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Change 0.7878... to a fraction.
Let n = 0.7878... Multiply both sides by 100 (by 100 when there are two repeating decimal digits)
100n = 78.7878... Subtract n from 100n.
99n = 78 Divide both sides by 99
n = 78/99 but n is 0.7878..., so
0.7878... = 78/99
Similarly with 0.66...
Let n = 0.66... Multiply both sides by 10 (by 10 when there is only one repeating digit)
10n = 6.66... Subtract n from 10n.
9n = 6 Divide both sides by 9.
n = 6/9 but n is 0.66..., so
0.66... = 6/9
Check using your calculator. 78/99 = 0.787878... and 6/9 = 0.6666...

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Unfortunately, you cannot find the exact value of because it is an irrational number that results in a non-repeating, non-terminating decimal.
Your calculator can give you an approximate value:
approximately.

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Let x = the deceased man's fortune.
To his wife, he leaves (1/2)x
To his son, he leaves (1/7) of (1/2)x
To his butler, he leaves (2/3) of ((1/7) of (1/2)x)
To his pig, he leaves the remaining $2,000.00
All these amounts should add up to $x, the deceased man's fortune.
(1/2)x + (1/7)(1/2)x + (2/3)(1/7)(1/2)x + $2,000.00 = $x Simplify and solve for x. Factor an x from the first three terms.
x(1/2 + 1/14 + 2/42) + $2,000.00 = $x Add up the fractions. The common denominator is 42.
x(21/42 + 3/42 + 2/42) + $2,000.00 = $x
x(26/42) + $2,000.00 = $x Simplify the fraction.
x(13/21) + $2,000.00 = $x Subtract x(13/21) from both sides.
$2,000.00 = $x - x(13/21)
$2,000.00 = $x(8/21) Multiply both sides by the multiplicative inverse of 8/21
$2,000.00(21/8) = $x
$x = $5,250.00 The deceased man's fortune.

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Well...I have never solved an equation with a cherry on top, but there's always a first time.

Cross-multiply;

Simplify and solve for x.

Subtract 14x from both sides.

Add 20 to both sides.

Divide both sides by 2.

Check:
= = =

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This means: If you had a polynomial (say, in x) and you solved it to find its roots and they turned out to be: x = -1/2 and x = 2/3, what would that polynomial look like?
Well, first, since there are two roots, the polynomial must be a quadratic function like: f(x) = ax^2 + bx + c
Now, if you have the roots to a quadratic function, you can form the factors of that function by recalling how you get the roots to a quadratic function in the first place.
If the roots are x = -1/2 and x = 2/3 then the factors would be:
(2x + 1) and (3x - 2)
Why is this?
Because to get the roots of the quadratic, you had to set each of the factors equal to zero, as in: (2x + 1) = 0, so x = -1/2 and (3x - 2) = 0, so x = 2/3
Now you have the factors of the quadratic function, you can find the function itself by multiplying these factors, (x + 1/2)(x - 2/3).
If you do this, you will get:

Check, using FOIL:

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Let x be the number of kids originally at the party.
Then:
(1/2)x + 18 + (1/3)x = x Simplify and solve for x.
(1/2)x + (1/3)x + 18 = x Add the fractions of x, common denominator is 6.
(3/6)x + (2/6)x + 18 = x
(5/6)x + 18 = x Subtract (5/6)x from both sides.
18 = x - (5/6)x
18 = (1/6)x Multiply both sides by 6.
x = 108 The number of kids originally at the party.
Check:
(1/2)(108) + 18 + (1/3)(108) = 54 + 18 + 36 = 108

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You have the right idea but the second ratio of your proportion is inverted.
Remember to keep like-items in the tops and like-items in the bottoms of the ratios.
Teachers/students = 4/52 You had this correctly shown.
Let x = # of students, so you want to see the number of teachers on top and the number of students (x) on the bottom.
27/x
The proportion looks like:
4/52 = 27/x Solve for x by cross-multiplying.
4x = (52)(27) Divide both sides by 4.
x = 1404/4
x = 351 This is the number of students.
Check:
4/52 = 27/351 Reduce the fractions.
1/13 = 1/13

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Well, you started out ok but when you got to the line: 8800 - x(.06) = 560, this is where your mistake lies.
It should have been: (8800 - x)(.06) This is 6% of the total amount minus that invested at 7%
So, you should have:
.07x + (8800 - x)(.06) = 560 Solve for x.
.07x + (.06)(8800) - .06x = 560 Simplify.
.07x - .06x + 528 = 560
0.1x = 32
x = $3200 The amount invested at 7%

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The equation of the line that passes through the point (2, -3) and is perpendicular to the line whose equation is: 5x + 3y = 9 can be found by recalling that perpendicular lines have slopes that are the negative reciprocal of each other.
So, you need to determine the slope of the given line and then find the negative reciprocal of it to get the slope of the new line.
Put the given equation into the slope-intercept form: y = mx + b
5x + 3y = 9 Subtract 5x from both sides.
3y = -5x + 9 Divide both sides by 3.
y = (-5/3) + 3 The slope of the given line is: m = -5/3 The negative reciprocal of -5/3 is 3/5 and this is the slope of the new line.
So, for the new line you can write: y = (3/5)x + b Now you need to find b, the y-intercept.
This done by substituting the x and y coordinates from the given point (2, -3) into y = (3/5)x + b and solving for b.
-3 = (3/5)2 + b Solve for b.
-3 = 6/5 + b Subtract 6/5 from both sides.
(-15/5)-(6/5) = b
-21/5 = b
Now you can write the equation of the new line: y = (3/5)x - 21/5
You can check this by seeing that the given point, (2, -3) satisfies this equation and by seeng that the slope of this line is the negative reciprocal of the slope of the given line.
This is left as an exercise for the student. If you have any further issues with this, please call again.

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Well, let the angle be A. The supplement of angle A is (180 - A) degrees. The problem says that the measure of angle A is 24 degrees less than (180-A).
So we have: A = (180-A) - 24. Solve for A
A = 180 - A - 24 Add A to both sides.
2A = 180 - 24
2A = 156 Divide both sides by 2.
A = 78 degrees.
A = (180-78)-24
A = (102)-24
A = 78 degrees.
Check:
A = (180-A)-24
A = (180-78)-24
A = (102)-24
A = 78

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Solve: x/(180 - x) = 1/8 or (x/180) - x = 1/8? It's probably x/(180-x) = 1/8
x/(180-x) = 1/8 Multiply both sides by (180-x)
x = (180-x)/8 Multiply both sides by 8.
8x = 180-x Add x to both sides.
9x = 180. Divide both sides by 9.
x = 20
Check:
20/(180-20) = 20/160 = 1/8

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Try this!
Let the two numbers be x and y.
Then:
x + y = 10 Rewrite as: x = 10 - y and substitute into equation below.
xy = -24.
(10 - y)Y = -24 Solve for y.
10y - y^2 = -24 Add 24 to both sides and put into standard form.
y^2 - 10y - 24 = 0 Solve the quadratic by factoring.
(y + 2)(y - 12) = 0 Apply the zero product principle.
y + 2 = 0, y = -2
y - 12 = 0, y = 12
Now substitute these values of y into the original equations:
x = 10 - y
x = 10 - (-2)
x = 12
And:
x = 10 -y
x = 10 - 12
x = -2
The two numbers are: -2 and 12
Check:
12 + (-2) = 10, and
12(-2) = -24

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Solve:
Add to both sides.
Square both sides.
Simplify.
Subtract 3x from both sides.
Subtract 2 from both sides.
Square both sides.
Simplify.
Subtract 48x from both sides.
Add 32 to both sides.
Divide both sides by 4.
Solve by factoring.
Apply the zero product principle.
x - 2 = 0, x = 2 or
x - 6 = 0, x = 6

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The simplest way to find the y-intercept is to let x = 0 and solve for y.
Let x = 0 Recall that the y-axis crosses the x-axis at x = 0.
Divide both sides by -5.

The y-intercept is at (0, 2)

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There are several ways to find the coordinates of the vertex of a parabola.
I'm assuming that there is a typographical error in your question.
You probably meant to type:

If not, your question is meaningless.
If this is the equation you intended, then read on.
1) You could graph the equation and find the location of the vertex of the parabola, or
2) You could put the equation into the form:

and read the coordinates of the vertex from: (h, k)
This method requires that you perform an operation known as "completing the square".
3) You could put your equation into standard form . Then find the x-coordinate of the vertex which is located at -b/2a.
Then you could get the y-coordinate by substituting the x-value into the equation and solve for y.
Here's method 2.
Set you equation equal to zero.
Subtract 2 from both sides.
Add the square of half the x-coefficient (-3)^2 = 9 to both sides.
Factor the left side.
Add 7 to both sides.
You now have the equation in the a(x-h)^2 + k form
h = 3 and k = -7
The coordinates of the vertex are: (h, k) or (3, -7)

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First, you'll need to find the coordinates of opposite corners of the rectangle. This is best done by sketching the rectangle on graph paper using the coordinates of the given vertices.
According to my sketch, the coordinates of the opposite corners are:
(-2, 1) & (3, 2)
(0, -1) & (1, 4)
Use the distance formula to find the distance between one pair of opposite corners:

Take either pair of coordinates and substitute the appropriate x, y values into the distance formula and solve for the distance, d. Let's take the second pair of points, (0, -1) & (1, 4)

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You'll notice that the given expression is the difference of two cubes:
x^3 and 2^3

The difference of two cubes is factored as follows:

In the given expression: A = x and B = 2

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You can check my previous answer to a similar problem, but again, all you need do is to substitute P = 37.55 (thousands)(that's 37,550) into the given formula and grind away.


Gallons per minute.

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You can do it this way:
First, find the amount of the raise by subtacting the old hourly rate from the new hourly rate.
$12.62 - $12.27 = $0.35 per hour.
Next, divide the amount of the raise by the old hourly rate and multiply the result by 100 to get the percent increase.
[($0.35)/($12.27)](100) = (0.285348...)(100) = 2.85% increase.

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Lacking a graphics capability, I'll try to explain in words how to go about this solving this problem. Perhaps you can draw the diagram yourself from the description below:
Draw a circle. Then draw two radii separated by some angle (the size of the angle is not crucial to solving).
Let's call the radius, R.
Now draw the chord connecting the two ends of the radii.
Let's call this C (for chord).
Now draw a third radius that bisects the chord.
So now each half of the chord is length C/2
The known distance from the chord to the circumference of the circle can be x.
So, what we need is an equation that relates the radius, R, to the length of the chord, C, and the distance, x.
If your diagram is correct, it should show inside the circle, a triangle formed by the first two radii and the chord.
The third radius exactly bisects this triangle into two congruent right triangles.
The hypotenuse of the right triangle is just radius, R.
The base of the right triangle is just half the length of the chord or C/2.
The height of the right triangle is R - x.
Now we can use the Pythagorean theorem to solve the problem.

Now we need to solve for R.
Simplifying:

Subtracting R^2 from both sides, we get:

Adding 2Rx to both sides:

Dividing both sides by 2x:

This simplifies to:

Since you know the length of the chord, C, and the distance, x, you can find the radius, R.

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To solve this problem, you need to substitute P=2250 (as P= 2.25 thousands) into the equation and grind away.




Gallons per minute.

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The remainder theorem states:
If a polynomial P(x) is divided by (x-a), where a is a constant, then the remainder is P(a).
Starting with your polynomial: and the given divisor: (w+3), you should be able to find the remainder by noting that, in this case, a = -3.
The remainder then should be P(-3) Let's find out.




To check this solution, you will need to perform the division, either by long division or by synthetic division.
I have done this and indeed, -119 is the remainder after dividing the given polynomial by w+3

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I would think that by "unit fractions", they would mean fractions having a numerator of 1.
So, I would separate the given fraction into the following:

Now replace each of the new fractions with their eqivalent fractions by simplifying each one of them.



The three unit fractions are:
, ,

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Try this:

Factor the numerator of the 1st fraction.

Now factor 2 from the numerator of the 2nd fraction.

After cancelling like factors in top and bottom you are left with:

Does this help?

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One of the methods of solving quadratic equations is known as "completing the square".
This method entails making a perfect square trinomial from the given quadratic equation or expression, then factoring the perfect square trinomial into its two identical factors.
This is best demonstrated by illustration:
Starting with your quadratic equation:

1) Move the constant to the right side of the equation, i.e. add 12 to both sides.

2) Add a new constant that is the square of one-half of the x-coefficient.
to both sides.

3) Factor the trinomial.
or

Now take the square root of both sides.

x + 2 = +/-4 Subtract 2 from both sides.
x = 4 - 2 = 2 or
x = -4 -2 = -6
In this case the roots are real.

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To find the value of c that makes a perfect square, just take the coefficient of the x-term (4), divide it by 2 and then square the result.

So, the trinomial becomes:
Now this factors into: or

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Solve: for r.
Simply divide both sides by 2(pi)

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Take a look at the coefficient of the x^2 term. Notice that it is negative: This indicates that the parabola opens downward, therefore, the vertex must be a maximum.
The equation is in the form:

The vertex is located at (h, k) or at ( 7, -2)

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You can find the radius of a circle when you know its circumference because:
C = 2(pi)r or
The circumference (C) is equal to two times pi (pi = 3.14) times the radius (r).
Rewrite this equation for r: r = C/2pi
2pi is aproximately 2(3.14) = 6.28
C = 6.8 meters.
So, r = 2(3.14)(6.8 meters) = 42.7 meters.
The radius is 42.7 meters, approximately.

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This kind of problem can be solved by asking "how much of the job is done in 1 hour?"
Let's call the two people Sam and Eric, or S and E for short.
So, S and E together can do the whole job in 4 hours. This means that they can do 1/4 of the job in 1 hour.
Now, Sam can do the whole job in 6 hours. This means that Sam can do 1/6 of the job in 1 hour.
To find find out how much of the job Eric can do in 1 hour, simply subtract what Sam can in 1 hour from what Sam & Eric can do together in 1 hour.
(S + E) - S = E
1/4 - 1/6 = 3/12 - 2/12 = 1/12
So, Eric can do 1/12 of the job in 1 hour, which means it will take him 12 hours to do the whole job.

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Recall that perpendicular lines have slopes that are the negative reciprocal of each other.
What does this mean?
Well...if you know the slope of a line, then you automatically know the slope of all lines that are perpendicular to it. For example: If the slope of a line is m = (-3/5), then the slope of the perpendicular would be the negative reciprocal of (-3/5) which is +5/3
So, the given line has the slope, m = 1.
How do you know this? The given equation is in the slope-intercept form:
y = mx + b Compare this with:
y = x + 1 and you'll see that m (the slope) is 1.
Now your problem doesn't state which form of the equation is required so you could use the slope-intercept form or the point-slope form.
The point-slope form looks like this: y - y1 = m(x - x1) and this would be easier to write since you have all of the required information at hand and no calculations are necessary. The x1 and y1 are just the x and y values of the given point (1, 3)
Now, what is the slope of the new line? It's just the negative reciprocal of 1 which is -1.
Now we can write the equation of the line (in the point-slope form) that is perpendicular to given line (y = x + 1) and which passes through the given point (1, 3).
y - 3 = -1(x - 1)