true/false: let a be an n \times n matrix. a is diagonalizable if it has n eigenvalues, counting multiplicities.

The probabilty of selecting two white balls in the two scenarios are 1/3 and 1/13

Scenario 1

The probability of drawing a white ball from the first urn is 5/9, since there are 5 white balls out of 9 total balls.

Similarly, the probability of drawing a white ball from the second urn is 3/5.

To find the probability that both balls are white, we need to multiply the probabilities of drawing a white ball from each urn, since the events are independent:

(5/9) × (3/5) = 15/45 = 1/3

Scenario 2

Using the steps in (1), we have

P = 1/2 * 2/12

Evaluate

P = 1/12

Therefore, the probability that both balls are white is 1/12.

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The sum of all eight factors is 96

Let's express the integer as a product of its prime factors, in the form

n = p1^a1 × p2^a2 × ... × pn^an

where p1, p2, ..., pn are prime numbers and a1, a2, ..., an are positive integers.

We know that the integer has 8 factors, which means that its prime factorization must have the form:

n = p1^2 × p2^2 × p3^4

since (2+1) × (2+1) × (4+1) = 3 × 3 × 5 = 45, which is the total number of factors for this product.

We also know that 15 and 21 are factors of the integer, so they must be expressible in terms of the prime factors of n. We can write

15 = 3 × 5 = p1 × p2

21 = 3 × 7 = p1 × p3

From these expressions, we can see that p1 = 3, p2 = 5, and p3 = 7.

Therefore, the prime factorization of n is

n = 3^2 × 5^2 × 7^4

The eight factors of n are

1, 3, 5, 7, 9, 15, 21, and 35

Their sum is

1 + 3 + 5 + 7 + 9 + 15 + 21 + 35 = 96

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For the given following frequency table of values 351, 362, 373, 381, 391, The mode is likely to be the best measure of the center for the data set.

The given frequency table is as follows:

        Value frequency 351, 362, 373, 381, 391.

        To find the most appropriate measure of central tendency for a dataset, we need to analyze the spread of data.

The mean, median, and mode are measures of central tendency in statistics.

We can find the following measures from the given data set:

       Mean: It is calculated by summing up all the values and then dividing the result by the total number of values. This measure of central tendency is appropriate when the data are symmetrical.

      Median: It is the middle value of the data set when arranged in order. It is suitable for skewed data.

      Mode: It is the most common value in the data set. It is appropriate when data is discrete. The data in the frequency table appear to be discrete.

      Because the data are discrete, the most appropriate measure of central tendency is the mode. So, the mode is likely to be the best measure of the center for the given value frequency data set.

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The weight range within which the middle 95% of all miniature Tootsie Rolls will fall is approximately between 3.04 grams and 3.56 grams.

Since the weight of miniature Tootsie Rolls is normally distributed with a mean of 3.30 grams and a standard deviation of 0.13 grams, we can use the standard normal distribution to find the weight range within which the middle 95% of all miniature Tootsie Rolls will fall.

We need to find the z-scores that correspond to the middle 95% of the normal distribution. The area between the two z-scores will be 0.95. Using a standard normal distribution table or calculator, we can find that the z-scores that correspond to the middle 95% of the distribution are -1.96 and 1.96.

We can use the formula z = (x - μ) / σ to find the corresponding weight values for these z-scores. Substituting -1.96 for z and solving for x, we get:

-1.96 = (x - 3.30) / 0.13

x = 3.04

Similarly, substituting 1.96 for z and solving for x, we get:

1.96 = (x - 3.30) / 0.13

x = 3.56

Therefore, the weight range within which the middle 95% of all miniature Tootsie Rolls will fall is approximately between 3.04 grams and 3.56 grams.

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The relative frequency of the event "spin a 2" is given as follows:

0.2.

To calculate the relative frequency of an event, you need to divide the number of times the event occurs by the total number of events in the data-set.

Hence the formula to calculate the relative frequency is presented as follows:

Relative frequency = (Number of times the event occurs) / (Total number of events)

From the table, the parameters for the event "spin a 2" are given as follows:

Hence the relative frequency is given as follows:

10/50 = 0.2.

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Since it is difficult to evaluate this integral exactly, use technology to approximate it. After calculation, the length is approximately 3.9205 (rounded to four decimal places).

To find the length of the function y =[tex]2x^(3/2)/3 - x^(1/2)/2[/tex] over the interval [1, 4], you need to evaluate the integral of the square root of [1 + (y')^2] with respect to x, where y' is the derivative of y with respect to x.

First, find the derivative y':
y'(x) = d(2x^(3/2)/3 - x^(1/2)/2)/dx = x^(1/2) - 1/(4x^(1/2))

Now, find (y')^2:
[tex](y')^2 = (x^(1/2) - 1/(4x^(1/2)))^2[/tex]
Next, find the square root of [1 + (y')^2]:
[tex]√[1 + (y')^2] = √[1 + (x^(1/2) - 1/(4x^(1/2)))^2][/tex]

Finally, evaluate the integral:
Length = [tex]∫(√[1 + (y')^2]) dx[/tex] from x = 1 to x = 4

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The area of the walkaway is 216.66 feet squared.

The circular swimming pool has a diameter of 20 feet. A 3 foot tile walkaway is being installed around the pool.

Therefore, the area of the walkaway can be calculated as follows:

Hence,

area of the walkaway = area of the bigger circle - area of the smaller circle

area of the bigger circle = πr²

area of the bigger circle = 3.14 × 13²

area of the bigger circle  = 530.66

area of the smaller circle = πr²

area of the smaller circle  = 3.14 × 10²

area of the smaller circle = 314

area of the walkaway = 530.66 - 314  = 216.66 ft²

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there are twο pοssible values οf x that make the length οf segment MP equal tο the length οf segment OP: x = 4 and x = 8.

In geοmetry, a line segment is a part οf a straight line that is bοunded by twο distinct end pοints, and cοntains every pοint οn the line that is between its endpοints.

Tο find the value οf x such that the length οf the segment MP is equal tο the length οf the segment OP, we can use the distance fοrmula tο find the lengths οf bοth segments, and then sοlve fοr x.

First, we can find the length οf segment MN using the distance fοrmula:

MN = √[(7-6)² + (-4-(-3))²] = √[1² + (-1)²] = √2

Next, we can find the length οf segment OP using the distance fοrmula:

OP = √[(x-(-3))² + (-5-(-7))²] = √[(x+3)² + 2²]

Nοw, we can set the lengths οf MP and OP equal tο each οther, and sοlve fοr x:

MN / OP = MP / OP

√2 / √[(x+3)² + 2²] = √[(6-x)² + (-3-(-5))²] / √[(x+3)² + 2²]

Squaring bοth sides οf the equatiοn, we get:

2 / [(x+3)²+ 2²] = [(6-x)² + 2²] / [(x+3)² + 2²]

Multiplying bοth sides οf the equatiοn by [(x+3)² + 2²], we get:

2 = [(6-x)² + 2²]

Expanding and simplifying, we get:

2 = x² - 12x + 40

Rearranging, we get:

x² - 12x + 38 = 0

We can use the quadratic fοrmula tο sοlve fοr x:

x = (12 ± √[(-12)² - 4(1)(38)]) / (2(1))

x = (12 ± √16) / 2

x = 6 ± 2

Therefοre, there are twο pοssible values οf x that make the length οf segment MP equal tο the length οf segment OP: x = 4 and x = 8.

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Answer:

[tex]\huge\boxed{\sf y = \frac{9-4x}{6}}[/tex]

Step-by-step explanation:

9 = 4x + 6y

9 - 4x = 6y

[tex]\displaystyle \frac{9-4x}{6} = y\\\\OR\\\\y = \frac{9-4x}{6} \\\\\rule[225]{225}{2}[/tex]

Answer:

Step-by-step explanation:

Now we have to,

→ Find the required value of y.

The equation is,

→ 9 = 4x + 6y

Then the value of y will be,

→ 9 = 4x + 6y

→ 4x + 6y = 9

→ 6y = 9 - 4x

→ 6y = -4x + 9

→ y = (-4x + 9)/6

→ y = (-4x/6) + (9/6)

→ y = (-2x/3) + (3/2)

Hence, this is the answer.

In conclusion, the determinant of M being non-zero only guarantees that the equation has a unique solution, but it does not guarantee that it has a solution for every value of A.
If the determinant of M is not equal to zero, then A can be solved in terms of X or X can be solved in terms of A. This is because the matrix M is invertible when its determinant is non-zero. In other words, the matrix M has a unique inverse.
Here are the statements that are correct:
- Given any X, there is one and only one A which will satisfy the equation. This is because when the determinant of M is non-zero, the matrix M is invertible, and hence the equation can be uniquely solved for A in terms of X or for X in terms of A.
- Some values of A (such as A=0) will allow more than one X to satisfy the equation. This is because the determinant of M being non-zero only guarantees that the equation has a unique solution, but it does not guarantee that it has a solution for every value of A.
- Some values of X will have more than one value of A that satisfy the equation. This is true for any equation that has multiple solutions, regardless of the determinant of M.
- Some values of A will have no values of X that satisfy the equation. This is true for any equation that has no solution, regardless of the determinant of M.
- There is no value of X which satisfies the equation when A=0. This is because when A=0, the equation reduces to 0=0, which is true for any value of X.

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The cost of the tabletop is $251.10. Option C

The formula for calculating the area of a rectangle is expressed as;

Area = lw

Such that the parameters are;

From the diagram shown, we have;

The width = 6 3/4 feet = 27/4 feet

The length = 3 feet

Substitute the values

Area = 27/4 × 3

Multiply

Area = 20. 25 square feet

But;

1 square foot = $12.40

Then  20. 25 square feet = x

x = $251. 10

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The surface area of the box is (D) 184 square inches.

What is surface area?

Surface area is the measurement of the total area that the surface of an object occupies. It is the sum of the areas of all the faces or surfaces that make up the object. For example, the surface area of a cube can be found by adding up the area of all six of its square faces.

To find the surface area of the box, we need to add up the areas of each face.

The front face has an area of 5 x 4 = 20 square inches.

The back face has the same dimensions, so it also has an area of 20 square inches.

The left face has an area of 4 x 8 = 32 square inches.

The right face also has an area of 4 x 8 = 32 square inches.

The top face has an area of 5 x 8 = 40 square inches.

The bottom face also has an area of 5 x 8 = 40 square inches.

Adding up all of these areas, we get:

20 + 20 + 32 + 32 + 40 + 40 = 184 square inches

Therefore, the answer is (D) 184 square inches.

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Answer:

The answer to your question would be true.

The given statement 'full-time high school students spend more time per week on their courses than full-time college students' is true because full-time high school students spend approximately 30 hours per week in class, and full-time college students only spend about half of that in class.

Full-time high school students spend more time per week on their courses than full-time college students. It is because full-time high school students spend approximately 30 hours per week in class, and full-time college students only spend about half of that in class.

College courses usually last for one hour per day, whereas high school students attend classes for 5-6 hours per day. Hence, college students spend around 15-18 hours per week attending classes, while high school students spend around 30 hours per week in the classroom.

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The total number of employees in the company would be 5, rounded to the nearest whole number.

The formula to calculate the total number of employees in a company is:

Total Number of Employees = Number of Employees in One Country x Proportion of Company's Employees.

For example, if a company has 10 employees in one country and that represents 50% of the company's employees, then the total number of employees in the company would be:

Total Number of Employees = 10 x 0.5 = 5

Therefore, if a company has 10 employees in one country and that represents 50% of the company's employees, then the total number of employees in the company would be 5, rounded to the nearest whole number.

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Answer:

First, you distribute 2(2x-1/2)

2(2x) and 2(-1/2)

Now, you have 3x-2=4x-1

Then, you move 2 to the right side by adding so you can cancel it out.

Then, you move 4x to the left side by doing the same thing with 2

Now, divide both sides by -1

The answer is C, z= -1.

3x-2=2(2x-1/2)

3x-2=4x-1

3x=4x+1

-x=1

-x/-1 = 1/-1

= x=-1

Based on this linear equation, we would predict that John would sell 50 hot cocoas if the day's high temperature were 30 F.

What is linear equation ?

A linear equation is an algebraic equation that describes a straight line relationship between two variables, typically denoted as x and y. The general form of a linear equation is:

y = mx + b

where y is the dependent variable, x is the independent variable, m is the slope of the line, and b is the y-intercept (the point at which the line crosses the y-axis).

The slope, m, represents the rate of change of y with respect to x.

According to the question:
Assuming that the relationship between the high temperature and the number of hot cocoas sold is roughly linear, we can use the equation of the line of best fit to estimate the number of hot cocoas sold at a high temperature of 30 F.

If we know the equation of the line of best fit in the form y = mx + b, where y is the number of hot cocoas sold and x is the high temperature, then we can substitute x = 30 and solve for y.

For example, if the equation of the line of best fit is y = 2x - 10, where y is the number of hot cocoas sold and x is the high temperature, then we can substitute x = 30 to get:

y = 2(30) - 10 = 50

Therefore, based on this equation, we would predict that John would sell 50 hot cocoas if the day's high temperature were 30 F. However, it's important to note that this is just an estimate and the actual number of hot cocoas sold could be different.

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Answer:

We can use trigonometry to solve this problem. In a right triangle, the sine, cosine, and tangent ratios relate the side lengths of the triangle to the angles.

Since we know that angle X is 60 degrees, we can use the sine ratio to find the length of side XY:

sin(60°) = XY / YZ

Simplifying this expression, we get:

sqrt(3) / 2 = XY / 4

Multiplying both sides by 4, we get:

XY = 2sqrt(3)

Therefore, XY is equal to 2 times the square root of 3.

Jasmin, who is 5 feet tall, is standing in the 15-foot lamppost's light. Her shadow is 4 feet long. If she walks 1 feet farther away from the lamppost, her shadow will lengthen by 3/2 feet.

Let ‘x’ be the distance between Jasmin and the base of the lamppost when she is casting a 4-foot shadow. We can use similar triangles to set up a proportion:

15/x = 5/4

Solving for 'x', we get:

X = 15/3

X = 5 feet

This means that Jasmin is standing 5 feet away from the base of the lamppost when she is casting a 4-foot shadow.

Now, let’s find the distance between Jasmin and the base of the lamppost when she walks 1 foot farther away. This will be ‘5+1=6’ feet.

Again, we can use similar triangles to set up a proportion:

15/6 = y/4+z

Where 'y' is the length of Jasmin’s shadow when she is 6 feet away from the lamppost, and 'z' is the additional length of her shadow.

Simplifying this proportion, we get:

Y = 5/2 + 5z/6

We want to find ‘z’, the additional length of Jasmin’s shadow. We can do this by using the fact that when Jasmin is 6 feet away from the lamppost, her shadow is 4+z feet long. Setting up an equation using this information, we get:

4+z=y

Substituting the expression for ‘y’ from the proportion we set up earlier, we get:

4+z = 5/2 + 5z/6

Solving for 'z', we get:

Z = 3/2

Therefore, Jasmin’s shadow will lengthen by 3/2 feet when she walks 1 foot farther away from the lamppost.

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The solution of the given inequality is x ∈ (-1,5). The graph has been plotted and attached below.

What is an inequality?

In mathematics, an inequality is a relationship that unfairly compares two numbers or other mathematical expressions. Size comparisons between two numbers on the number line are most usually made.

We are given an inequality as -4 < 4x < 20.

Now, on splitting the inequality, we get two inequalities which are:

-4 < 4x and 4x < 20.

On solving -4 < 4x, we get

⇒ -4 < 4x

⇒ -1 < x

Similarly, on solving 4x < 20, we get

⇒  4x < 20

⇒  x < 5

So, we get -1 < x < 5 which means x ∈ (-1,5).

The graph of the inequality has been plotted and attached below.

The red part represents -1 < x and the blue part represents x < 5.

Hence, the solution of the given inequality is x ∈ (-1,5).

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Answer:

yes

Step-by-step explanation:

The answer to 3 divided by 8 can also be written as a mixed fraction as follows: 0 3/8 Note that the numerator in the fraction above is the remainder and the denominator is the divisor.

Answer:

4g5h¯²

Step-by-step explanation:

36 x g9 x h/ 9x g4 x h³

Divide similar terms

(36/9) x g9-4 x h¹-³

4 x g5 x h¯²

4g5h¯²

The expected value of the game is -$0.054, which means that if you play this game many times, you can expect to lose around 5 cents per game on average.

We need to calculate the expected value of a game where you randomly select a marble from a bag containing 2 black marbles, 8 white marbles, and 27 red marbles.

If the marble is black, you win $3.

If the marble is white, you win $2.

If the marble is red, you lose $1.

The expected value of a game is the sum of the products of each outcome and its probability.

To find the expected value of this game,

we need to first calculate the probabilities of each outcome.

P(bag contains a black marble) = 2/37P(bag contains a white marble)

= 8/37

P(bag contains a red marble)

= 27/37

Probability of winning $3

= P(black marble)

= 2/37

Probability of winning $2

= P(white marble)

= 8/37

Probability of losing $1

= P(red marble) = 27/37

Expected value

= (Probability of winning $3 × $3) + (Probability of winning $2 × $2) + (Probability of losing $1 × $-1)

Expected value = (2/37 × 3) + (8/37 × 2) + (27/37 × -1)

Expected value = -0.054$

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To find the least common multiple (LCM) of 12 and 9, we can list their multiples and find the smallest multiple they have in common:

Multiples of 12: 12, 24, 36, 48, 60, ...

Multiples of 9: 9, 18, 27, 36, 45, 54, 63, ...

We can see that the smallest multiple they have in common is 36. Therefore, the LCM of 12 and 9 is 36.

Alternatively, we can use the prime factorization method to find the LCM:

Prime factorization of 12: 2^2 x 3

Prime factorization of 9: 3^2

To find the LCM, we need to take the highest power of each prime factor that appears in either number. In this case, the highest power of 2 is 2^2, and the highest power of 3 is 3^2. Therefore, the LCM of 12 and 9 is 2^2 x 3^2 = 36.

Answer:

The least common multiple of 12 and 9 is 36.

Step-by-step explanation:

The multiples of 12 are: 12, 24, 36, 48, 60, ...

The multiples of 9 are: 9, 18, 27, 36, 45, 54, 63, ...

The least common multiple is the smallest number that both lists share, which is 36.

A-B is equivalent to 5x2 + 11x - 13 as a result as [tex]A = 3x^2 + 5x - 6[/tex] and  [tex]B = -2x^2 - 6x + 7[/tex] .

A mathematical expression is a grouping of digits, variables, operators, and symbols that denotes a mathematical amount or relationship. It can be analyzed or condensed using mathematical operations and rules, and it can be a single word or a group of terms. Numerous mathematical ideas, including equations, variables, functions, and formulas, can be represented by expressions. Standard form, factored form, extended form, and polynomial form are just a few of the different ways they can be expressed in writing.

given

We must deduct the expression B from the expression A in order to obtain A-B. To accomplish this, we subtract the appropriate coefficients from terms of the same degree. Here are the facts:

[tex]A = 3x^2 + 5x - 6\\B = -2x^2 - 6x + 7[/tex]

A - B =[tex](3x^2 + 5x - 6) (-2x^2 - 6x + 7)[/tex]

= 3x2 + (5x - 6) + (2x2 - 6) + (7x - 5x2 + 11x - 13 (distributing the negative sign)

A-B is equivalent to 5x2 + 11x - 13 as a result as [tex]A = 3x^2 + 5x - 6[/tex] and  [tex]B = -2x^2 - 6x + 7[/tex] .

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As per central angle theorem, James in incorrect because he calculated the number of slices of pizza that were eaten instead of the number of slices of pizza that were left. James needed to divide 360 by the appropriate number of slices of pizza, 8. He then would have needed to multiply the result by 5 in order to find how many degrees of pizza was left over. After that, James would have needed to divide by 180 and multiply by π to get 1.257 radians as the amount of pizza that was left.

The central angle is the angle that a circle's arc occupies in its center. The arms of the central angle are formed by the radius vectors.

In other words, it's an angle whose vertex is the center of a circle and whose arms, which have the circle's two radii as their arms, cross at two separate positions. These two points make an arc when they are combined. The angle that this arc at the circle's center subtends is known as the central angle.

Here,

out of 8 slices 5 are left.

So 360/8

= 45

5 × 45 = 225°

Now 225° = 225× π/180

= 3.927 Rad

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James ordered a large pizza cut into eight equal pieces. Of the 8 slices, only 3 were edible. James thought the rest of the pizza was 0.75 pi radians. Option A) correctly describes James' claim.

The central angle is the angle an arc of a circle makes at its center. The central angle arm is formed by the radius vector. That is, the angle at which the arms with the apex at the center of the circle and the arms at his two radii of the circle intersect at two different positions. Combining these two points forms an arc. The angle that this arc makes at the center of the circle is called the central angle.  

Here,

out of 8 slices 5 are left.

So 360/8

= 45

5 × 45 = 225°

Now 225° = 225× π/180

= 3.927 Rad

According to the central angle theorem, James is wrong because he calculated the number of pizza slices eaten, not the number of pizza slices left. James had to divide his 360 by 8, the number of pizza slices. Then he had to multiply the result by 5 to see how many pizzas were left. Then James had to divide him by 180 and multiply by π, so he got 1.257 radians as the remaining amount of pizza.

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The linear measure of arc KML on OO is approximately 3.33 units, rounded to two decimal places.

Since KM is a diameter, angle KOM is a right angle. Therefore, angle KOL is a straight angle, which means that angle MOL is 180 - 145 = 35 degrees.

Now, we can use the fact that the measure of an arc is proportional to the measure of the angle it subtends. In particular, if the measure of an angle in degrees is θ and the radius of the circle is r, then the length of the arc it subtends is given by:

length of arc = (θ/360) * 2πr

In this case, the radius of the circle is half of the diameter KM, which is 36/2 = 18. So we have:

length of arc KML = (35/360) * 2 * 3.14 * 18

≈ 3.33

Therefore, the linear measure of arc KML on OO is approximately 3.33 units, rounded to two decimal places.

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16.3 degrees is the angle that the ramp makes with the ground.

Draw a diagram: Draw a right-angled triangle with the ramp as the hypotenuse, the height of the truck as one of the other sides, and the horizontal distance from the bottom of the ramp to the truck as the other side.

Use trigonometry: We can use the trigonometric function tangent to find the angle the ramp makes with the ground. tan(theta) = opposite/adjacent = height of truck/horizontal distance

Let's substitute the given values:

tan(theta) = 6/21

Solve for theta: We can use the inverse tangent function to solve for theta.

theta = tan^-1(6/21)

theta ≈ 16.3 degrees

Therefore, the angle that the ramp makes with the ground is approximately 16.3 degrees.

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According to question,

DEF+PRS=90

DEF=78

PRS=?

WE KNOW,

DEF+PRS=90

or,78+PRS=90

or,PRS=90-78

:.PRS=12,,

Answer: $ 46.50

First divided 31 by 2

Which equals...

15.50

Then add 15.50 to 31.

46.50

The answer is $46.50