Decide how many solutions this equation has:
x2 - 2x + 1 = 0

The equation of the function in vertex form is f(x) = -2·(x - 8)² + 6

Function, in mathematics, an expression, rule, or law that defines a relationship between one variable (the independent variable) and another variable.

here, we have,

The given values are

x,            f(x)

6,             -2

7,              4

8,              6

9,              4

10,            -2

The equation of the function in vertex form is given as follows;

f(x) = a × (x - h)² + k

To find the values of a, h, and k, we proceed as follows;

When x = 6, f(x) = -2

We have;

-2 = a × (6 - h)² + k = (h²-12·h+36)·a + k.............(1)

When x = 7, f(x) = 4

We have;

4 = a × ( 7- h)² + k = (h²-14·h+49)·a + k...........(2)

When x = 8, f(x) = 6...........(3)

We have;

6 = a × ( 8- h)² + k

When x = 9, f(x) = 4.

We have;

4 = a × ( 9- h)² + k ..........(4)

When x = 10, f(x) = -2...........(5)

We have;

-2 = a × ( 10- h)² + k

Subtract equation (1) from (2)

4-2 =  a × ( 7- h)² + k - (a × (6 - h)² + k ) = 13·a - 2·a·h........(6)

Subtract equation (4) from (2)

a × ( 9- h)² + k - a × ( 7- h)² + k

32a -4ah = 0

4h = 32

h = 32/4

  = 8

From equation (6) we have;

13·a - 2·a·8 = 6

-3a = 6

a = -2

From equation (1), we have;

-2 = -2 × ( 10- 8)² + k

-2 = -8 + k

k = 6

The equation of the function in vertex form is f(x) = -2·(x - 8)² + 6

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The requried properties are true for all the parallelograms are,
1. Opposite sides are parallel     2. Opposite sides are congruent

A parallelogram is a quadrilateral consisting of pairs of parallel sides.

The following properties are true for all parallelograms,
The opposite sides are parallel.
The Opposite sides are congruent.

All parallelograms have four congruent sides (property 1), four right angles (property 2), or exactly one pair of parallel sides (property 3). These properties only apply to specific types of parallelograms, such as rectangles or squares.

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There was 19% percent error of the baker's prediction.

Percent error is the difference between the estimated value and the actual value in comparison to the actual value and is expressed as a percentage.

Given that, a bakery sold 105 cupcakes in one day, the head baker predicted he would sell 85 cupcakes that day.

We are asked to find the percent error of the baker's prediction,

Percent error = |expected value - exact value| / exact value × 100 %

The expected value is the prediction of the Chef = 85

The exact value = 105

Percent error = |85-105| / 105 × 100 %

= 20/105 × 100 %

= 19%

Hence, there was 19% percent error of the baker's prediction.

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The coordinates of both positions of point B is (2√5,0) and B2: (0,2√5).

The intersection of the graph of the function with the y-axis gives y-intercept of that function. The y-intercept is the value of y on the y-axis at which the considered function intersects it.

Assume that we've got: y = f(x)

At y-axis, we've got x = 0, so putting it will give us the y-intercept.

Thus, y-intercept of y = f(x) is y = f(0)

We are given that;

Point b=  2 root 5 units

Now,

Since point B is 2√5 units away from point A, we can draw two circles with radius 2√5 centered at point A. The two intersections of these circles with the gridlines will give us the two possible positions of point B.

For example, if point A has coordinates (0,0) and the gridlines are the x-axis and y-axis, then the two possible positions of point B are:

B1: (2√5,0) - the intersection of the circle with the positive x-axis

B2: (0,2√5) - the intersection of the circle with the positive y-axis

If the gridlines are different, then the positions of point B will be different.

Therefore, by the intercept the answer will be (2√5,0) and B2: (0,2√5).

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Sets of parametric equations for the rectangular equation are x=t and y=8t+7.

A parametric equation has the variables x and y expressed in terms of t which is the third independent variable.

The rectangular equation is [tex]y = 8x + 7[/tex].

If we denote the variable x by t.

Then, we can say that x=t

The parametric equation can be formed by replacing x with t in the rectangular equation [tex]y=8x+7[/tex].

Therefore, sets of parametric equations for the rectangular equation are x=t and y=8t+7.

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Answer:

The slope is 3.

Step-by-step explanation:

You can find this by using rise/run.

The line goes from (1,2) to (2,5).

The rise for this is 3 and the run is 1.

3/1 is 3, therefore the slope is 3

The equivalent ratio to the given expression 3mm : 1cm as required in the task content is; 3 : 10.

It follows from the task content that the equivalent ratio of 3mm : 1cm is to be determined .

Recall, 1cm is equivalent to 10 mm.

On this note, the given ratio can be written as; 3mm : 10 mm.

Consequently, the ratio which is equivalent to the given expression is; 3 : 10.

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Answer:

2/3, 8/12

Step-by-step explanation:

(4 divided by 2)/(6 divided by 2) = 2/3

(4 times 2)/(6 times 2)= 8/12

Multiply/Divide the numerator and the denominator by the same number

There are infinte equivalent fractions to 4/6! 4000/6000 is one!

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The number of times the group need to fill the measuring spoon in order to perform the experiment is 12 times

Equations are mathematical statements with two algebraic expressions flanking the equals (=) sign on either side.

It demonstrates the equality of the relationship between the expressions printed on the left and right sides.

Coefficients, variables, operators, constants, terms, expressions, and the equal to sign are some of the components of an equation. The "=" sign and terms on both sides must always be present when writing an equation.

Given data ,

Let the equation be represented as A

Now , the value of A is

Let the amount of teaspoon of salt added to the solution be = ( 3/8 ) of spoon

Let the number of students = 4 students

The group only has ( 1/8 ) teaspoon measuring spoon

Substituting the values in the equation , we get

So , the amount of teaspoon of salt added to the solution by one student = amount of teaspoon of salt added to the solution / ( 1/8 )

On simplifying the equation , we get

The amount of teaspoon of salt added to the solution by one student = ( 3/8 ) / ( 1/8 )

The amount of teaspoon of salt added to the solution by one student = 3 times

And , the number of times the salt is added by 4 students = 4 x 3 = 12 times

Hence , the number of times is 12

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The equation of the plane that passes through (-1, 3, 2) and is perpendicular to the planes x + 2y + 2z = 11 and 3x + 3y + 2z = 15 is -2x + 4y - 3z - 17 = 0

In geometry, a plane is a flat, two-dimensional surface that extends infinitely in all directions. A plane can be defined by an equation in the form of ax + by + cz = d, where a, b, and c are coefficients that determine the plane's orientation and d is a constant.

To find the equation of a plane, we need to know its normal vector, which is a vector perpendicular to the plane. We can find the normal vector of the plane we are looking for by taking the cross product of the normal vectors of the two given planes.

The normal vector of the first plane, x + 2y + 2z = 11, is (1, 2, 2), and the normal vector of the second plane, 3x + 3y + 2z = 15, is (3, 3, 2). To take their cross product, we can use the following formula:

n = (a₂b₃ - a₃b₂, a₃b₁ - a₁b₃, a₁b₂ - a₂b₁)

where n is the normal vector, and a and b are the two given normal vectors. Plugging in the values, we get:

n = (2(2) - 2(3), 2(3) - 1(2), 1(3) - 2(3)) = (-2, 4, -3)

This means that the normal vector of the plane we are looking for is (-2, 4, -3). We also know that the plane passes through the point (-1, 3, 2), so we can use the point-normal form of the equation of a plane, which is:

a(x - x₀) + b(y - y₀) + c(z - z₀) = 0

where (x₀, y₀, z₀) is the given point, and a, b, and c are the coefficients of the normal vector. Plugging in the values, we get:

-2(x + 1) + 4(y - 3) - 3(z - 2) = 0

Simplifying, we get:

-2x + 4y - 3z - 17 = 0

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The position vs. time connection is best described by equations is [tex]& -(0.15 \mathrm{~m}) \cos (3.65) t[/tex] .

From the given data,

A student attaches a 3.0 kg mass to a spring with a spring constant of 40 N/m.

The student compresses the spring to the left by 0.15 m and then releases the mass allowing it to oscillate.

The position of the mass for the spring- mass oscillatory system is,

Where

The angular frequency of the system is,

The angular frequency refers to the angular displacement of any wave element per unit of time or the rate of change of the waveform phase.

Therefore, equation (1) changes as follows

[tex]x & =A \cos \left(\sqrt{\frac{k}{m}}\right) t \\[/tex]

[tex]& =-(0.15 \mathrm{~m}) \cos \left(\sqrt{\frac{40 \mathrm{~N} / \mathrm{m}}{3.0 \mathrm{~kg}}}\right) t \\[/tex]

[tex]& =-(0.15 \mathrm{~m}) \cos (3.65) t[/tex]

Therefore, the position vs. time connection is best described by equations is [tex]& -(0.15 \mathrm{~m}) \cos (3.65) t[/tex] .

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The surface area of the composite figure is 415. 25 in²

From the image shown, we can see that the figure is made up of a cuboid and a semicircle.

The formula for calculating the surface area of a cuboid is given as;

Surface area =2lw+2lh+2hw

Given that, l is the length, h is the height and w, the width.

Surface area = 2(6×10) + 2(6×8) + 2(8 × 10)

expand the brakect

Surface area = 120 + 96+ 160

Surface area = 376 in²

Surface area of a semicircle = 1/2(πr2 ),

Substitute the values

Surface area = 1/2 ×22/7 ×5² = 39.25 in²

Hence, the value is the sum of the surface areas

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The rate of return is an important measure of an investment's performance and is calculated by dividing the net investment income by the initial investment.

The formula for calculating a rate of return is expressed as the net investment income divided by the initial investment (ROI = (Gain from Investment – Cost of Investment) / Cost of Investment). To calculate the rate of return, you need to first determine the net investment income by subtracting the cost of investment from the gain from investment. Then, divide this number by the cost of investment and multiply it by 100 to convert it into a percentage. For example, if you invest $1000 and you make a total of 1200 back, your rate of return is 20%. This is calculated by subtracting the cost of investment (1000) from the gain (1200), resulting in a net income of 200. Then, divide $200 by the cost of investment (1000) to get 0.2 and multiply it by 100 to get a rate of return of 20%.

The rate of return is an important measure of an investment's performance and is calculated by dividing the net investment income by the initial investment.

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Answer: 17x = 54

x represents chau's savings before it increased

Your expected winnings from playing this game are 2 pounds.

A game is an activity or a form of play, often with a set of rules and goals, that is undertaken for enjoyment, competition, or skill development.

Let's analyze this game to see what your expected winnings are.

If the first roll is 1, 2, or 3, the game continues and you have a 3 in 6 chance (or 1/2 chance) of continuing to roll the die. Each subsequent roll has the same probabilities and outcomes as the first roll.

Let's start with the case where you win on the first roll with a probability of 1/2. In this case, your winnings are 1 pound.

If you don't win on the first roll, the game continues with a probability of 1/2, and your expected winnings from that point on are the same as your expected winnings from the beginning of the game (since the probabilities and outcomes are the same for all rolls).

Therefore, the expected winnings from the start of the game are:

E = 1/2 * 1 + 1/2 * E

Solving for E, we get:

E = 1 + E/2

E/2 = 1

E = 2

Therefore, your expected winnings from playing this game are 2 pounds.

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The edge of the cube is equal to 2 m³.

There are different quadrilaterals, for example: square, rectangle, rhombus, trapezoid and parallelogram.  Each type is defined accordingly to its length of sides and angles. For example,  in a square, all angles are 90° and all sides present the same value.

A square is a 2D figure, when the square is associated with a 3D figure, the figure is called a cube.

The cube presents 6 faces with length (l), height (h) and width (w) equal. Thus, the cube presents congruent edges. The volume is given from the formula V=a³, where a is a congruent edge.

The exercise gives the volume of the cube 8m³. As presented previously, the edges of the cube are congruent and the formula by the volume is V=a³. Thus,

V=a³

8=a³

[tex]\sqrt[3]{a^3}[/tex]=[tex]\sqrt[3]{8}[/tex]

a=[tex]\sqrt[3]{2^3}[/tex]

a=2 m

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Answer:

  September, October, November

Step-by-step explanation:

When monthly rainfall in centimeters is represented by A(t) = 2.3sin(πt/6)+9.25 and A(0) represents July's rainfall, you want to know the months in which average rainfall is at least 10.5 cm.

We want to find the values of t that make A(t) ≥ 10.5:

  2.3sin(πt/6) +9.25 ≥ 10.5

  2.3sin(πt/6) ≥ 1.25 . . . . . . subtract 9.25

  sin(πt/6) ≥ 0.543478 . . . . divide by 2.3

  πt/6 ≥ 0.574575 . . . . . . . take inverse sine

  t ≥ 1.097

The sine function is symmetrical about π/2, so this also means solutions will be of the form ...

  πt/6 ≤ π -0.574575

  t ≤ 6 -1.097 ≈ 4.903

The month numbers that will have rainfall at least 10.5 cm will fall in the range ...

  1.097 ≤ t ≤ 4.903

  t ∈ {2, 3, 4}

If July is month 0, then these months are September, October, November.

Jacy's hometown will get at least 10.5 cm of rain in September, October, and November.

__

Additional comment

The attachment confirms this result. We have shifted the rainfall function so it can use conventional month numbers. It shows months 9, 10, 11 have rainfall above 10.5 cm.

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a) The pet shelter has 40% cats (20% black, 5% adopted), and 60% dogs (10% black, 10% adopted). 50% are adopted in non-black cats and dogs, b) 0.45, c) 0.19, d) 0.15, e) 0.375, f) 0.6.

a) The probability tree would seem like this:

P(Cat) = 40%

P(Black Cat) = 20%

P(Adopted Black Cat) = 5%

P(Not Adopted Black Cat) = 95%

P(Not Black Cat) = 80%

P(Adopted Not Black Cat) = 50%

P(Not Adopted Not Black Cat) = 50%

P(Dog) = 60%

P(Black Dog) = 10%

P(Adopted Black Dog) = 10%

P(Not Adopted Black Dog) = 90%

P(Not Black Dog) = 90%

P(Adopted Not Black Dog) = 50%

P(Not Adopted Not Black Dog) = 50%

b) P ( Adopted & Not Black Dog ) = P(Not Black Dog) x P(Adopted | Not Black Dog) = 0.90 * 0.50 = 0.45

c) P(Not Adopted & Black Cat) = P(Black Cat) x P(Not Adopted | Black Cat) = 0.20 * 0.95 = 0.19

d) P(Adopted) = P(Adopted Black Cat) + P(Adopted Not Black Cat) + P(Adopted Black Dog) + P(Adopted Not Black Dog) = 0.05 + 0.50 + 0.10 + 0.50 = 0.15

e) P(Adopted | Cat) = P(Adopted & Cat) / P(Cat) = (0.05 + 0.50) / 0.40 = 0.375

f) P(Cat | Adopted) = P(Cat & Adopted) / P(Adopted) = (0.05 + 0.50) / 0.15 = 0.6.

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a. The average American spend on Food & Beverage is $152.2. b. The average American spend on Entertainment/Recreation is $98.93. c. They spent $53.27 more on food than on entertainment.

In order to show mathematical issues, the "pie chart," often referred to as a "circle chart," divides the circular statistical visual into sectors or portions. A proportionate amount of the entire is indicated by each sector. The Pie-chart is now the most effective method for determining the composition of something. Pie charts typically take the role of other graphs, such as bar graphs, line plots, histograms, etc.

Given that, the average American spent $761.00.

a. Money spent on Food and Beverages is:

F = 761(20/100) = $152.2

b. Money spent on Entertainment is:

E = 761(13/100) = $98.93

c. The money spent on food is more than entertainment by:

A = 152.2 - 98.93 = $53.27

Hence, a. The average American spend on Food & Beverage is $152.2. b. The average American spend on Entertainment/Recreation is $98.93. c. They spent $53.27 more on food than on entertainment.

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The solution is, the product is 9.25* 10^83.2.

In mathematics, multiplication is a method of finding the product of two or more numbers. It is one of the basic arithmetic operations, that we use in everyday life.

here, we have,

2.5\times 10^22.5×10^2 and 3.7 \times 10^53.7×10^5

=2.5* 10^22.5×10^2 × 3.7 * 10^53.7×10^5

=9.25* 10^24.5*10^58.7

=9.25* 10^83.2

Hence, The solution is, the product is 9.25* 10^83.2.

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1. To determine if the pair of planes 3x + 2y + z = 8 and x + 3y – z = 5 are parallel or intersecting, we can check if their normal vectors are parallel or not.

The normal vector of the first plane is <3, 2, 1>, and the normal vector of the second plane is <1, 3, -1>. These normal vectors are not parallel, so the planes intersect at a line.

To find a parametrization for the line of intersection, we can set one of the variables to be a parameter and solve for the other two. Let's choose z as the parameter.

From the second plane equation, we have z = x + 3y - 5. Substituting this into the first plane equation, we get: 3x + 2y + (x + 3y - 5) = 8

Simplifying, we get: 4x + 5y = 13

Solving for y in terms of x, we get: y = (13 - 4x) / 5

So a parametrization for the line of intersection is: x = t

y = (13 - 4t) / 5, z = t + 3((13 - 4t) / 5) - 5 = (2t + 2) / 5

2. To determine if the pair of planes 2c – y + z = 4 and 4c – 2y + 2z = 3 are parallel or intersecting, we can again check if their normal vectors are parallel or not.

The normal vector of the first plane is <2, -1, 1>, and the normal vector of the second plane is <4, -2, 2>. These normal vectors are parallel (in fact, they are scalar multiples of each other), so the planes are parallel.

To find the distance between the two parallel planes, we can take any point on one plane and find its distance to the other plane. Let's choose the point (0, 0, 4) on the first plane. The distance between this point and the second plane is: |4c - 2(0) + 2(4) - 3| / sqrt(4^2 + (-2)^2 + 2^2) = |4c - 5| / 2sqrt(6)

So the distance between the two parallel planes is |4c - 5| / 2sqrt(6).

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The equation of the line is y = (2/3)x + 20/3.

The equation of a line is given by:

y = mx + c

where m is the slope of the line and c is the y-intercept.

Example:

The slope of the line y = 2x + 3 is 2.

The slope of a line that passes through (1, 2) and (2, 3) is 1.

We have,

The equation has:

Slope = 2/3

Passes through the line (-4, 4).

Now,

The equation of the line is y = mx + c.

m = 2/3

Consider (-4, 4) = (x, y)

So,

4 = (2/3)(-4) + c

4 = -8/3 + c

c = 4 + 8/3

c = (12 + 8) / 3

c = 20/3

Now,

y = (2/3)x + 20/3

Thus,

The equation of the line is y = (2/3)x + 20/3.

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Answer:

The answer to the question is Sixteen.

a) The exponential function that represents the population of the bacteria after t hours is given as follows:

y = a(2)^(t/12). (a is the initial amount).

b) The initial amount is given as follows: a = 2.

The standard definition of an exponential function is given as follows:

y = a(b)^(x/n).

In which:

Considering that the colony doubles in 12 hours, the parameters b and n are given as follows:

b = 2, n = 12.

Hence the function is defined as follows:

y = a(2)^(t/12).

After 4 days = 96 hours, there are 400 bacteria, hence the initial amount is obtained as follows:

400 = a(2)^(96/12)

256a = 400

a = 400/256

a = 2. (rounding).

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Answer:

w = 12

Step-by-step explanation:

To simplify the equation, we have to isolate 'w'. To isolate 'w',

     [tex]\dfrac{w}{2}-21=-15\\\\\\\dfrac{w}{2 }-21+21=-15+21 \ \text{\bf (Addition property of equality)}\\[/tex]

                    [tex]\dfrac{w}{2}= 6[/tex]

                [tex]2*\dfrac{w}{2}=6*2 \ \text{\bf (Multiplication property of equality)}\\\\\\[/tex]

                     w = 12

Answer:

Step-by-step explanation:

The line segment joining two points P and Q can be represented by the equation of a straight line in the form y = mx + b, where m is the slope and b is the y-intercept.

To find the equation of the line, we need to find the slope, which can be calculated using the formula:

m = (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are the coordinates of the points P and Q, respectively.

In this case, the coordinates are:

P = (-3, 2) and Q = (5, 7)

So, the slope is:

m = (7 - 2) / (5 - (-3)) = 5 / 8

Next, we can use either of the points to find the y-intercept. Let's use point P:

b = y - mx, where y and x are the y and x coordinate of the point, respectively.

In this case,

b = 2 - m * (-3) = 2 - (5/8) * (-3) = 2 + 15/8 = 89/8

So, the equation of the line joining the points P and Q is:

y = (5/8)x + 89/8

Now, to find the point where the line crosses the y-axis, we need to find the x-coordinate of the point where y = 0.

So, we have:

0 = (5/8)x + 89/8

Solving for x, we get:

x = -(89/8) / (5/8) = -89 / 5

This means that the line crosses the y-axis at the point (-89/5, 0). To find the ratio in which the line segment is divided by the y-axis, we need to find the ratio of the distance from the y-axis to point P to the distance from the y-axis to point Q.

Let's call the point of intersection with the y-axis R. The distances are then:

PR = (3, 2) and QR = (5 - (-89/5), 7)

The ratio of the distances is then:

PR / QR = (3, 2) / (5 - (-89/5), 7) = 3 / (5 + 89/5) = 3 / (94/5) = 15/47

So, the line segment joining the points P and Q is divided by the y-axis in the ratio 15:47.

Answer:

64°

Step-by-step explanation:

the sum of the 3 angles in a triangle = 180°

let the third angle be x , then

x + 90° + 26° = 180°

x + 116° = 180° ( subtract 116° from both sides )

x = 64°

the other angle is 64°

As a result, the answer is (e) 2.4 square inches, which is the difference between Richard's measured and real circle area.

In mathematics, area is a measure of the amount of space occupied by a two-dimensional object, such as a rectangle, triangle, circle, or any other shape. It's a scalar quantity that describes the size of a region in two-dimensional space. The units of area are typically square units, such as square inches, square centimeters, square meters, etc. In general, the area of a shape is a measure of how much space it occupies, and it is an important concept in geometry, engineering, and many other fields.

Here,

The formula for the area of a circle is given by:

A = πr²

Where r is the radius of the circle.

Using the incorrect radius of 3.6 inches, the calculated area would be:

A = π * (3.6 inches)² = 40.44 square inches

Using the correct radius of 3.5 inches, the actual area would be:

A = π * (3.5 inches)² = 38.5 square inches

So, the difference between Richard's measured area of the circle and the actual area of the circle would be:

40.44 square inches - 38.5 square inches = 1.94 square inches

Therefore, the answer is (e) 2.4 square inches that is the difference between Richard’s measured area of the circle and the actual area of the circle.

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Mark had 480 bear stickers and 400 bird stickers at first.

A word problem in math is a math question written as one sentence or more that requires children to apply their math knowledge to a 'real-life' scenario.

Given that, Mark had 5/6 as many birds stickers as bear stickers at first. After buying another 20 stickers of each type, he has 7/8 as many bird stickers as bear stickers.

We are asked to find the number of each type of stickers did mark have at first.

Let Mark have x bear stickers at first,

Therefore,

Bird sticker = 5x/6

5x/6 + 20 = 7x/8

Multiply the equation by 48

40x + 960 = 42x

2x = 960

x = 480

5x/6 = 400

Hence, Mark had 480 bear stickers and 400 bird stickers at first.

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