One weekend, Nate compared his history homework to his younger sister Caroline’s social studies homework. He found that for every 2 1−3 pages he had to read for homework, Caroline had to read one page for homework. How many pages did Nate read for every three pages Caroline read?

The value of a number such that The difference between three-fifths of that number and 7 is -36 is -145/3.

Arithmetic operators are four basic mathematical operations in which summation, subtraction, division, and multiplication involve.,

Summation = addition of two or more numbers or variable

Subtraction = Minus of any two or more numbers with each other called subtraction.

Let's say that number is x.

Three-fifth of x ⇒ (3/5)x

The difference of (3/5)x and 7 will be,

(3/5)x - 7 = -36

Add 7 on both sides

(3/5)x = -29

Multiply 5/3 on both sides

x = -29(5/3)

x = -145/3

Hence "The value of a number such that The difference between three-fifths of that number and 7 is -36 is -145/3".

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Answer: The slope (m) of the line using two points where the line goes through is:

m = - 4 (negative 4).

Solution:

m=(y2-y1) / (x2-x1)

Taking two points in the graph where the line goes through:

P1=(0,2)=(x1,y1)→x1=0, y1=2

P2=(1,-2)=(x2,y2)→x2=1, y2=-2

Replacing in the formula of the slope:

m=(-2-2) / (1-0)

m=(-4) / (1)

m=-4

Answer:

As given : Graph in the coordinate plane with title Julia’s volleyball digs. Vertical axis goes from 0 to 60 in increments of 5 and is labeled number of digs. Horizontal axis goes from 0 to 10 in increments of 1 and is labeled games played. A ray on the graph starts at point begin ordered pair 0 comma 25 end ordered pair and goes through points begin ordered pair 4 comma 37 end ordered pair and begin ordered pair 8 comma 49 end ordered pair.

As the ray passes through (0,25) and (4,37) and (8,49).

Equation of line passing through (a, b) and (p,q) is given by:

 →  [tex]\frac{y-b}{x-a}=\frac{q-b}{p-a}[/tex]

→Equation of line passing through (0,25) and (4,37) is :

[tex]\frac{y-25}{x-0}=\frac{37-25}{4-0}=3\\\\y -25=3x\\y =3x + 25[/tex]

As , y= 3 x + 25,  also passes through (8,49) also.

Equation of line is : y = 3 x + 25 or in Standard form : - 3 x + y = 25

Option (D) is the standard form of line which is : - 3 x + y = 25

the correct order from greatest to least is:

0.863, 0.6833, 0.386, 0.3

A linear equation or function is always a representation of a straight line in the coordinate plane. The equation of the line that passes through points (4, 5) and (−3, −1) is y = (6/7)x + 9/7.

A straight line on the coordinate plane is represented by a linear function.

The formula for a linear function is f(x) = ax + b, where a and b are real values.

The equation of a straight line is given by, y = mx + c

Where m is the slope while c is the y-intercept.

The slope of line passing by two points (x₁, y₁) and (x₂, y₂)

Slope(m) = (y₂ - y₁)/(x₂ - x₁)

The slope of the line that passes through  (4, 5) and (−3, −1) will be,

m = (-1 -5)/(-3-4)

m = -6/-7 = 6/7

So equation converts as, y = (6/7)x + c

Substitute, (4,5)

5 = (6/7)4 + c

c = 9/7

So, y = (6/7)x + 9/7 will be the equation.

Hence "The equation of the line that passes through points (4, 5) and (−3, −1) is y = (6/7)x + 9/7".

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A yard is a unit of length. Hence, the Rebeccas have  [tex]\frac{11}{20}[/tex] a yard of ribbon.

What is the yard?

A yard is a unit of length in both US Customary and British Imperial Systems of Measurement.

Here, Rebecca had [tex]\frac{9}{10}[/tex] a yard of ribbon, and Rebecca gives [tex]\frac{7}{20}[/tex] of a yard of a friend.

Now, we have to subtract [tex]\frac{7}{20}[/tex] from the original [tex]\frac{9}{10}[/tex].

Since the denominator is the same as the rules of fractions.

Now, we multiply [tex]\frac{9}{10}[/tex] by [tex]\frac{2}{2}[/tex] so that the denominator [tex]10[/tex] and [tex]20[/tex] would be the same.

Therefore,

[tex]\frac{9}{10}[/tex]×[tex]\frac{2}{2}[/tex]=[tex]\frac{18}{20}[/tex].

Here, both the denominator are the same, we can subtract [tex]\frac{7}{20}[/tex] from the [tex]\frac{18}{20}[/tex].

[tex]\frac{18}{20}-\frac{7}{20}\\=\frac{11}{20}[/tex]

Hence, the Rebeccas have  [tex]\frac{11}{20}[/tex] a yard of ribbon.

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Final answer:

Rebecca has 11/20 of a yard of ribbon left after subtracting the 7/20 yard she gave to a friend from her initial 9/10 yard of ribbon by converting both fractions to have a common denominator.

Explanation:

Rebecca started with 9/10 of a yard of ribbon and gave away 7/20 of a yard. To find out how much ribbon she has left, we need to subtract the amount she gave away from the amount she started with. This can be done following these steps:

Ensure that both fractions have a common denominator. The least common multiple of 10 and 20 is 20, so multiply the numerator and the denominator of the first fraction (9/10) by 2 to get the equivalent fraction with a denominator of 20, which gives us 18/20.

Now subtract the second fraction (7/20) from the first fraction (18/20) since they now have a common denominator. This gives us 18/20 - 7/20 = 11/20.

Therefore, Rebecca has 11/20 of a yard of ribbon left.

The values of a and b are 1/2 and 7/4 respectively. The sum of a and b is 9/4.

To find the values of a and b, we need to solve the system of equations formed by the given lines. Let's start by substituting the given coordinates (1, 2) into the equations:

x = (1/4)y + a -> 1 = (1/4)(2) + a -> 1 = 1/2 + a -> a = 1 - 1/2 = 1/2

y = (1/4)x + b -> 2 = (1/4)(1) + b -> 2 = 1/4 + b -> b = 2 - 1/4 = 7/4

Therefore, a = 1/2 and b = 7/4. To find the sum of a and b, we add them: a + b = 1/2 + 7/4 = 2/4 + 7/4 = 9/4

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Final answer:

Thomas will earn $2415 per month after a 15% raise on his current salary of $2100, which results in a raise amount of $315.

Explanation:

Thomas currently earns $2100, and with a 15% raise, we need to calculate the increase in his monthly pay. To find the amount of the raise, we calculate 15% of his current earnings. That is, we multiply his earnings by the raise percentage expressed in decimal form: $2100 × 0.15 = $315.

After finding the raise amount, we add it to his current salary to find his new monthly earnings. $2100 + $315 = $2415 per month. Therefore, after a 15% raise, Thomas will make $2415 per month.

rate of change for 2x+4y=10

here rate of change can be found by finding slope

the general form for slope is y=mx+b

isolating y on the left side:

2x+4y=10

4y=[tex] y=\frac{-2x+10}{4}

[/tex]-2x+10

from here slope is -1/2

Rate of change =-1/2

The simplifying value of the expression is,

⇒ 3q - 6

What is an expression?

Mathematical expression is defined as the collection of the numbers variables and functions by using operations like addition, subtraction, multiplication, and division.

Given that;

The expression is,

⇒ 4p + 12 - 3q - 4p - 18 + 6q

Now,

Simplify the expression as;

⇒ 4p + 12 - 3q - 4p - 18 + 6q

⇒ 4p - 4p - 3q + 6q + 12 - 18

⇒ 0 + 3q - 6

⇒ 3q - 6

Thus, The simplifying value of the expression is,

⇒ 3q - 6

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The rate at which the water level is rising when the water is 8 inches deep in the trough is 13/30 ft/min.

To find how fast the water level rises when the water is 8 inches deep, we need to use the equation V = 30h^2, which represents the volume of the trough as a function of its height h. We are given that the trough is being filled at a rate of 13 ft^3/min, so we can set dV/dt (the rate of change of volume) equal to 13.

Substituting the given values into the equation dV/dt = 60h * dh/dt, we get 13 = 60 * h * dh/dt. Since h = 0.5 (in feet), we can solve for dh/dt which turns out to be 13/30 ft/min.

Therefore, the water level is rising at a rate of 13/30 ft/min when the water is 8 inches deep.

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The expression that uses the greatest common factor and the distributive property to rewrite the sum 75 + 45 is B. 15(5 + 3)

The distributive property states that an expression of the form A(B + C) can be solved by a multiplication over addition operation that is

A(B + C) = AB + AC

This property is known by distributive law and applies to subtraction also.

Given; the distributive property to rewrite the sum 75 + 45

First find the greatest common factor.

75: 1, 3, 5, 15, 25, 75

45: 1, 3, 5, 9, 15, 45

Their GCF is 15.

To get 75, we will multiply 15 by 5.

To get 45, we will multiply 15 by 3.

Therefore, 15(5 + 3) = 75 + 45.

The expression that uses the greatest common factor and the distributive property to rewrite the sum 75 + 45 is B. 15(5 + 3)

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