Please help!! Worth 50 points

The distance from frank's house to john's house and then back to frank's house is less than a mile by 1/5 units.

While travelling from one location to another and back again just multiplying the distance between the two places will give you the total distance travelled when travelling from one spot to another and returning to the beginning position.

The distance from frank's house to john's house = 4/10.

The distance back to Frank's house = 4/10.

Total distance = 4/10 + 4/10

Total distance = 8/10

The distance is thus,

1 - 8/10

Take the LCM:

(10 - 8) / 10

2/10 = 1/5

Hence, the distance from frank's house to john's house and then back to frank's house is less than a mile by 1/5 units.

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Answer:

26

Step-by-step explanation:

10(2) -2(-3)=20+6=26

Answer:

Step-by-step explanation:

[tex]\tt 10a-2b[/tex]

[tex]\tt a=2[/tex]

[tex]\tt b=-3[/tex]

______

Simplify: 10×2 = 20

Simplify: 2×-3= -6

Simplify:-

_____________________

Hope this helps! :)

A. The equations representing plant height after a given number of days are:

B. To graphically represent the system of equations, we can plot the points that correspond to the height of each plant after a given number of days. For example, after 5 days, the first plant is 15 cm tall (2(5) + 5 = 15) and the second plant is 18 cm tall (3(5) + 3 = 18).

You can draw this graph as a coordinate plane with the horizontal axis representing the number of days since the initial measurement and the vertical axis representing the height of the plants in centimeters.

C. To find the height of each plant after 15 days, we can plug t = 15 into the equations we encountered in letter A:

It is a set of two or more equations that share common variables and are solved jointly.

Therefore, a system of equations is used when one wants to find the values of variables capable of satisfying all the equations of the system, whose solutions represent the points where the graphs of the equations in the Cartesian plane.

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The width of the rectangular prism shaped toolbox is 10 inches.

A prism is a three-dimensional geometric shape that has two parallel and congruent bases that are connected by a set of parallelograms or rectangles. A prism is named according to the shape of its base, such as triangular prism, rectangular prism, or hexagonal prism.

A prism is characterized by its length, width, and height, and it has a volume equal to the product of its base area and its height. The lateral faces of a prism are parallelograms or rectangles, and the edges of a prism where the lateral faces meet the bases are perpendicular to the bases.

Now,

To find the width of the toolbox, we can use the formula for the volume of a rectangular prism, which is:

Volume = length x width x height

We are given that the volume of the toolbox is 4,500 cubic inches, the length is 30 inches, and the height is 15 inches.

Then

4,500 = 30 x width x 15

Simplifying the right side, we get:

4,500 = 450 * width

Dividing both sides by 450, we get:

width = 10 inches

Therefore, the width of the toolbox is 10 inches.

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Sketch of  ABC and drawn image after the translation (x,y) (x+4, y + 2) is mentioned below.

In mathematics, translation refers to the process of moving a geometric shape or object from one location to another without changing its size, shape, or orientation. This movement is typically described in terms of the distance and direction of the translation.

For example, a point on a coordinate plane can be translated by adding or subtracting a constant value to its x and y coordinates. Similarly, a line segment or polygon can be translated by moving each of its vertices by the same distance and in the same direction.

Mathematical translations can be represented using transformation matrices or vectors. A translation matrix is a square matrix that describes the movement of a point or object in a specific direction and distance. A translation vector is a one-dimensional vector that describes the direction and distance of the translation.

To sketch the triangle ABC, we first plot the three given points on a coordinate plane:

We connect the three points to form triangle ABC:

    |

 6  |        C(0,5)

    |              /\

 3  |           /    \

    |          /        \

    |        /            \

    |      /                 \

    |    /__________\

    |A(-3,3)       B(1,-1)

    |

    |

    |

    |

    |

    |

        -3     -1    1     3

To find the image of the triangle under the translation (x,y) → (x+4, y+2), we add 4 to the x-coordinates and 2 to the y-coordinates of each point. This gives us the following new coordinates:

A(-3,3) → A'(-3+4, 3+2) = A'(1,5)

B(1,-1) → B'(1+4, -1+2) = B'(5,1)

C(0,5) → C'(0+4, 5+2) = C'(4,7)

We can now plot these new points and connect them to form the image of triangle ABC under the given translation:

    |

 9  |       C'(4,7)

    |             / \

 7  |          /     \

    |         /         \

 5  |      /             \

    |      /                 \

    |    /__________\

 3  |A'(1,5)       B'(5,1)

    |        

    |        

    |        

 1  |        

    |

    |

        -3     -1    1     3     5

This is the final image of triangle ABC after the translation.

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The vertices of the translated figure are A'(1, 5), B'(5, 1), and C'(4, 7).

What is translation?

A translation in mathematics does not turn a shape; instead, it moves it left, right, up, or down. They are congruent if the translated shapes (or the image) seem to be the same size as the original shapes. They have only changed their direction or directions.

To translate the figure by (x+4, y+2), we add 4 to the x-coordinate and 2 to the y-coordinate of each vertex.

So the new coordinates of A' would be -

x-coordinate: -3 + 4 = 1

y-coordinate: 3 + 2 = 5

Therefore, A' has coordinates (1, 5).

The new coordinates of B' would be -

x-coordinate: 1 + 4 = 5

y-coordinate: -1 + 2 = 1

Therefore, B' has coordinates (5, 1).

The new coordinates of C' would be -

x-coordinate: 0 + 4 = 4

y-coordinate: 5 + 2 = 7

Therefore, C' has coordinates (4, 7).

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[tex] \: [/tex]

[tex] \: [/tex]

Solution:-

[tex] \: [/tex]

[tex] \: [/tex]

[tex] \: [/tex]

[tex] \: [/tex]

━━━━━━━━━━━━━━━━━━━━━━━━━━

hope it helps! :)

The probability that exactly three plants don't grow is given as follows:

0.0054 = 0.54%.

The mass probability formula, giving the probability of x successes, is of:

[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]

[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]

The parameters are given by:

The parameter values for this problem are given as follows:

The probability that exactly three don't grow is P(X = 3), hence:

P(X = 3) = C(8,3) x (0.05)^3 x (0.95)^5 = 0.0054.

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Answer:

5a + 3b = 13

7a + 6b = 20

And solve for the values of "a" and "b" using algebraic methods like substitution or elimination.

Step-by-step explanation:

Solving for a and b in the system of equations:

5a + 3b = 13

7a + 6b = 20

We can multiply the first equation by 2 to get:

10a + 6b = 26

Then, we can subtract the second equation from this to eliminate b:

(10a + 6b) - (7a + 6b) = 26 - 20

3a = 6

Solving for a, we get:

a = 2

Substituting this value into the first equation, we can solve for b:

5a + 3b = 13

5(2) + 3b = 13

10 + 3b = 13

3b = 3

b = 1

Therefore, the solution to the system of equations is:

a = 2

b = 1.

The equation of the circle x² + y² - 8x - 6y + 16 = 0 in standard form is (x - 4)² + (y - 3)² = 3².

Given the equation of the circle ;

x² + y² - 8x - 6y + 16 = 0

To write the equation of a circle in standard form, we need to express it as:

(x - h)² + (y - k)² = r²

where (h, k) is the center of the circle and r is the radius.

To do this, we need to complete the square for both x and y terms in the given equation.

Starting with the x terms:

x² - 8x = x² - 8x + 16 - 16

= (x - 4)² - 16

Similarly, for the y terms:

y² - 6y = y² - 6y + 9 - 9

= (y - 3)² - 9

Substituting these results into the original equation, we get:

x² + y² - 8x - 6y + 16 = 0

(x - 4)² - 16 + (y - 3)² - 9 + 16 = 0

Simplifying, we get:

(x - 4)² + (y - 3)² - 9 = 0

(x - 4)² + (y - 3)² = 9

So the equation of the circle in standard form is:

(x - 4)² + (y - 3)² = 3²

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The value of the car after 7 years is approximately $18,994.7.

A linear equation is defined as an equation in which the highest power of the variable is always one.

Let V be the value of the car after 7 years.

In the first year, the value of the car depreciates by 5%, so the value after the first year is:

V1 = 0.95 × 27,200

In the second year, the value of the car depreciates again by 5%, so the value after the second year is:

V2 = 0.95 × V1

Similarly, the value of the car after the third year is:

V3 = 0.95 × V2

We can continue this process for each of the 7 years:

V4 = 0.95 × V3

V5 = 0.95 × V4

V6 = 0.95 × V5

V7 = 0.95 × V6

So, the equation that represents the value of the car after 7 years is:

V = 0.95⁷ × 27,200

Simplifying this expression, we get:

V ≈ 18,994.7

Therefore, the value of the car after 7 years is approximately $18,994.7.

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Answer:

12

Step-by-step explanation:

The probability that Hector captures the golden brick and chooses the correct door is approximately 0.2, or 20%.

What is probability ?

Probability can be defined as ratio number of favourable outcomes, total number of outcomes.

A represents the event that Hector captures the golden brick (with a probability of 0.7), B represents the event that Hector chooses the correct door (with a probability of 0.4), and C represents the intersection of A and B (the event that Hector captures the golden brick and chooses the correct door, with a probability of 0.2).

The probability of event C can be calculated using the formula:

P(C) = P(A and B) = P(A) * P(B|A)

where P(A) is the probability of event A (capturing the golden brick), and P(B|A) is the conditional probability of event B (choosing the correct door) given that event A has occurred (i.e., Hector has captured the golden brick). We are given that P(A) = 0.7, P(B) = 0.4, and P(A and B) = 0.2, so we can calculate:

P(B|A) = P(A and B) / P(A) = 0.2 / 0.7 ≈ 0.286

Therefore, the probability of event C (capturing the golden brick and choosing the correct door) is:

P(C) = P(A and B) = P(A) * P(B|A) = 0.7 * 0.286 ≈ 0.2

Therefore, the probability that Hector captures the golden brick and chooses the correct door is approximately 0.2, or 20%.

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Answer:

Step-by-step explanation:

To find the vertical and horizontal components of the muscular force, we can use trigonometry. Let F be the force exerted by the muscle, and θ be the angle it makes with the horizontal.

Horizontal component of the force Fx = F cos θ

Vertical component of the force Fy = F sin θ

Given that the muscular force is 75kg and the angle of pull is 15 degrees, we have:

Fx = 75kg x cos(15) = 72.62 kg (approx.)

Fy = 75kg x sin(15) = 19.34 kg (approx.)

To determine which force would be responsible for the rotation of the bone around the joint, we need to consider the torque or moment created by the force. The torque is given by the force times the lever arm, which is the perpendicular distance from the line of action of the force to the axis of rotation. In this case, the axis of rotation is the joint, and the lever arm is the distance from the joint to the point where the muscle attaches to the bone.

The torque due to the horizontal component of the force is zero, since it acts perpendicular to the lever arm. The torque due to the vertical component of the force is:

Torque = Fy x d

where d is the lever arm. If we assume that the lever arm is 5 cm, we have:

Torque = 19.34 kg x 0.05 m = 0.967 Nm

Therefore, the vertical component of the force would be responsible for the rotation of the bone around the joint.

To determine which force would cause a compressive force at the joint, we need to consider the direction of the force. A force that acts perpendicular to the bone would cause compression at the joint. In this case, the vertical component of the force is perpendicular to the bone, while the horizontal component is parallel to it. Therefore, the vertical component of the force would cause a compressive force at the joint.

Here is a diagram of the situation:

     |\

Fy | \          /|

     |  \        / |

     |   \      /  |

     |    \    /   |

     |     \  /    |

     |      \/     |

     |      /\     |

     |     /  \    |

     |    /    \   |

     |   /      \  |

     |  /        \ |

     | /          \|

     |/______\

        Fx

In the diagram, Fx is the horizontal component of the force, Fy is the vertical component of the force, and the bone is represented by the vertical line. The angle θ is the angle between the force and the horizontal. The lever arm d is the distance between the joint and the point where the muscle attaches to the bone.

Answer:

65

Step-by-step explanation:

All 4 sides of a polygon equals 360 degrees. We are given the equation of 2 sides to find the degree.

12x + 19 + 8x + 1 = 180

(We put it equal to 180 instead of 360 since we are only given 2 sides)

Subtract 19 from both sides

12x + 8x + 1 = 161

Subtract 1 from both sides

12x + 8x = 160

Add 12x + 8x

20x = 160

Divide 20x by 160

x = 8

Plug in x value to (8x + 1)

(8(8) + 1)

= 65

(-x+18)(x+4)

-x(x+4) + 18 (x+4)

-(x^2+4x) + 18(x+4)

-x^2-4x+18x+72

−x^2+14x+72

I hope this helps! :)

Equivalence class of {x} is the set of all elements of the set {A} which are related to {x} i.e. → [x]R={y ∈ A ∣ xRy}.

A binary relation which is symmetric, transitive and reflexive is called a equivalence relation.

Given is to find the equivalence classes of the equivalence relations.

If it is given that {R} is an equivalence relation on {A} and {x} ∈ A, then we can write the equivalence class of {x} is the set of all elements of the set {A} which are related to {x} i.e. →

[x]R={y ∈ A ∣ xRy}.

Therefore, equivalence class of {x} is the set of all elements of the set {A} which are related to {x} i.e. → [x]R={y ∈ A ∣ xRy}.

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The dimensions of the poster are 10 inches by 14 inches.

The area of Rectangle is length times of width.

Given that length of a rectangle poster is nine more inches than half it’s width.

Length = 1/2 width +9

The area of the poster is 140 square inches.

Area=Length×Width

140=(W/2+9)W

140=W²/2 +9W

W²+18W-240=0

(w + 28)(w - 10) = 0

W=-28 and w=10

We take only positive value so width is 10

Length=10/2+9

=5+9=14 inches

Hence, the dimensions of the poster are 10 inches by 14 inches.

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Using linear equations, Tiffany was charged $10.77 for 8 minutes at the self-serve car wash.

Solving a linear equation involves determining the answer to a linear equation with one, two, three, or more variables. The answer to a linear equation is the value or values of the variable(s) that are part of the equation. In the elimination procedure, any coefficient is first made equal, then deleted. After elimination, the equations are solved to provide the other equation.

Given that, the car wash charges $5.25 to use its facilities, plus an additional $0.69 for each minute.

C = 5.25 + 0.69x

For 8 minutes:

C = 5.25 + 0.69(8)

C = $10.77

Hence, Tiffany was charged $10.77 for 8 minutes at the self-serve car wash.

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When compared to the value denoted by the same digit with in hundreds place, the value in the thousands place is 10 times greater.

We can determine the place value from every digit by looking at a place value chart, which is a highly helpful table format.

In the given number: 4,434.

thousands place : 4

hundreds place: 4

Thus, when compared to the value denoted by the same digit with in hundreds place, the value in the thousands place is 10 times greater.

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The measurements of the diagonal distance across the middle school gym is given by the following persons and is rounded to the nearest hundredths

The calculation of Arturo = 31.75 yards

The calculation of Bonnie = 32.98 yards

The calculation of Courtney = 31.42 yards

The calculation of Dawson = 31 yards

There are basically two rules while rounding up numbers

If the number you are rounding is followed by 0, 1, 2, 3, or 4, round the number down and if the number you are rounding is followed by 5, 6, 7, 8, or 9, round the number up.

Non-zero digits are always significant

Zeros between non-zero digits are always significant

Leading zeros are never significant

Trailing zeros are only significant if the number contains a decimal point

Given data ,

Let the measurements of the diagonal distance across the middle school gym be represented as A

Now , the measured values are

a)

The measurement of Arturo = 6√28 yards

The measurement of Arturo = 6 x 5.2915

On rounding , we get

The measurement of Arturo = 31.75 yards

b)

The measurement of Bonnie = 8√17 yards

The measurement of Bonnie = 8 x 4.123

On rounding , we get

The measurement of Bonnie = 32.98 yards

c)

The measurement of Courtney = 10π yards

The measurement of Courtney = 10 x 3.142

On rounding , we get

The measurement of Courtney = 31.42 yards

d)

The measurement of Dawson = 3.1√100 yards

The measurement of Dawson = 3.1 x 10

On rounding , we get

The measurement of Dawson = 31 yards

Hence , the measurements are rounded

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Answer:

Let's start by defining some variables to represent the width and length of the pool:

Let w be the width of the pool (in feet).

Let l be the length of the pool (in feet).

We know from the problem statement that the length of the pool is 5 feet more than 3 times its width:

l = 3w + 5

We also know that the deck surrounding the pool is 2 feet wide. This means that the overall width of the pool and the deck combined is:

w_total = w + 2 + 2 = w + 4

Similarly, the overall length of the pool and the deck combined is:

l_total = l + 2 + 2 = l + 4

We are given that the area of the deck is 196 square feet. The area of the deck can be calculated by subtracting the area of the pool from the area of the pool and deck combined:

Area of deck = Area of pool and deck - Area of pool

196 = w_total * l_total - w * l

We can substitute the expressions for w_total and l_total in terms of w and l:

196 = (w + 4) * (l + 4) - w * l

Simplifying this expression gives:

196 = wl + 4w + 4l + 16

We can substitute the expression for l in terms of w:

196 = w(3w + 5) + 4w + 4(3w + 5) + 16

Simplifying this expression gives:

196 = 13w + 36

Solving for w, we get:

w = 14

Therefore, the width of the pool is 14 feet.

( 2,-9) points lie on the same line.

Line point intercept is defined.

Quantifying soil cover, including vegetation, litter, rocks, and biotic crusts, quickly and accurately can be done using the line-point intercept approach.

                                  These metrics deal with the erosion caused by wind and water, water infiltration, and the site's capacity to withstand damage and recover from it.

k = -5 - 3/1 - (-1)  = -8/2 = -4

the formula of the line is  

                                    y - 3 = -4(x+ 1)

 when x  = -3 , y = 11 ≠ 12 the points aren't on line.

 when x = -5, y = 19 ≠ 1

  when x = 4, y = -7 ≠ 15

   when x = 2, y = -9 ≠ 7

    when x = 0, y = -1 ≠ -2

  ( 2,-9) are on the same line.

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Answer: 8

Step-by-step explanation:

Since 1 batch uses 1/8 of a bag of flour, I would use equivalent fractions to find how many 8ths are in 3/4.  3/4=?/8

4 times 2 equals 8, so I would multiply 3 by 2 also.  3/4=6/8.  If 1 batch takes 1/8, then 6 batches would use 6/8 or 3/4 of a bag of flour.

The domine of function = {0, 1, 2, 3}  

The range of function = {-3, -1, 1, 3}

The domain of a function is the set of all possible input values for which the function is defined. It can be restricted by certain conditions, such as division by zero, square roots of negative numbers, or logarithms of non-positive numbers.

The range of a function is the set of all possible output values that the function can produce. It depends on the nature of the function and the restrictions on the domain.

Here we have a table

x -    0    1     2     3  

f(x)  -3   -1     1      3    

Here x values are indicates input values for the function f(x)

Hence, the domine of function = {0, 1, 2, 3}  

The values of f(x) represent the output values for each input value of 'x'

Hence, the range of function = {-3, -1, 1, 3}

Therefore,

The domine of function = {0, 1, 2, 3}  

The range of function = {-3, -1, 1, 3}

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Jessica sold the drinks for $1 each to raise money as she is running for class president.

The elimination method is a technique used to solve a system of linear equations in two variables. In this method, we multiply one or both equations by appropriate constants to obtain equivalent equations in which one of the variables has the same coefficient but with opposite signs. Then we add the equations to eliminate one of the variables, and solve for the remaining variable.

Let the price of one candy bar be represented by x, and the price of one drink be represented by y.

From the first equation, we know that:

50x + 40y = 65

From the second equation, we know that:

100x + 50y = 100

We want to find the value of y, so we can solve for y using either substitution or elimination method. Here, we will use elimination method.

Multiplying the first equation by 2, we get:

100x + 80y = 130

Subtracting the above equation from the second equation, we get:

100x + 50y - (100x + 80y) = 100 - 130

Simplifying the left side and right side of the equation, we get:

-30y = -30

Dividing both sides by -30, we get:

y = 1

Therefore, Jessica sold the drinks for $1 each.

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The probability that there are not enough seats on the flight is approximately 0.9735 or 97.35%.

The probability of a passenger missing a flight is 0.0993. Therefore, the probability that a passenger does not miss the flight is:

P(not missing the flight) = 1 - 0.0993 = 0.9007

The plane has a capacity for 250 passengers, but 270 tickets are sold. Assuming that the probability of a passenger missing a flight is independent of other passengers, we can model the number of passengers who show up for the flight as a binomial distribution with n = 270 and p = 0.9007.

Let X be the number of passengers who show up for the flight. We want to find the probability that there are not enough seats on the flight, which means that X > 250. We can use the complement rule to find this probability:

P(X > 250) = 1 - P(X ≤ 250)

To find P(X ≤ 250), we can use a binomial probability table or a calculator with a binomial probability function. Using a calculator, we get:

P(X ≤ 250) = binomcdf(270, 0.9007, 250) ≈ 0.0265

Substituting this into the complement rule formula, we get:

P(X > 250) = 1 - 0.0265 ≈ 0.9735

Therefore, the probability that there are not enough seats on the flight is approximately 0.9735 or 97.35%.

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For the differential equation y = [(x² - 1) sin x] / (sin x + 1), the derivative is   dy/dx = [(x² + 2x) sin² x] / (sin x + 1)².

What is differential equation?

Any equation with one or more terms and one or more derivatives of the dependent variable with respect to the independent variable is referred to as a differential equation.

To find dy/dx, we need to use the quotient rule of differentiation, which states that -

d/dx [f(x)/g(x)] = [g(x) × f'(x) - f(x) × g'(x)] / [g(x)]²

Applying this rule to the given function, we get -

y = [(x² - 1) sin x] / (sin x + 1)

y' = [(sin x + 1) d/dx[(x² - 1) sin x] - (x² - 1) sin x d/dx[sin x + 1]] / (sin x + 1)²

Now we need to find the derivatives of the two terms in the numerator separately -

d/dx[(x² - 1) sin x] = (2x sin x + (x² - 1) cos x) sin x

d/dx[sin x + 1] = cos x

Substituting these values in the formula for y', we get -

y' = [(sin x + 1) * (2x sin x + (x² - 1) cos x) - (x² - 1) sin x * cos x] / (sin x + 1)^2

Simplifying this expression, we get -

y' = [(x² + 1) sin² x + 2x sin x cos x - (x² - 1) sin x cos x - (x² - 1) sin x cos x] / (sin x + 1)²

y' = [(x² + 2x) sin² x] / (sin x + 1)²

Therefore, the derivative of y with respect to x (dy/dx) is -

dy/dx = [(x² + 2x) sin² x] / (sin x + 1)²

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Based on the information, I can notice that mathematics is used in various situations in daily life. Additionally, I wonder how to apply algebra in daily life.

Based on what we have learned about mathematics, I have noticed that it is a very broad area that covers various types of operations and mathematical problems that allow us to solve various problems. For example: It helps us to know how to distribute a budget in our daily life, it helps us to divide the birthday cake in equal parts and to find the area of our house and our room.

What we ask ourselves about mathematics is how we can apply the problems and formulas of algebra and other areas derived from mathematics in daily life. Sometimes it seems that these complex problems do not have an application in daily life.

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Answer:

13

Step-by-step explanation:

1. 20/2 = n - 3

2. 10 = n - 3

3. 10 + 3 = n

4. 13 = n