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Each at the end of it all the wine is diluted by 50%. And since each each iteration of the process will dillute the wine by the same factor each time we know that the dilution factor z is z = (0.5)^(1/3) which turns out to be z = 0.7937.

And the dilution of the first iteration is simply the wine left divided by the entire volume.(B = Barrel) z = wineleft / B

And wine left is just B - 3 cups. z = (B-3cups)/B. z = 1- 3cups/B. rearranging for B:

zB = B - 3 c 

zB -B = -3c 

B*(z-1) = -3c

B= -3c/(z-1)

And putting in the values you get 

B = 14.512 cups

Man writing this out took me longer than actually calcualting it

You have a ratio of:

1/4 : 4  multiply both sides by 4 and you have:

1 : 16    Then multiply both sides by 25 and you have:

25 inches =16 x 25 = 400 miles - the actual distance.

Very nice, both guests.....this was a "stumper"....!!!!

9 1/4 days to 3 weeks. This is the same as:

9 1/4 days to 21 days. Now we can turn this into "hours" by multiplying both sides by 24:

222 : 504 hours. If you want smaller units(minutes in this case), just multiply both sides by 60 and so on.

14^15 mod 15  = 14. Use the calculator here. First calculate: 14^15 and then press "mod" key followed by 15. You should get 14 as the remainder.

Let N be the number of cups of liquid in the barrel.  This is pure wine to start with.

At any point in time, the concentration of wine is defined as the amount of pure wine in the barrel divided by N.

Let C1 be the concentration of wine in the barrel at the beginning of the first day before anything is removed that day.

Let C2 be the concentration of wine in the barrel at the beginning of the second day before anything is removed that day.

Let C3 be the concentration of wine in the barrel at the beginning of the third day before anything is removed that day.

We know that C1=1.

During the first day, three cups of liquid are removed from the barrel.  Those three cups contain 3 *C1 cups of pure wine.

That means the concentration of wine in the barrel at the end of the first day (and the beginning of the second day) is

C2=(N - 3*C1) / N

During the second day three more cups of liquid (this time it is diluted) are removed from the barrel.  Those three cups contain 3*C2 cups of pure wine.  That means the concentration of wine in the barrel at the end of the second day (and the beginning of the third day) is

C3 = (N - 3*C1 - 3*C2) / N

During the third day, three more cups of diluted liquid are removed from the barrel.  Those three cups contain 3*C3 cups of pure wine.  That makes the concentration of wine in the barrel drop to

C4 = (N – 3*C1 – 3*C2 – 3*C3) / N

which we are told should be 0.5

Plug the equation for C1 into the one for C2, then plug that in to the equation for C3 then plug that into the equation for C4 .

Set C4 equal to 0.5 and solve the resulting cubic to get 14.542 cups of wine in the barrel to start with.

Let us start with a barrel of wine.
We have: Wine amount(in cups)=W, and its concentration is 100%. Then he removes 3 cups from diluted wine, so we have: W-3 and its concentration would be:[W-3] / W, or [1 - 3/W]. Then he removes 3 more cups of diluted wine, so we have:W-3-3[1-3/W]=W-6+9/W, and the concentration will be={W-6+9/W} / W =[1-3/W]^2. Then he removes 3 more cups of diluted wine, so we have:W-6+9/W-3[1-3/W]^2=[W-9+27/W-27/W^2] and the concentration would be:[1-3/W]^3. So, now we know the final concentration=
[1-3/W]^3 = 1/2, or 50%. Will solve for W.
[1-3/W^3] =1/2,
= [1-3/W]=[(1/2)^1/3]. Then we have:
W =[3(2)^1/3] / [(2)^1/3 - 1]
W = ~14.5420 cups of wine in the barrel.
                                   

    6xy + 10x²y                    We can  " reverse " distribute this by factoring out a   2xy  .

=  2xy * (3 + 5x)

There will be          \(21\frac58-3\frac34\)          yards left.

\(\,21\frac58-3\frac34 \\~\\ =(21+\frac58)-(3+\frac34) \\~\\ =21+\frac58-3-\frac34 \\~\\ =21-3+\frac58-\frac34 \\~\\ =18+\frac58-\frac68 \\~\\ =18-\frac18 \\~\\ =17\frac78\)

Since the absolute value of anything is always positive, there is no way that the left side can be less than negative 8 .

So, there is no value of  x  that makes the inequality true.

The table top must look something like this:

So we can see that the overall dimensions of the table top will be

\((19\frac12+1\frac34+1\frac34)\,\text{inches}\qquad\text{by}\qquad(22+1\frac34+1\frac34)\,\text{inches} \\~\\ 23\,\text{inches}\qquad\text{by}\qquad 25\frac12\,\text{inches} \)

She drove a total of     39,333 – 38,948  =  385 miles

Assuming the tank was emptied on the 7th day,    it took  11  gallons of gas to drive  385  miles.

She can drive at a rate of 385 miles per 11 gallons of gas.

\(\frac{385\,\text{miles}}{11\,\text{gallons}}\,=\,\frac{385\,\text{miles}\,\div11}{11\,\text{gallons}\,\div11}\,=\,\frac{35\,\text{miles}}{1\,\text{gallons}}\)

That is, 35 miles per gallon

He earns....

$275   plus   $22   for each contract.

$275      +     $22     *       4

=  $275  +  $88

=  $363

Shift and the number 6 button above the letters. You can also press the x^y button on the calculator. Just make sure you input what x and y stands for because it can't solve it without a problem, lol.

If repeats are allowed(such as 111111), then you have:

18^6 = 34,012,224

If repeats are not allowed, then you have:

18P6 = 13,366,080

You would have to specify what "numbers" you are referring to....

(f/g)(x)   =  f(x)  / g(x)  =   [4x  - 5]  / [ 3x  ]

x + y  = 2     square both sides →  x^2 + 2xy + y^2  =  4

Since  xy  = 23,, then  2xy  = 46

So we have that

x^2 + 46 + y^2  = 4       subtract 46 from both sides

x^2 + y^2  =  -42

There are no "real" solutions, just "complex" ones:

x = 1 - i sqrt(22) ≈ 1.00000 - 4.69042 i and y = 1 + i sqrt(22) ≈ 1.00000 + 4.69042 i

x = 1 + i sqrt(22) ≈ 1.00000 + 4.69042 i and y = 1 - i sqrt(22) ≈ 1.00000 - 4.69042 i

So that: ( 1 + i sqrt(22))^2 + (1 - i sqrt(22))^2 

Simplify the following:
(i sqrt(22) + 1)^2 + (-(i sqrt(22)) + 1)^2

(i sqrt(22) + 1)^2 = 1 + i sqrt(22) + i sqrt(22) - 22 = 2 i sqrt(22) - 21:
2 i sqrt(22) - 21 + (-(i sqrt(22)) + 1)^2

(-(i sqrt(22)) + 1)^2 = 1 - i sqrt(22) - i sqrt(22) - 22 = -2 i sqrt(22) - 21:
-21 + 2 i sqrt(22) + -2 i sqrt(22) - 21

-21 + 2 i sqrt(22) - 21 - 2 i sqrt(22) = -42:
Answer: |  -42

Positive difference = 49m - 35n. where m and n are integers.

The greatest common factor of 49 and 35 is 7, so this is the smallest positive debt possible.

For example, if Devon owed Laura 7 dollars, he could give her 3 Harry Potters = 147 dollars, and she could give him 4 Twilights = 140 dollars in return.

Lol, nice one. It can't really be boring, it's just boring if you choose to leave it that way. I'm not saying start doing crazy things that might blow up the school or whatever, just try to make it fun.

Technically, they don't flag everything. Only the inappropriate things . . .

That's right, CPhill! There is no such thing as not doing anything, there is something always happening.