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when they are done trading CD's, they each have an equal number of CD's.
since the total number of CD's is equal to 39, this means that they each have 13 CD's.
Let H = number of CD's that Ho has when they started trading.
Let L = number of CD's that Lori has when they started trading.
Let S = number of CD's that Sophia has when they started trading.
you need to tabulate all the changes from when they started to get their starting numbers.
For Ho:
H - 6 + 2 + 3 = H - 1
H - 1 = 13 means that H = 14 when they started.
For Lori:
L + 6 - 3 = L + 3
L + 3 = 13 means that L = 10 when they started.
For Sophia:
S - 2 = S - 2
S - 2 = 13 means that S = 15 when they started.
you get:
Ho had 14 when they started.
Sophia had 15 when they started.
Lori has 10 when they started.
This is summarized as:
Ho 14 Sophia 15 Lori 10
Ho gave 6 CDs to Lori.
Tally is now:
Ho 8 Sophia 15 Lori 16
Sophia gave 2 CD's to Ho
Tally is now:
Ho 10 Sophia 13 Lori 16
Lori gave 3 CD's to Ho
Tally is now:
Ho 13 Sophia 13 Lori 13
They're done and they all have the same number of CD's.

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population mean is equal to 500
population standard deviation is equal to 100
sample of 400 students has a sample mean of 511.
sample mean standard deviation = population standard deviation / square root of sample size.
this becomes;
sample mean standard deviation = 100 / sqrt(400) which is equal to 5.
99% confidence level is achieved within +/- 2.576 standard deviations.
2.576 standard deviations * 5 is equal to +/- 12.88
this means that, at 99% percent confidence level, the sample mean of 511 infers a population mean between 498.12 and 523.88
this also means that, if you took many samples of 400 students each, 99% of the time your sample mean would be between 498.12 and 523.88.
that's my understanding of what they are looking for here.

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without getting into the derivation of it, the formula for doubling 8000 assuming you are earning interest compounded at 2% per day is:
y = x * (1.02)^(log(2) / log(1.02))
if x is 8000, then y equals 16000
if x is 16000, then y equals 32000, etc.
since log(2) / log(1.02)) is equal to 35.00278878, you can use an exponent of:
35 rounded to the nearest integer.
35.003 rounded to the nearest 3 decimal places.
35.0028 rounded to the nearest 4 decimal places.
35.00279 rounded to the nearest 5 decimal places.
suppose you used 35 (that would be 35 days).
your formula would be:
8000 * 1.02^35 = 15999.11642
that's pretty close.
let's say that z = log(2)/log(1.02)
your formula becomes:
y = x*1.02^z
that's the formula you use.
you can make z as accurate as you want it.
the most accurate is to plug solve for z = log(2)/log(2.01) in your calculator and then stores the result in memory and then use that memory time you need z.
the daily interest rate is 2%
the money you make each day is compounded and calculated as the principal for the interest calculated for the next day, etc.

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let x = total income.
20% of that was spend for materials.
that means that 80% of that was spent for other purposes.
that means that:
.80*x = 78000
divided both sides of this equation by .8 to get:
x = 78000 / .8 = 97500
total income equals x which is equal to 97500
20% of that is spend for materials which makes expenditures for materials equal to .20 * 97500 = 19500.
80% of that is spent for other purposes which make expenditures for other purposes equal to .80 * 97500 = 78000
add the 2 pieces together to get 19500 + 78000 = 97500 which equals the total expenditure.

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3000 rads per hour is equal to 3000 / 3600 rads per second which is equal to .83... rads per second.
since there are 2*pi rads in one complete revolution of 360 degrees, then you have .83 / (2*pi) revolutions per second which comes out to be .1326291192 revolutions per second.
i believe this is what you had
your equation of 3000 rads/hour / 3600 / 2*pi) equals .1326291192 revolutions per second, same as i got above.
to see if this is good, let's go in reverse.
.1326291192 revolutions per second * 3600 = 477.4648293 revolutions per hour.
477.4648293 revolutions per hour * 2*pi rads per revolution equals 3000 rads per hour.
it all checks out so the answer appears to be good.
your answer, however, is not the same answer i god.
your answer was 1.3 revolutions per second.
you divided 3000 rads/hour find to get rads per second.
you divided that further by 2pi which was good.
only your arithmetic was off.
you got 1.32..... when you should have gotten .132.....
i see what you did.
you divided 3000 by 7200 and then multiplied that by pi.
that gets you 3000/7200 = .416666 which was then multiplied by pi to get 1.308999 which you then rounded to 1.3
you needed to divide 3000 by (7200*pi)
first you muiltiply 7200 * pi and then you divide 3000 by the result.

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addition and subtraction are 2 sides of the came coin.
5 - (+7) = -2
5 + (-7) = -2


y - 3x = -4 is your equation
apparently you want to solve for y.
if so, then this is what you do:
add 3x to both sides of this equation to get:
y = 3x - 4
you have just solved for y.
asuming you had wanted to solve for x instead, this is what you do:
equation is:
y - 3x = -4
subtract y from both sides of the equation to get:
-3x = -y - 4
divide both sides of the equation by -3 to get:
x = y/3 + 4/3
your 2 equations are:
y = 3x - 4
x = y/3 + 4/3
when x = 1:
y = 3x - 4 yields y = 3 - 4 = -1
when y = -1:
x = y/3 + 4/3 yields x = -1/3 + 4/3 = 3/3 = 1
the 2 equations are equivalent too each other.
y = equation solves for given values of x.
x = equation solves for given values of y.

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jane is 27
linda is 12
here's how:
let x = jane's age now.
let y = linda's age now.
7 years ago, jane was 4 times linda's age.
equation for that is:
x-7 = 4*(y-7)
simplify to get:
x-7 = 4y-28
solve for x to get:
x = 4y-21
when jane was 2 times linda's age, linda was 1/3 * jane's age.
this happened k years ago (we use letter k because we don't know how many years ago at this point in time).
equation for this is:
x-k = 2y
y-k = x/3
solve both these equations for k to get:
k = x-2y
k = y-x/3
since both equations are equal to k, you can set the expressions on the right of each equation equal to each other to get:
x-2y = y-x/3
multiply both sides of this equation by 3 to get:
3x-6y = 3y-x
add 6y to both sides of this equation and add x to both sides of this equation to get:
4x = 9y
divide both sides of this equation by 4 to get:
x = 9y/4
you now have 2 equations for x.
you have:
x = 4y-21
x = 9y/4
since both equations are equal to x, you can set the expressions on the right of each equation equal to each other to get:
4y-21 = 9y/4
multiply both sides of this equation by 4 to get:
16y-84 = 9y
subtract 9y from both sides of this equation and add 84 to both sides of this equation to get:
7y = 84
divide both sides of this equation by 7 to get:
y = 12
you should be able to use any of the equations to solve for x to get x = 27.
the one i used was the original one of:
x = 4y-21
this yielded x = 27
x = 9y/4 yielded x = 27 also.
you have x = 27 and y = 12
that's your answer.
to confirm, do the following from the original problem statement of:
when i was twice as old as you are now said Jane you were a third as old as i am now. Linda thought for a moment thats right she agreed and seven years ago you were 4 times as old as i was.
jane is 27
linda is 12.
7 years ago jane was 20 and linda was 5 which made jane 4 times as old as linda.
jane was twice as old as linda is now when jane was 24.
that was 3 years ago.
3 years ago, linda was 9 and and jane was 24 which made linda 1/3 as old as jane.
all portions of the problem statement check out so the answer is good.
CORRECTION TO PREVIOUS STATEMENT
3 years ago jane was 24 which is 2 times as old as linda is now (12).
3 years ago linda was 9 which is 1/3 times as old as jane is now (27).
answer is correct but last statement might have been confusing and so was corrected.

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start with:
sqrt(5x+5) = x+1
square both sides of this equation to get:
5x+5 = (x+1)^2
simplify to get:
5x+5 = x^2+2x+1
subtract 5x and subtract 5 from both sides of the equation to get:
0 = x^2-3x-4
this is the same as:
x^2-3x-4 = 0
factor this quadratic equation to get:
(x-4)*(x+1) = 0
solve for x to get:
x=4, x=-1
both of these solutions are good as you will see when you substitute for x in the original equation using each in turn.

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z = (x-m)/s
x = data value
m = mean
s = standard deviation.
in your problem:
x = 77
m = 82
s = 3
z = (77-82)/3 = -5/3 = -1.6666667 which rounds to -1.67
to find the area to the left of that z-score, you would go to the z-score table and look for -1.6 in the left hand column of that table which tells you what row you answer will be in. you then look for .07 in the top row of that table which tells you what column your answer will be in.
the intersection of column and row is the area to the left of the z-score of -1.67.
answer you should get if you look in the z-score table is .0475
this means that a z-score of -1.67 is better than 4.75% of the population z-score.
that's not very good.
here's the z-score table that i used.
http://lilt.ilstu.edu/dasacke/eco148/ztable.htm
this assumes that the distribution of scores is normal.

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this is a truth table type of problem as far as i can tell.
your truth table would look like this:

      p     q     ~q     p^~q
      T     T      F     F
      T     F      T     T
      F     T      F     F
      F     F      T     F

when q is true, ~q is false.
p^~q is true only if both p and ~q are true, otherwise it is false.

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if you knew the length of the aquarium, you could solve this easily.
assume the length is 3 feet.
your aquarium is 3 feet high 1 feet wide and 3 feet long.
the volume is 3*1*3 = 9 cubic feet.
now you put a rock in the water and the water rises by 1/2 of an inch.
1/2 of an inch is equal to 1/24 of a foot because there are 12 inches to a foot and therefore 24 1/2 inches to a foot.
so the water rises by 1/24 of a foot.
your volume is now 3+1/24 feet high, 1 feet wide, 3 feet long.
the new volume is 73/24 * 1 * 3 = 9.125 cubic feet.
the volume of the water that is displaced by the rock is equal to .125 cubic feet.
if you had a different length, you would get a different answer.
assume the length of the aquarium was 6 feet.
the original volume is now 3*1*6 = 18 cubic feet.
add 1/24 of a foot to the height and you get dimensions of 3+1/24 feet high, 1 feet wide, 6 feet long.
the new volume is 73/24 * 1 * 6 = 18.25 cubic feet.
this means the volume of the rock is equal to .25 cubic feet.
when the aquarium was 3 feet long, the volume of the water displaced by the rock was .125 cubic feet.
when the aquarium was 6 feet long, the volume of the water displaced by the rock was .25 cubic feet.
the difference was in the length of the aquarium.
since volume is equal to length * width * height, it stands to reason if the length of the aquarium is doubled, then the additional 1/2 inch in height has consumed double the volume.
1/2 inches equals 1/24 of a foot.
1/24 * 1 * 3 = 3/24
1/24 * 1 * 6 = 6/24
assuming that the rock raised the water by 1/2 inch in both cases, the same rock displaces double the volume when the length of the aquarium is doubled.
bottom line is you need the 3 dimensions to determine the volume displaced by the rock.
since you only have 2, you can't measure the volume of the rock until you know the length of the aquarium
note that if we are talking about the same rock, then the rock would have raised the height of the water only 1/4 of an inch if the length of the aquarium was 6 feet rather than 3.
that's because the volume of the rock doesn't change, so if the dimnensions of the aquarium change, then the change in height that the rock causes has to change.
rock causes a change in height of 1/2 inch when the length of the aquarium is 3 feet (assumed).
to keep the same volume, the rock would cause a change in height of 1/4 inch when the length of the aquarium is 6 feet.
73/24 * 1 * 3 = 9.125 from 3*1*3 = 9 caused a change in volume of .125 cubic feet.
145/48 * 1 * 6 = 18.125 from 3*1*6 = 18 causes a change in voolume of .125 cubic feet.
the change in volume is the same as it should be since the volume of the rock hasn't changed.
note that 73/24 is equal to 3 + 1/24 feet and 1/24 feet = 1/2 inch.
note that 145/48 i8s equal to 3 + 1/48 feet and 1/48 feet = 1/4 inch.
bottom line again:
you need the length of the aquarium to measure the volume of the rock.
that measurement is missing.

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hypotenuse is 5 inches.
1 leg is 2 inches.
from the pythagorean formulla, a^s + b^2 = c^2 here a and b are legs and c is the hypotenuse.
you get:
2^2 + b^2 = 5^2 which simplifies to":
4 + b^2 = 25
solve for b^2 to get:
b^2 = 25 - 4 = 21
b equals sqrt(21)
your triangle has legs of 2 and sqrt(21) and a hypotenuse of 5 where:
a = 2
b = sqrt(21)
c = 5
angle A is opposite side a
angle B is opposite side b
angle C is opposite side c
angle C is equal to 90 degrees.
now you use the trigonometric formulas for sine, cosine, and tangent to get:
sin(A) = opp/hyp = 2/5
sin(B) = opp/hyp = sqrt(21)/5
you can use your calculator to find A and B.
C is equal to 90 degrees.
A is equal to sin^-1(2/5) which is equal to 23.57817848 degrees.
B is equal to sin^-1(sqrt(21)/5) which is equal to 66.42182152
23.57817848 + 66.42182152 = 90 degrees as they should since the sum of the angles of a triangle = 180 degrees and 90 + 90 (angle C) is equal to 180.

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I = interior angle
E = exterior angle
the sum of I and E is equal to 180 (they are supplementary).
the ratio of I to E is 4:1
this means that I = 4E
since I + E = 180 and I = 4E, you get:
4E + E = 180 which simplifies to:
5E = 180 which results in:
E = 36 degrees
since the sum of the exterior angles of a polygon is equal to 360 degrees, you divide 360 by 36 to get the number of sides of the polygon.
360/36 = 10 which means the polygon has 10 sides which makes it a decagon.
working back from the knowledge that the polygon is a decagon, we can use the formula for the interior angle of a polygon to find the interior angle which should be equal to 180 - 36 = 144 degrees.
the formula for the interior angles of a polygon is:
I = 180(n-2)/n
since the decagon has 10 sides, this formula becomes:
I = 180(8)/10 which is equal to 1440/10 which is equal to 144 degrees.
Since I = 144, E = 180 - 144 = 36 degrees.
Decagon is confirmed.
One last confirmation:
ratio of I to E is 4:1
144:36 is the same as 4:1 because 4*36 = 144.
We're good all around.

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original investment = 450 with annual return of 11%.
add $800, some at 12% and some at 6%.
total return on entire $1250 equals 9%.
assume x at 12% and assume 800-x at 6%
sum of all investment is equal to 800 + 450 = 1250
total interest = 1250 * .09
your formula is:
.11*450 + .12*x + .06(800-x) = .09*1250
simplify to get:
.11*450 + .12*x + .06*800 - .06*x = .09*1250
simplify further to get:
49.5 + .12*x + 48 - .06*x = 112.5
combine like terms to get:
97.5 + .06*x = 112.5
subtract 97.5 from both sides of the equation to get:
.06*x = 112.5 - 97.5 which simplifies to:
.06*x = 15
divide both sides by .06 to get:
x = 15/.06 which simplifies to:
x = 250
when x = 250, 800 - x = 550
your formula of:
.11*450 + .12*x + .06(800-x) = .09*1250 becomes:
.11*450 + .12*250 + .06*550 = .09*1250 which simplifies to:
49.5 + 30 + 33 = 112.5
combine like terms to get:
112.5 = 112.5
this confirms the results of x = 250 and 800-x = 550 as good.
you need to invest 450 at 11% and 250 at 12% and 550 at 6% and you'll get a total combined interest rate of .09 on the total of 1250.

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m/n = 4 looks like it is direct.
multiply both sides by n and you get:
m = 4n
the 4 could represent k in the formula m = kn which is a direct variation formula.
if it was direct, then an increase in m would correleate to an increase in n.
let's see if that's true.
allow m = 1
you get 1/n = 4 and you solve for n to get n = 1/4
your ratio becomes 1/(1/4) = 4 which is true.
now allow m = 2
you get 2/n = 4 and you solve for n to get n = 2/4
your ratio becomes 2/(2/4) = 4 which is also true.
as m increased, n increases, so the variation is direct.
allow m to be 100.
the formua says that 100/n = 4
multiply both sides by n to get 100 = 4n
divide both sides by 4 to get n = 25
your ratio is 100/25 = 4 which is true
as m increased, n increased.
direct variation again.
direct is m = kn which leads to m/n = k
inverse is m = k/n which leads to mn = k
joint is m = kn/p which leads to k = mp/n
i'd go for direct.
combined appears to be a combination of any of the above.
here's one answer from the web:
http://answers.yahoo.com/question/index?qid=20070726203353AAgDenC

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original equation to be solved is:
9*log(5,x) = 25*log(x,5)
log(5,x) means log of x to the base of 5.
log(x,5) means log of 5 to the base of x.
convert everything to base of 5 using the base conversion formula of log(b,x) = log(c,x)/log(c,b).
original equation of 9*log(5,x) = 25*log(x,5) becomes:
9*log(5,x) = 25*log(5,5)/log(5,x)
since log(5,5) is equal to 1, this equation becomes:
9*log(5,x) = 25/log(5,x)
multiply both sides of this equation by log(5,x) and divide both sides of this equation by 9 to get:
log(5,x) * log(5,x) = 25/9
this can be written as:
(log(5,x))^2 = 25/9
take the square root of both sides of this equation to get:
log(5,x) = +/- sqrt(25/9) which becomes:
log(5,x) = +/- 5/3
your possible answers are:
log(5,x) = 5/3
log(5,x) = -5/3
from the basic definition of logarithms, you get:
log(b,x) = c if and only if b^c = x
apply this to your first possible answer and you get:
log(5,x) = 5/3 if and only if 5^(5/3) = x
that's one of your possible answers.
x = 5^(5/3)
apply this to your second possible answer and you get:
log(5,x) = -5/3 if and only if 5^(-5/3) = x
that's the other of your possible answers.
5^(5/3) resolves to 14.62008869
5^-(5/3) resolves to 1/5^(5/3) which resolves to .068399038
to see if these answers are good, you need to substitute them into your original equation by substituting them in place of x.
before you do that, however, you need to convert equation to base of 10 so you can solve it using your calculator.
your original equation of:
9*log(5,x) = 25*log(x,5) becomes:
9*LOG(x)/LOG(5) = 25*LOG(5)/LOG(x)
all you need to do now is substitute.
when x = 14.62008869, your formula becomes:
9*LOG(14.6200869)/LOG(5) = 25*LOG(5)/LOG(14.6200869)
solve this using your calculator and you get:
15 = 15, so x = 14.62008869 is good.
remember 14.62008869 is equivalent to 5^(5/3)
when x = .068399038, your formula becomes:
9*LOG(.068399038)/LOG(5) = 25*LOG(5)/LOG(.068399038)
solve this using your calculator and you get:
-15 = -15
remember .068399038 is equivalent to 5^-(5/3) which is equialent to 1/5^(5/3)
both answers are good.

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4 + x = 2
subtract 4 from both sides of the equation to get:
4 + x - 4 = 2 - 4
simplify to get:
x = -2
second problem:
6 + n = 4
subtract 6 from both sides of the equation to get:
6 + n - 6 = 4 - 6
simplify to get:
n = 4 - 6
simplify to get:
n = -2
-----
when you are solving equations, you can do the following:
add the same amount to both sides of the equation.
subtract the same amount from both sides of the equation.
multiply both sides of the equation by the same amount.
divide both sides of the equation by the same amount.
-----
doing any of these thing will not change the equality of the equation.
if the left side was equal to the right side before the operation, it will still be equal to the right side after the operation.
-----
you do these things to isolate the variable that you want to solve for on one side of the equation.
usually it's the left side of the equation but doesn't have to be.
once the variable has been completely isolated to one side of the equation it has been solved.
you were left with x = -2 which means that x was solved for.
you were left with n = -2 which means that n was solved for.

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multiply both sides of the first equation by -2 to get:
6x - 8y = -2
add the second equation to the first equation to get:
6x - 8y = -2
plus:
-6x + 6y = -3
equals:
-2y = -5
divide both sides by -2 to get:
y = 5/2
substitute for y in second equation to get:
-6x + 6(5/2) = -3
simplify to get:
-6x + 15 = -3
subtract 15 from both sides of the equation to get:
-6x = -18
divide both sides by -6 to get:
x = 3
you have:
x = 3
y = 5/2
substitute in first original equation to get:
-3x+4y=1 becomes:
-3(3) + 4(5/2) = 1 which simplifies to:
-9 + 20/2 = 1 which further simplifies to:
-9 + 10 = 1
subtract 10 from both sides of this equation to get:
-9 = -9 confirming that x = 3 and y = 5/2 are solutions to the system of equations.

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55 minutes is the average.
120 minutes is equal to 2 hours.
3% take over 120 minutes.
this looks like a binomial distribution type of problem.
binomial distribution formula is:
p(x) out of n = C(n,x) * p^x * q^(n-x)
p(x) = probability of number of events that are successful
n = total number
p = probability that commute time will be greater than 120 minutes (success)
q = probability that commute time will be less than or equal to 120 minutes (failure)
c(n,x) is the combination formula for x events out of n = n! / (x! * (n-x)!)

based on that:

i) If 50 staff are surveyed on a particular day, what is the probability that 5 will report the travel of over two hours?

p(5) out of 50 = C(50,5) * (.03)^5 * (.97)^45 = .013074229

(ii) If the company has 3000 staff, what is the expected number of staff who has the travel of over two hours?

expected number would be equal to n*p = 50 * .03 = 90

(iii) If the company has 3000 staff, what are the variance and standard deviation of the number of staff who has the travel of over two hours?

based on the binomial distribution formula:

variance is equal to n*p*q = 3000 * .03 * .97 = 87.3
standard deviation is equal to square root of variance is equal to 9.343446901

a reference is here:
http://www.stat.yale.edu/Courses/1997-98/101/binom.htm

i think these are the answers you are looking for but i'm not totally sure.
check them out and see if they agree with the answers you were supposed to have.

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i calculated as well and got the same answer after reducing the fractions to their simplest terms.
i'd say you did good.

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let x be your daughter's age now.
today you are 4 times your daughter's age.
that means you are 4*x years old.
7 years ago, you were 11 times her age.
7 years ago you were 4*x - 7 years old.
7 years ago, your daughter was x-7 years old.
you were then 11 times her age which means that:
4*x - 7 = 11*(x-7)
solve this equation to get the value of x.
4x - 7 = 11*(x-7)
simplify to get:
4x - 7 = 11x - 77
add 7 to both sides of this equation to get:
4x = 11x - 77 + 7 which simplifies to:
4x = 11x - 70
subtract 11x from both sides of this equation to get:
4x - 11x = -70 which simplifies to:
-7x = -70
divide both sides of this equation by -7 to get:
x = 10
since x is your daughter's age today, she is now 10 years old.
you are 4 times her age so you are 40.
7 years ago you were 33.
7 years ago she was 3
33 is 11 * 3
the value of x is good.
your aqe is now 40.

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the problem is:
(1/2+1/3)/(2/3-1/12)
you solve the expression within the inner set of parentheses first.
there is no set of inner parentheses here, so you solve the expression within the leftmost set of parentheses first and then solve the expression within the right set of parentheses next.
the expression within the first set of parentheses is:
(1/2+1/3)
working from left to right, you perform multiplies and divides first and then go back and do pluses and minuses.
this expression becomes:
(1/2) + (1/3)
since the division has already been simplified as far as it can go, you then do pluses and minuses.
in this case, you need to get a common denominator.
the common denominator is 6, so you multiply 1/2 by 3/3 and you multiply 1/3 by 2/2 to get:
(3/6) + (2/6) which you can now combine to become equal to:
(5/6)
you resolve the next set of parentheses next.
that set is:
(2/3 - 1/12)
since your multiplies and divides have been simplified as far as they can go, you then do your pluses and minuses.
you need to get a common denominator.
the smallest common denominator is 12.
you multiply 2/3 by 4/4 and you multiply 1/12 by (1/1) which means you don't have to multiply 1/12 by anything to get a denominator of 12 because it already has one.
you get:
(8/12 - 1/12) which becomes:
(7/12)
you are left with:
(5/6) / (7/12) which is the same as:
(5/6) * (12/7) which is the same as:
(5*12) / (6*7) which becomes:
60 / 42 which can be simplified to:
10/7
---------
in your scientific calculator, you would enter your expression as follows:
((1/2)+(2/3))/((2/3)-(1/12))
the calculator should give you an answer of 1.428571429
multiply this answer by 7 and you will get an answer of 10.
1.428571429 is equivalent to 10/7.
-------
the order of operations is:
powers and roots first scanning from left to right.
multiplies and divides next scanning from left to right.
adds and subtracts next scanning from left to right.
you repeat these steps within each set of parentheses working from the innermost set of parentheses first and then proceeding to the outer-most set of parentheses.
------
you might think that 1/2 hasn't been resolved as far as it will go and you would be correct.
1/2 is equal to .5
once you get .5 you have resolved the division as far as it will go.
you can, however, leave it in fraction form as we did above and assume it's been resolved as far as it will go because there's no other operation other than simplification that might be needed to reduced it even further.
-----
working with fractions is where the waters get muddy and you simply have to know and understand whether or not you need to work the operation any further.
in your problem, if i took a strict definition, i would have done the following:
(1/2+1/3)/(2/3-1/12) simplifies to:
(.5 + .333333333) / (.666666667 - .0833333333)
lots of endless fractional parts in this expression but the calculator can handle them well if you save the intermediate results into memory.
this now becomes:
.83333333333 / .583333333333
which becomes:
1.428571429 which is the decimal equivalent of 10/7.

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(-1,2) is equidistant with (4,4) and (x,4)
the y value has to be 4 which is on the line y = 4
the x value can be any value as long as the y value is 4.
there is only one place where it can be, however.
that place is when the distance between the point (-1,2) and the new value of (x,4) is equal to the distance between the point (-1,2) and (4,4).
if the distance is the same, they are equidistant.
so, the problem is find the new point on (x,4) that has the same distance between it and the point (-1,2) as the old distance between the point (4,4) and the point (-1,2).
the distance between the point (-1,2) and the point (4,4) is found using the following formula:
d = sqrt((4+1)^2 + (4-2)^2) which is simplified to:
d = sqrt(5^2 + 2^2) which is further simplified to:
d = sqrt(29)
that has to be the distance between the point (-1,2) and the new point of (x,4)
using the same formula, we get:
d = sqrt((x+1)^2 + (4-2)^2) which is simplified to:
d = sqrt((x+1)^2 + 2^2) which is further simplified to:
d = sqrt((x+1)^2 + 4)
since d is equal to sqrt(29), this formula becomes:
sqrt(29) = sqrt((x+1)^2 + 4)
if we square both sides of this equation, we get:
29 = (x+1)^2 = 4
subtract 4 from both sides of this equation to get:
(x+1)^2 = 25
take the square root of both sides of this equation to get:
x+1 = +/- 5
when x+1 = 5, we get x = 4
when x+1 = -5, we get x = -6
the answer is that x can be either 4 or -6.
since we already have the point (4,4), we then assume the other point is equal to (-6,4)
to confirm, we calculate the distance between the point (-1,2) and the point (-6,4).
that distance is calculated as:
d = sqrt((-6-(-1))^2 + (4-2)^2) which simplifies to:
d = sqrt((-6+1)^2 + (2)^2 which further simplifies to:
d = sqrt((-5)^2 + 4) which further simplifies to:
d = sqrt(25+4) which further simplifies to:
d = sqrt(29).
that's the same distance from the point (4,4) so we're good.
the new point is (-6,4).
if you create a right triangle from the point (-1,2) to the point (4,4), you will see that the absolute value of the difference in the x values and the y values is 5 for x and 2 for y.
if you create a right triangle from the point (-1,2) to the point (-6,4), you will see that the absolute value of the difference in the x values and the y values is 5 for x and 2 for y.
what you have is 2 right triangles with the same leg lengths which makes their hypotenuse the same.
the hypotenuse is equal to square root of (leg1^2 + leg2^2)
that's the formula of length of line segment equals sqrt ((x1-x2)^2 + (y1-y2)^2
a reference is shown below:
http://www.regentsprep.org/regents/math/geometry/GCG3/Ldistance.htm

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the formula tells you so.
a^2 - b^2 = (a-b) * (a+b)
that's the product of the sum and the difference of the same 2 terms and it is equal to the difference of each term squared.
if you multiply (a-b) * (a+b) you will get:
a^2 + ab - ab - b^2
the 2 middle terms cancel out and you are left with:
a^2 - b^2
it's nothing more than that.
it just puts into words what the formula is showing you.
with multiplication of (a+b) * (a-b) you are using the distributive property of multiplication.
(a-b) * (a+b) is equal to:
a * (a+b) - b * (a+b) which is equal to:
a*a + a*b - b*a - b*b which results in:
a^2 + ab - ab - b^2 which results in:
a^2 - b^2 because ab and -ab cancel out.

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the pyramid has a square base.
your dimensions are:
length of base is equal to 5 feet.
slant height is equal to 6 feet.
you want to find the surface area.
the slant height of the pyramid is the apothem which is the height of each triangular face.
you have all the information you need.
the area of the base is 5*5 = 25 square feet.
the area of one of the triangular faces of the pyramid is equal to 1/2 * 5 * 6 = 15.
multiply that by 4 and you get a total of 60 square feet for the triangular faces.
add all those together and you get a total surface area of 60 + 25 = 85 square feet.
a reference can be found here:
http://mathworld.wolfram.com/SlantHeight.html

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i did this by brute force because i didn't know any formula that would allow me to find it directly.
there may be one but i don't know what it is.
the brute force method used excel.
i simply multiplied 3456 by numbers starting from 1 and going up and then checked to see if i got a perfect square or not.
the answer is 6
3456 * 6 is a perfect square and multiplying by 6 is the smallest integer i could multiply 3456 by before getting a perfect square as the result.
here's the results of the brute force method.

3456	1	58.78775383	
3456	2	83.13843876	
3456	3	101.8233765	
3456	4	117.5755077	
3456	5	131.4534138	
3456	6	144                   this is a perfect square
3456	7	155.5377768	
3456	8	166.2768775	
3456	9	176.3632615	
3456	10	185.9032006	
3456	11	194.9769217	
3456	12	203.646753	
3456	13	211.9622608	
3456	14	219.9636334	
3456	15	227.6839915	
3456	16	235.1510153	
3456	17	242.3881185	
3456	18	249.4153163	
3456	19	256.249878	
3456	20	262.9068276	
3456	21	269.3993318	
3456	22	275.739007	
3456	23	281.936163	
3456	24	288                   this is a perfect square
3456	25	293.9387691

if there is a rule, i don't know it, as i said before.
it's interesting that the next perfect square is multiplying by a multiple of 6, specifically 4 * 6.
note that 6 by itself is not a perfect square even though 4 by itself is a perfect square.
no real relationship there that i can figure out.
if you're looking for a formula to determine the smallest multiple to get a perfect square i don't have it, but you do have your answer is that's all you wanted.
i did a search of the web and i did find a way to determine it.
the link i found is here:
http://answers.yahoo.com/question/index?qid=20090616105542AAKgSOq
basically, you break the number down into factors and each factor has to be a perfect square.
if there's a number that's not a perfect square, then that number is the number you have to multiply your original number by in order to get it to be a perfect square.
applying that concept to your number, we get the following:
3456 / 144 = 24 which means that:
3456 = 144 * 24
144 is a perfect square, but 24 is not.
we can, however break down 24 into a multiple of perfect squares.
24 = 2*12 = 3*8 = 4*6
4 is a perfect square, so we get:
3456 = 144*4*6
since 144*4 is equal to 576 and 576 is a perfect square (24*24), we can reduce this to:
3456 = 576*6
6 is not a perfect square, but if we multiply it by 6 we will get a perfect square, then that's the smallest we can get.
this is because:
6 is not a perfect square but it can be broken down into:
6 = 2*3 = 1*6 neither of which is a perfect square.
that makes 6 the smallest multiple of 3456 to get a perfect square.
brute force method confirms that:

6	1	2.449489743	
6	2	3.464101615	
6	3	4.242640687	
6	4	4.898979486	
6	5	5.477225575	
6	6	6                this is a perfrect square

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12 = r^y
what is the value of 12r?
what you are doing is multiplying both sides of this equation by r to get:
12r = r^y*r
by the laws of exponents
x^a*x^b = x^(a+b)
since:
12r = r^y*r is the same as:
12r = r^y*r^1, then this becomes:
12r = r^(y+1)
an example will show this to be true.
we start with:
12 = r^y
we'll take a value of r at random.
let's assume that r = 3
our equation becomes:
12 = 3^y
we can use logs to solve this to get:
y = 2.261859507
you can use your calculator to confirm that:
12 = 3^(2.261859507)
we now multiply both sides of this equation by 3 to get:
3*12 = 3*3^(2.261859507)
using the rule above that states that:
12r = r^(y+1), we get:
3*12 = 3^(3.261859507)
using our calculator again, we get:
36 = 36, confirming the rule.
an easier example to understand would be as follows:
suppose the problem were:
100 = 10^2
y = 2 in this case.
r = 10
100 = 10^2 is the same as:
100 = r^y because r = 10 and y = 2
we now multiply both sides of this equation by r which means we multiply both sides of this equation by 10
we get:
100*10 = 10*10^2 which is the same as:
1000 = 10^1*10^2 which is the same as:
1000 = 10^(1+2) which is the same as:
1000 = 10^3 which is the same as:
1000 = 1000
this confirms the rule.
summary of all of this is that:
12 = r^y becomes:
12r = r(y+1) when you multiply both sides of that equation by r.

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-3x^6-7x^3+4x+3/7
number of terms is 4
each term is separated from each other term by a + sign or a - sign.
separated out, this expression looks like this:
(-3x^6) - (7x^3) + (4x) + (3/7)
the highest exponent is 6 so the degree of the expression is 6.
the coefficients of the expression are:
-3, 7, 4
-3 is the coefficient of x^6
7 is the coefficient of x^3
4 is the coefficient of x
the constant term is (3/7)
note that the constant term can also be viewed as (3/7)*x^0 because x^0 is equivalent to 1 and (3/7)*1 is equivalent to (3/7).
if you look at it that way, then the constant term of (3/7) could also be considered a coefficient of the x^0.
i would just call it a constant term unless somebody you need to answer to is a stickler for details like that.
note that if there were 2 variables in the expression, then the degree of the expression would be the term with the highest sum of exponents.
example:
x^6y^3 term has a degree of 6+3 = 9
since you're only dealing with x, that fact doesn't come into the picture.
here's a reference on degrees of a polynomial.
http://www.basic-mathematics.com/degree-of-a-polynomial.html

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complex number are at the top of the food chain.
every number is a complex number
a complex number is composed of a real part and an imaginary part.
the real number system is the real part of a complex number with the imaginary part equal to 0.
the imaginary number system is the imaginary part of a complex number with the real part equal to 0.
the hierarchy of the number system shows this.
see:
http://zonalandeducation.com/mmts/miscellaneousMath/typesOfNumbers/typesOfNumbers.html
http://www.purplemath.com/modules/complex.htm

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z equals what he pays each month.
x equals the price he pays each month.
he buys 12 apples at a cost of x.
the equation becomes:
z = 12x
the price drops 40%.
the new price is now .6x per apple.
he still pays the same each month, so the new equation is:
z = y*.6x where y is the new number of apples he can now buy.
since both expressions on the right side of the equations equal z on the left side of the equations, you can set the expressions equal to each other to get:
12x = y*.6x
divide both sides of this equation by .6x and you get:
12x / .6x = y
x cancels out and you are left with:
y = 12/.6 which simplifies to:
y = 20
if the price drops 40% he can buy 20 apples each month rather than 12 apples each month.
assume he pays 1.00 per apple.
12 * 1.00 = 12.00 what he pays each month.
the price drops by 40% which means the price per apple is now 60 cents.
20 * 60 cents = 12.00 dollars.
he pays 12.00 for 12 apples at the old price and he pays 12.00 for 20 apples at the new price.

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3000 * .035 = 105 interest each year.
210 / 105 = 2
he can take as many as 2 years to pay the loan.
3000 * .035 * 2 = 210