex]x=30-y\ \ \ ....(1)[/tex]

Now, the total sales of mowers are:

[tex]249.99(x)+329.99(y)=8379.70[/tex]

Substituting the value of [tex]x[/tex] from equation (1) we get:

[tex]249.99(30-y)+329.99y=8379.70[/tex]

[tex]7499.7-249.99y+329.99y=8379.70[/tex]

[tex]7499.7+80y=8379.70[/tex]

Subtracting both sides by 7499.7 we get:

[tex]80y=880[/tex]

Dividing both sides by 80 we get:

[tex]y=11.[/tex]

The number of large mower = 11.

Now, to get the number of small mowers we substitute the value of [tex]y[/tex]  in equation ( 1 ):

[tex]x=30-y\\x=30-11\\x=19.[/tex]

The number of small mower = 19.

Therefore, the number of small mowers are 19 and the large mowers are 11.

Answer:

Direct variation is constantly proportional (constant multiple)

Answer:

a) y = - 1/2 x

Step-by-step explanation:

The equation that represents a direct variation is y = -3x. So, correct option is B.

A direct variation is a relationship between two variables where one is a constant multiple of the other. In other words, as one variable increases or decreases, the other also increases or decreases in a consistent manner. The equation for direct variation is typically of the form y = kx, where k is the constant of variation.

Let's analyze each given equation:

y = 2/x - This equation does not represent a direct variation because y is not a constant multiple of x.

y = -3x - This equation does represent a direct variation because y is a constant multiple of x. Here, k (the constant of variation) is -3.

y = x² - 2x + 5 - This equation is a quadratic equation, not a direct variation, as it involves x raised to a power greater than 1.

y = √x - This equation does not represent a direct variation as y is not a constant multiple of x.

So, the equation that represents a direct variation is option 2) y = -3x. It has a direct relationship between y and x with a constant of variation of -3.

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The probability that both the woman and her husband fail to get the correct brand of paint is 4/9.

To determine the probability that both the woman and her husband fail to get the correct brand of paint, we need to consider the probability of each event happening independently. Since the woman selects two brands at random out of the three available, the probability of her not getting the correct brand is 2/3. Similarly, the husband also selects two brands at random out of the three, so his probability of not getting the correct brand is also 2/3. Since the events are independent, we can multiply the probabilities together to get the probability that both fail to get the correct brand. Therefore, the probability is (2/3) * (2/3) = 4/9.

The husband's selections are independent of his wife's selections because each brand he chooses is not influenced by the brands his wife chooses. Regardless of what she selects, he still has three brands to choose from and selects two at random. This means that his probability of not getting the correct brand is the same regardless of his wife's choices.

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Answer:5 sqrt(2)

Step-by-step explanation:

All the sides of the square are congruent by the markings. Therefore they all measure 5 and because all 4 sides are congruent it is a square. Because it is a square the corners are right angles and you can use the Pythagorean Theorem. a^2 +b^2 = c^2.

5^2 + 5^2 = x^2

25 + 25 = x^2

50= x^2

X = square root of 50

X= square root of 25 Times Square root of 2

X= 5square root of 2

Answer: 5 sqrt (2)

Step-by-step explanation:

ABCD is a square

BCD is a triangle

BC=DC=5

To find DB

Let BC= a, DC=b, BD=c.

using Pythagoras Theorem in Triangle BCD

c2 = a2+b2

c2 = 52+52

c2= 25+25

c= √(50)

c2= √(25)×√ (2)

c= √(5× 5) √(2)

c=  5√(2)   or    c=5sqrt(2)  

Rmbr,  Let BC= a, DC=b, BD=c.

The statement "Every elementary row operation is​ reversible" is true because interchanging can be reversed by​ scaling, and scaling can be reversed by replacement (Option A is correct).

The statement "Every elementary row operation is​ reversible" is true.

The correct choice is: A. True, because interchanging can be reversed by​ scaling, and scaling can be reversed by replacement.

- Interchanging rows (row swapping) can be reversed by another interchange.

- Scaling a row by a non-zero scalar can be reversed by scaling it by the reciprocal of that scalar.

- Replacement operations (adding or subtracting multiples of one row from another) can also be reversed by adding or subtracting the same multiples in the opposite direction.

So, all three elementary row operations (replacement, interchanging, and scaling) are reversible, which makes option A the correct choice.

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The statement 'Every elementary row operation is reversible' is true. All the three types of elementary row operations, i.e., scaling, interchanging, and replacement, can be reversed using appropriate methods.

The statement 'Every elementary row operation is reversible' is indeed true. The three types of elementary row operations, i.e., scaling, interchanging, and replacement, are all reversible. Scaling can be reversed by multiplying the row by the reciprocal of the scale factor. Interchanging rows can be undone by simply interchanging them again. Replacement can be reversed by applying a replacement operation with the opposite sign.

For example, if you multiply a row by a factor of 3 (scaling), you can reverse this by multiplying the row by 1/3. If you interchange row 1 and row 2, you can reverse this by interchanging these two rows again. Finally, if you replaced row 1 by adding 2*row 2 to it, you could reverse this by replacing row 1 by subtracting the same 2*row 2 from it.

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P(A) =0.54

P(B)= 0.68

P'(A)= 1-0.54 = 0.46

P'(B)= 1- 0.68 = 0.32

The probability of neither of both event will occur:

= P'(A)×P'(B)

=0.46 × 0.32

=0.1472

Answer:

A) Be sure the task is understood.

Step-by-step explanation:

The principle "Make sure the mission is understood, performed, and achieved."

Another way we talk about this principle in the Navy is through the idea of "intrusive leadership." In some respects both "micromanagement" and "intrusive leadership" sound terrible.

Think about certain great managers and leaders you have had in your career yet again. Probability are they will be the ones who asked you those difficult questions, too.

They moved everyone to new technical levels, and eye for detail. When you said you knew what you were doing or when you announced the progress of a project, they didn't necessarily take it to face value.

Yo sup??

From Newton's 2nd law of motion

F*t=Δp

=mv-mu

mu=0.4*15

=6

t=0.2

mv=0

Therefore

F*0.2=6

F=30 N

Hope this helps.

Answer: The density of the paper clips is 0.64 g/ml.

Step-by-step explanation:

Given : Volume of water = 10.0 mL

mass of water =  25.0 g

After 51 clips added to water , the total mass = 28 g

Total volume = 14.7

Mass of clips =  total mass - mass of water

= 28 g-25g = 3g

Volume of clips = Total volume- Volume of water

= 14.7- 10.0 = 4.7 ml

Density of paper clips =[tex]\dfrac{\text{Mass of paper clips}}{\text{Volume of paper clips}}[/tex]

[tex]=\dfrac{3}{4.7}=0.638297\approx0.64\ g/ml[/tex]

Hence, the density of the paper clips is 0.64 g/ml.

Answer:

The density of the paper clips is 0.64 g/m

Step-by-step explanation:

hope this helps :)

This is an incomplete question, here is a complete question.

A hurricane wind blows across a 7.00 m × 12.0 m flat roof at a speed of 150 km/h.

What is the pressure difference Δp = p(inside)-p(outside)? Use 1.28 kg/m³ for the density of air. Treat the air as an ideal fluid obeying Bernoulli's equation.

Answer : The pressure difference will be, [tex]1.11\times 10^3Pa[/tex]

Step-by-step explanation :

As we are given:

Speed = 150 km/h = 41.66 m/s

Density = [tex]\rho=1.28kg/m^3[/tex]

Area = A = 7.00 m × 12.0 m

Formula used :

[tex]\Delta P=\frac{1}{2}\times \rho \times v^2[/tex]

Now put all the given values in this formula, we get:

[tex]\Delta P=\frac{1}{2}\times (1.28kg/m^3)\times (41.66m/s)^2[/tex]

[tex]\Delta P=1.11\times 10^3Pa[/tex]

Thus, the pressure difference will be, [tex]1.11\times 10^3Pa[/tex]

Answer:

Step-by-step explanation:

Please refer to the picture below

Answer

7.5 pounds

Step-by-step explanation:

Since adding x grams of salt will bring th percentage of the salt to 25

Hence. 10% of the 50 gram gives. 5 gram initial salt before it is added.

5+ x/ 50 * 100= 25/

5+x /50 = 0.25

5+x = 12.5

X= 12.5- 5

X= 7.5pounds