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door breaks down 7 times an hour and idle time on the door is 30 dollars per hour.
30 dollars per hour idle time equates to 30/60 = .5 dollars per minute.
bob can fix 10 doors an hour.
it takes him 60/10 = 6 minutes per door.
bill can fix 13 doors an hour.
it takes him 60/13 = 4.615384615 minutes per door.
since the door breaks down 7 times an hour, then:
with bob, the total down time per hour would be 7 * 6 = 42 minutes at a cost of .5 dollars per minute for a cost of 21 dollars per hour.
with bill, the total down time per hour would be 7 * 4.615384615 = 32.30769231 minutes at a cost of .5 dollars per minute for a cost of 16.15384615 dollars per hour.
total cost to the company for bob equals 21 + 21 = 42 dollars per hour.
total cost to the company for bill equals 27 + 16.15384615 = 43.15384615 per hour.
it's close, but as long as the door doesn't break down more than 7 times per hour, bob is the cheaper choice.
the equation for the total cost with bob is:
y = 21 + .5 * 6 * x
the equation for the total cost with bill is:
y = 27 + .5 * 4.615384615 * x
thee lines will intersect when x = 8.6666667 for a cost of 47 per hour.
when x is less than 8.6666667, bob is cheaper.
when x is greater than 8.6666667, bill is cheaper.
the graph is shown below:

the line that intersects the y-axis at 27 is bill's equation.
the line that intersects the y-axis at 21 is bob's equation.
the horizontal line at y = 47 shows where the break even point is.

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probability that all 4 tires will last at least 60,000 miles is .95^4 = .81450625
probability that all 4 tires will not last at least 60,000 miles is 1 - .81450625 = .18549375
probability that none of the tires will last at least 60,000 miles is .05^4 = 6.25 * 10^-6
the total probabilities are shown below:
p(4 last 60,000) = 4C0 * .95^4 * .05^0 = 1 * .95^4 * .05^0 = .81450625
p(3 last 60,000) = 4C1 * .95^3 * .05^1 = 4 * .95^3 * .05^1 = .l71475
p(2 last 60,000) = 4C2 * .95^2 * .05^2 = 6 * .95^2 * .05^2 = .0135375
P(1 last 60,000) = 4C3 * .95^1 * .05^3 = 4 * .95^1 * .05^3 = 4.75 * 10^-4
p(0 last 60,000) - 4C4 * .95^0 * .05^4 = 1 * .95^0 * .05^4 = 6.25 * 10^-6
add all these probabilities together and you get a total probability of 1.
the probability that 4 last 60,000 is.81450625.
the probability that 0 last 60,000 is 6.25 * 10^-6.
the probability that 4 do not last 60,000 is equal to 1 - .81450625 = .18549375.
this is the probability that 0 and 1 and 2 and 3 last 60,000 which is the total probability minus the probability that 4 last 60,000 miles.
you were asked:
a. (10 pts) What is the probability that all 4 tires will last at least 60,000 miles?
answer is.81450625
b. (10 pts) What is the probability that none of the tires will last at least 60,000 miles?
answer is 6.25 * 10^-6 which is equivalent to .00000625

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what kind of trouble?
are you talking about solving equations simultaneously though the addition method?
give me one of your problems and i'll show you how to solve it.

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7 men and 3 women.
2 selected at random.
both are the same gender.
let p(a) be the probability that they are both women.
let p(b) be the probability that they are both the same gender.
the probability that they are both men is 7/10 * 6/9 = 42/90
the probability that they are both women is 3/10 * 2/9 = 6/90
the probability that they are the same gender is equal to 42/90 + 6/90 = 48/90.
the probability that they are the same sex and are both women at the same time is the same as the probability that they are both women.
the formula for conditional probability is:
p(a|b) = p(a^b)/p(b)
this formula means:
probability of a given b equals the probability of a and b occurring at the same time divided by the probability of b.
probability of a and b occurring at the same time is the probability that they are the same gender and are both women at the same time which is equal to 6/90.
the probability of b occurring is the probability that they are both of the same gender which is equal to 48/90.
p(a|b) = p(a^b)/p(b) becomes:
p(a|b) = (6/90) / (48/90) = (6/90) * (90/48) = 6/48 = .125

this might be easier to see in a smaller problem.
assume 5 people and 2 are picked at random.
3 are men and 2 are women.
probability that 2 people picked will be men is 3/5 * 2/4 = 6/20.
probability that 2 people picked will be women is 2/5 * 1/4 = 2/20.
probability that 2 people picked will be the same gender is equal to 6/20 + 2/20 = 8/20.
probability that 2 people picked will be the same gender and will be women at the same time is equal to 2/20 which is the same as the probability that 2 women picked will be both women.
probability that 2 people picked will be 2 women given that the people picked are the same gender is equal to probability they are the same gender and they are both women at the same time divided by the probability they are both of the same gender.
this becomes (2/20) / (8/20) which equals (2/30) * (20/8) which is equal to (2/8) which i equal to .25.
there is a .25 probability that, if the 2 people picked are the same gender, that they will be women.
with 5 people, it's easier to show you how it works rather than with 10.
here's the breakdown.
assume men are numbers 1, 2, and 3
assume women are letters a and b.
the possible 2 person combinations are:
1,2 ***
1,3 ***
1,a
1,b
2,3 ***
2,a
2,b
3,a
3,b
a,b ***
the persons of the same gender have an asterisk next to their selection.
there are 4 out of 10 possible selections that are the same gender.
out of these 4, 1 is both women.
the probability that, given they are the same gender, that they will both be women is equal to 1 pair of women out of 4 total possible pairs of same gender.
the same formula applies to 10 people where 7 are men and 3 are women as shown above.

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your formula is:
bx + a(m-x) = abm/c
a = miles per gallon of the first car.
b = miles per gallon of the second car.
c = combined mileage of both cars.
m = total miles traveled by both cars.
x = total miles traveled by first car.
m-x = total miles traveled by second car.
number of combined gallons consumed is equal to m/c
number of gallons consumed by first car is x/a
number of gallons consumed by second car is (m-x)/b
let's see how it works.
assume both cars traveled 1500 miles.
assume first car gets 20 miles per gallon.
assume second car gets 35 miles per gallon.
assume the combined mileage is 25 miles per gallon.
you want to know how many gallons was consumed by first car and how many gallons was consumed by second car.
the formula is:
bx + a(m-x) = abm/c
you are given:
a = 20
b = 35
c = 25
m = 1500
substitute in the equation to get:
35x + 20(1500-x) = 20*35*1500/25
simplify this equation to get:
35x + 30000 - 20x = 42000
combine like terms to get:
15x + 30000 = 42000
subtract 30000 from both sides of this equation to get:
15x = 42000 - 30000
simplify to get:
15x = 12000
divide both sides of this equation by 15 to get:
x = 800
since m = 1500 and x = 800, this means that:
m-x = 700
you now have:
x = 800
m-x = 700
x is the number of miles traveled by the first car.
m-x is the number of miles traveled by the second car.
the number of gallons consumed by the first car is x/a which is equal to 800/20 which is equal to 40.
the number of gallons consumed by the second car is (m-x)/b which is equal to 700/35 which is equal to 20.
total gallons consumed by both cars is 40 + 20 = 60.
both cars traveled 1500 miles total at a combined efficiency of 25 miles per gallon.
combined gallons consumed is equal to m/c which is equal to 1500/25 which is equal to 60 gallons.
the calculations for gallons consumed are the same so the values for x and m-x must be good.
the formula works for any combined miles and combined miles per gallon as long as you know the miles per gallon of the first car and the miles per gallon of the second car as well.

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2x + 4 > 1/3
subtract 4 from both sides of the equation to get:
2x > 1/3 - 4
multiply 4 by 3/3 to get:
2x > 1/3 - 12/3
simplify to get:
2x > -11/3
divide both sides of the equation by 2 to get:
x > -11/(3*2)
simplify to get:
x > -11/6
since this can't be simplified any further, that' your answer.
confirm by substituting for x in your original equation.
pick a value of x > -11/6
pick a value for x that is greater than -11/6.
try -10/6
your original equation is:
2x + 4 > 1/3
substitute -10/6 for x to get:
2 * (-10/6) + 4 > 1/3
simplify to get:
-20/6 + 4 > 1/3
multiply 4 by 6/6 to get:
-20/6 + 24/6 > 1/3
combine like terms to get:
4/6 > 1/3
simplify to get:
2/3 > 1/3
since this equation is true, your value for x is probably good.
to make a complete test, you would need to test out when x = -11/6 and when x < -11/6.
both those tests should lead to a false equation (one that is not true).
for example:
when x = -11/6, the original equation of:
2x + 4 > 1/3 becomes:
2 * (-11/6) + 4 > 1/3
simplify to get:
-22/6 + 4 > 1/3
multiply 4 by 6/6 to get:
-22/6 + 24/6 > 1/3
combine like terms to get:
2/6 > 1/3
simplify to get:
1/3 > 1/3
this equation is false because 1/3 is equal to 1/3 and not greater than it.
the value of x = -11/6 is not valid.
this is correct because the solution to the equation said that x > -11/6, not x = to -11/6 and not x >= -11/6.

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your first equation is already in slope intercept form.
slope intercept form is y = mx + b
m is the slope and b is the y intercept.
the y intercept is the value of y when x = 0.
your second equation is in standard form.
standard form is ax + by = c
you need to convert the equation from standard form to slope intercept form before you do anything else.
start with 4x + ky = 7
subtract 4x from both sides of this equation to get:
ky = -4x + 7
divide both sides of the equation by k to get:
y = (-4/k)x + (7/k)
your 2 equations are now:
y = (-5/2)x + 2
y = (-4/k)x + (7/k)
if the lines are perpendicular, then their sloped are negative reciprocals of each other,.
the slope of the first equation is equal to (-5/2).
the negative reciprocal of (-5/2) is equal to (2/5)
since the line determined by your second equation is perpendicular to the line determined by your first equation, this means that:
(-4/k) = (2/5)
you can solve for k by doing the following:
equation is:
-4/k = 2/5
multiply both sides of this equation by k to get:
-4 = 2k/5
multiply both sides of this equation by 5/2 to get:
-4*5/2 = k
this makes k equal to -10
the slope of your second equation becomes:
slope = -4/k = -4/-10 = 4/10 = 2/5
2/5 is the negative reciprocal of -5/2 so we got the slope right.
your second equation becomes:
y = (2/5)x + 7/k
since k is equal to -10, your equation becomes:
y = (2/5)x - (7/10)
you can now graph these equations to see how they look.
the equations are:
y = -(5/2)x + 2
y = (2/5)x - (7/10)

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the square root of 596 squared is equal to 596.
the square root of anything squared is anything.
example:
the square root of 4^2 = 4
the square root of 6^2 = 6
you can prove this for yourself by squaring 4 and squaring 6
you get 16 and 36
square root of 16 = 4
square root of 36 = 6
in your problem you want to take the square root of 596^2.
that equals 596.
if you use your calculator and square 596 you'll get 355216
if you use your calculator and take the square root of 355216 you'll get 596.

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the formula is 16 * x^2 because the measurement was done in feet and seconds.
if the measurement was done in some other measure, like inches in half a second intervals, the factors would have been different.
the more generalized formula would be:
d = k * x^2 where:
k is the distance the object falls in 1 unit of time.
x is the unit of time.
for example:
assume the distance was in inches and the time in seconds.
the formula would then have become:
d = (16*12 = 192) * x^2 where:
192 is the distance in inches and x is the number of seconds.
assume the distance was in feet and the time interval was half a second.
the formula would then have become:
d = 4 * x^2 where:
4 is the number of feet the object fell in half a second and x is the number of half seconds.
the formula works the same either way.
the following chart should demonstrate that.
a is from the equation d = 16 * x^2 where 16 is the the number of feet in 1 second and x is the number of seconds.
b is from the equation d = 192 * x^2 where 192 is the number of inches in 1 second and x is the number of seconds.
c is from the equation d = 4 * x^2 where 4 is the number of feet in half a second and x is the number of half seconds.

seconds     half seconds        a              b             c
1               2               16             192           16
2               4               64             768           64
3               6               144            1728          144
15              30              3600           43200         3600

note that the results are the same regardless of the units used as long as you did the conversions correctly.
3600 feet * 12 inches per foot equals 43200 inches.
16 * 15^2 = 3600
4 * 30^2 = 3600
the formula is valid regardless of the units used.
the reason why the formula you are looking at is d = 16 * x^2 where 16 is the number of feet it dropped in 1 second and x is the number of seconds is because those are the units of measure that were used in galileo's observation.
from what i read, he may have actually used meters.
the formula would have been d = 4.9 * x^2 where 4.9 is the number of meters in 1 second.
4.9 meters is roughly equivalent to 16 feet which is why you see 16 feet.

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probabilities for service ratings are:

very difficult        .11
difficult             .16
average               .35
easy                  .28
very easy             .10

presumably these ratings are independent of each other, i.e. a laser printer can be difficult to service or it can be easy to service but it can't be both at the same time.
this makes p(a or b) equal to p(a) + p(b)
if they were not independent of each other, then p(a or b) would be equal to p(a) + p(b) - p(a and b)
as you may know, p(a and b) is equal to p(a) * p(b).
if you don't yet, you will soon enough.
anyway, assuming these are independent, then p(a or b) is equal to p(a) + p(b).
based on this, the answers to your questions are:

servicability rating                 calculation             probability
difficult or very difficult          .11 + .16                   .27
difficult or average or easy         .16 + .35 + .28             .79
average or better                    .35 + .28 + .10             .73

average or better presumes average or easy or very easy.

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the problem is:
-3x * (x-3)
you distribute as follows:
(-3x * x) + (-3x * -3)
you then simplify this to:
(-3x^2) + (9x) which becomes:
-3x^2 + 9x
your answer is:
-3x*x - -3x*3
this simplifies to:
-3x^2 + 9x
looks good to me.

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let a be one of the numbers.
let b be the other of the numbers.
your equation is a * b is equal to 0.
assume a is equal to 0 and be is equal to any other number except 0.
your equation becomes 0 * b = 0
assume b is equal to 0 and a is equal to any other number except 0.
your equation becomes a * 0 = 0
assume both a and b are equal to 0.
your equation becomes 0 * 0 = 0
your product will be equal to zero in all cases.
at least one of the numbers has to be equal to 0.
both can also be equal to 0.
the value of the number that is not equal to 0 can be anything because anything * 0 will always equal 0.

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let x = the width of the border.
the total area including the border is equal to 315 square feet.
the area of the border is equal to 117 square feet.
subtract 117 from 315 and you get 198 square feet for the garden.
the length of the garden will be equal to 21 - 2x.
the width of the garden will be equal to 15 - 2x.
since the area of the garden is equal to the length times the width, then the equation for the area of the garden will be:
(21 - 2x) * (15 - 2x) = 198 square feet.
we multiply these factors out to get:
4x^2 - 72x + 315 = 198
we subtract 198 from both sides of this equation to get:
4x^2 - 72x + 117 = 0
this doesn't factor well so we use the quadratic formula to get:
x = 16.19374728 or x = 1.806252715
x = 16.19374728 doesn't work because 21 - 2*16 is negative.
the answer has to be x = 1.806252715 or there is no solution.
we go back to our equation for the area of the garden.
it is:
(21 - 2x) * (15 - 2x) = 198 square feet.
substituting 1.806252715 for x gets us:
21 - 2*1.806252715) * (15 - 2*1.806252715) = 198
simplifying this equation gets us:
17.38749457 * 11.38749457 = 198
simplifying this further gets:
198 = 198.
this confirms the value for x is good.
we confirm even further by determining the area of the border.
that will be:
4 * x^2 plus 2 *(21 - 2x) * x plus 2 * (15-2x) * x
substituting for x in this equation gets us 117 square feet.
the area of the garden is 198 square feet.
the area of the border is 117 square feet.
the value of x = 1.806252715 feet.
x is the width of the border.
the length of the garden will be 21 - 2x = = 17.38749457
the width of the garden will be 15 - 2x = 11.38749457
the area of the garden is equal to (21-2x) * (15-2x) = 198 square feet.
the equation we solved using the quadratic formula is:
4x^2 - 72x + 117 = 0
in that equation:
a = 4
b = -72
c = 117
the quadratic formula is:
x =
your answer need to be to the nearest tenth of a foot.
that makes your answer:
the width of the border is equal to 1.8 feet.
the length of the garden is equal to 17.4 feet
the width of the garden is equal to 11.4 feet.

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76,222 is equivalent to:
70,000 plus 6,000 plus 200 plus 20 plus 2.

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there would be 6! = 6*5*4*3*2*1 = 720
if only 2 numbers (4 and 6), then the number of 2 digit numbers would be equal to 2! = 2 * 1 = 2 and the numbers would be 46, 64.
if only 3 numbers (4 and 6 2), then the number of 3 digit numbers would be equal to 3! = 3 * 2 * 1 = 6 and the numbers would be 462, 426, 642, 624, 246, 264.
720 numbers with 6 digits is much harder to show, but you can see that the formula applies to any number of digits.

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you problem states that y varies directly as x and inversely as z^2.
the formula for that would be:
y = kx/z^2
you got started right because this agrees with what you showed.
if x and y are each reduced by 20%, then you get:
x - .2x = .8x
z - .2z = .8z
you would replace x with .8x and you would replace z with .8z to get:
y = k(.8x)/(.8z)^2
you would simplify this to:
y = .8kx / .64z^2 which you can further simplify to:
y = 1.25kx/z^2
let's see how this would pan out.
assume x = 64 and assume z = 8 and assume k = 1.
that makes the math simple.
your original equation of:
y = kx/z^2 becomes:
y = 1*64/8^2 which becomes:
y = 64/64 which becomes:
y = 1.
now assume that x and y are both decreased by 20%.
your equation of y = kx/z^2 becomes y = 1*(.8x) / (.8z)^2 which becomes:
y = .8*64 / (.8*8)^2 which becomes:
y = 51.2 / 6.4^2 which becomes:
y = 51.2 / 20.96 which becomes:
y = 1.25
we said that your equation became:
y = 1.25kx/z^2
since your original x = 64 and your original y = 8, this equation becomes:
y = 1.25(64)/(8^2) which becomes:
y = 80/64 which becomes:
y = 1.25
the answer comes out the same so the equation is good.
your answer is:
y = 1.25kx/z^2
a good reference on this type of problem can be found here:
http://regentsprep.org/Regents/math/algtrig/ATE7/indexATE7.htm

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i'm not exactly sure what you're asking.
you may be asking:
7 * sqrt(t) = 360
in that case, you would divide both sides of that equation by 7 to get:
sqrt(t) = 360/7
you would then square both sides of this equation to get:
t = (360/7)^2 which is equivalent to:
t = 360^2 / 7^2 which would be equivalent to:
t = 129600 / 49.
that can't be simplified further so that's your anser.
in decimal form, that answer would be:
t = 2644.897959
if that's not what you were looking for, then send me an email with what you were really looking for in english.,
something like:
7 times the square root of (t) = 360 would be an english equivalent of what i thought the problem was.

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the ratio of horizontal pixels to vertical pixels is 4:3.
let x = the number of horizontal pixels.
let y = the number of vertical pixels.
the ratio of x to y is 4 to 3 which can be shown as:
x/y = 4/3
cross multiply to get:
3x = 4y
solve for y to get:
y = 3x/4
the total number of pixels is 10.1 million.
since the total number of pixels is the number of horizontal pixels times the number of vertical pixels, then your equation would be:
x * y = 10.1
since y = 3x/4, you can substitute for y in this equation to get:
x * 3x/4 = 10.1
simplify this equation to get:
3x^2/4 = 10.1
multiply both sides of this equation by 4 to get:
3x^2 = 40.4
divide both sides of this equation by 3 to get:
x^2 = 13.46666667
take the square root of both sides of this equation to get:
x = 3.669695719
since x * y = 10.1, substitute for x in this equation to get:
3.669695719 * y = 10.1
divide both sides of this equation by 3.669695719 to get:
y = 2.752271789
this suggest that:
the number of horizontal pixels across the screen is 3.669695719 million.
the number of vertical pixels across the screen is 2.752271789.
since the ratio of horizontal pixels to vertical pixels is 4/3, then that ratio should also be equal to 3.669695719 / 2.752271789.
your equation to see if that's true would be:
4/3 = 3.669695719 / 2.752271789.
you would cross multiply to get:
4 * 2.752271789 = 3 * 3.669695719
you would simplify that to get:
11.00908716 = 11.00908716 which suggest that the ratios are the same which means you probably have the right numbers of horizontal pixels and vertical pixels.
you would test that out by multiplying them together to get:
3.669695719 * 2.752271789 = 10.1.
all the numbers check out so that's your answer.

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Σ means "the sum of".
example:
the sum of (2 + 1) = 3
the sum of (0 + 1) = 1
the sum of (4 + 5) = 5
the sum of (2 + 1) = 3
usually that symbol is used to get the sum of a bunch of numbers in a list or table as shown below:

i                            value
1                            2
2                            0
3                            4
4                            2

i is the name of the subscript that identifies the position of each element in the list or table.
x is the name of the variable used for each element.
each value in the table is identified by the variable name followed by the subscript.
you would then identify each unique value in the table by the variable name followed by the subscript.
in this table, that would be shown as:
x[1] = 2
x[2] = 0
x[3] = 4
x[4] = 2
if you want to find the total value in the table, you would then be asked to provide the sum of x[i] for i = 1 to 4.
this would be shown as Σ (x[i]) for i = 1 to 4
when i = 1, x[i] = 2
when i = 2, x[i] = 0
when i = 3, x[i] = 4
when i = 4, x[i] = 2
the Σ (x[i]) for i = 1 to 4 would then be equal to 2 + 0 + 4 + 2 = 8.
judging by the way the question is posed, i suspect that all they want to know is the sum of (x + 1) when x is equal to the value shown.
your answer would, in that case, be:
A. Σ(2 + 1) = 3
B. Σ(0 + 1) = 1
C. Σ(4 + 1) = 5
D. Σ(2 + 1) = 3

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if you are looking for the shortest distance between each of these, then i believe the table should look like this:

	E	NNE	NE	SE	WNW
N	90	22.5	45	135	67.5
ENE	22.5	45	22.5	67.5	135
NW	135	67.5	90	180	22.5
WSW	157.5	135	157.5	112.5	45
SSW	112.5	180	157.5	67.5	90

in some cases you had the right distance, but it was the longer distance between them rather than the shorter distance.
the shorter distance always had to be less than 180.
i based my number on the following diagram:

each interval is equal to 22.5 degrees.
to calculate the distance between selected points, just count the number of dots
for the shortest distance between them to come up with the difference.
example:
WSW to ESE is 6 dots
6 * 22.5 = 135 degrees.
WSW is at 247.5 degrees
ESE is at 112.5 degrees
247,5 - 112.5 = 135 degrees.
another example:
NNW to NNE is 2 dots
2 * 12.5 = 45 degrees
337.5 - 22.5 = 315 degrees
this is actually the long way around because it is greater than 180 degrees.
to convert it to the shortest way around, subtract it from 360 degrees to get 45 degrees.
i saw only one distance where you actually calculated it wrong.
i believe that was SSW to SE.
you showed 247.5 and i calculated 67.5 the short way there and 292.5 the long way.
anyway, that's my view of it.
you can use the chart to double check your work as well.

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C = 15x + 54000
R = 18x
break even is when R = C
this means that:
18x = 15x + 54000
subtract 15x from both sides of this equation to get:
3x = 54000
divide both sides of this equation by 3 to get:
x = 18000
the breakeven point is when x = 18000
R = 18x = 18 * 18000 = 324000
C = 15x + 54000 = 15 * 18000 + 54000 = 270000b + 54000 = 324000

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if i understand this correctly, you got a 12 and an 18.
since the 12 is worth 60% and the 18 is worth 40%, then your weighted score would be 60% * 12 + 40% * 18 = 14.4 which would put you in the C14 category if they categorize you to the nearest integer.
it appears they are scoring from 0 to 20 where 0 = 0 and 20 = 100%
this means that the ratio of score units to grade percentage points is equal to 5.
this means that if you score a 12, then your grade percentage for the test was 12 * 5 = 60, and if you score an 18, then your grade percentage for the test was 18 * 5 = 90.
60% * 60 and 40% * 90 equals an overall grade percentage of 72.
based on the ratio, an overall grade percentage of 72 results in a score of 14.4 units because 72/5 = 14.4.
that's what i think anyway.
your overall score would be 14.4 which puts you closer to C14 than to B15 which would round you into the C14 category.
that's what i think anyway.

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root(3,135) / root(3,5) is equal to root(3,135/5)
this says that:
the cube root of 135 divided by the cube root of 5 is equal to the cube root of (135 divided by 5).
in radical form, this looks like:

you can confirm that this is correct by using your calculator to calculate each side of the equation.
= 3
= 3
they're the same.

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your graph i shown below:

this graph will repeat itself every 2*pi intervals on the x-axis.
this means that:
if you have a value at -pi/2, then you will have the same value at pi/2
if you have a value at -2p, then you will have the same value at 0 and you will have the same value at 2p.
the value of the equation at selected intervals is:

x value         y value
-2pi            -2/3
-pi             +2/3
0               -2/3
pi              +2/3
2pi             -2/3

this graph will have asymptotes at:
x = -3pi/2
x = -pi/2
x = pi/2
x = 3pi/2
these asymptotes are represented by the vertical lines in the following graph of the same equation.

they may not look totally vertical, but they're as vertical as i can make them.

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i'm not an educator, but this sounds like an awfully difficult problem to be giving to a 7 year old.

the problem is:
Madam five wants to buy some poultry from the market. The seller told her that a chicken and a duck cost $12, a duck and a turkey cost $20, a chicken and a turkey cost $16. How much does each poultry cost?

if you look at what's presented, you might be able to logically deduce as follows:

chicken and duck equal 12 dollars
turkey and duck equal 20 dollars
chicken and turkey equal 16 dollars

what you have is:
chicken and turkey equal 16 dollars
chicken and duck and turkey and duck equals chicken and turkey and 2 ducks and the total of that is equal to 12 dollars plus 20 dollars which equals 32 dollars.

since you know that chicken and turkey equals 16 dollars, then you can take away chicken and turkey from chicken and turkey and 2 ducks which leaves you with 2 ducks.

since chicken and turkey and 2 ducks equals 32 dollars and chicken and turkey equals 16 dollars, when you take away chicken and turkey you are taking away 16 dollars from 32 dollars and you are left with 16 dollars.

this means that 2 ducks equals 16 dollars which means that 1 duck is equal to 8 dollars.

now that you know that 1 duck is equal to 12 dollars, you can take chicken and duck equals 12 dollars and subtract duck and are left with 4 dollars.

similarly, you can take turkey and duck equal to 20 dollars and subtract 8 dollars to get turkey equals 12 dollars.

you get:
chicken equals 4 dollars
duck equals 8 dollars
turkey equals 12 dollars

checking your answer out to see if it's correct, you get:
chicken and duck costs 12.00
4 dollars for the chicken and 8 dollars for the duck totals 12.00 so that part is ok.
turkey and duck costs 20.00
8 dollars for the duck and 12.00 for the turkey totals 20.00 so that part is ok.
turkey and chicken costs 16.00
12 dollars for the turkey and 4 dollars for the chicken totals 16.00 so that part is ok.

everything checks out so the answer is good.

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i can solve this using trigonometry
i don't know how to solve it otherwise,
using trigonometry i would do the following:
the area of the larger circle is equal to 12*pi
since the area of a circle is equal to pi*r^2, this means that:
the radius of the circle = sqrt(12)
the radii of the big circle in the diagram are:
DB, DC, DA
the radii of the small circle in the diagram are:
DE, DF, DG
these radii are also the apothems of the triangle.
the apothems of the triangle intersect the sides of the triangle at a right angle.
this forms 8 right triangles.
they are:
DBF, DFC, DCG, DGA, DAE, DEB
the hypotenuse of each of these 8 triangles are the radii of the large circle.
this makes their length equal to sqrt(12)
the angle of each of these triangles with a vertex at the center of the circle is equal to 60 degrees.
using trigonometry, you can calculate the distance of the legs of these triangles that form the perimeter of the larger triangle which is an equilateral triangle and is labeled ABC.
using right triangle DBF:
the hypotenuse is DB with a length of sqrt(12)
the angle is BDF which is 60 degrees.
the sine of angle BDF is equal to opposite / hypotenuse which is equal to BF / BD
since BD is equal to sqrt(12), then we get:
sine (BDF) is equal to BF / sqrt(12)
multiply both sides of this equation by sqrt(12) to get:
BF = sqrt(12) * sine(BDF)
since angle BDF is 60 degrees, we get:
BF = sqrt(12) * sine (60) which becomes:
BF = 3
2 * 3 is equal to the length of one of the sides of the equilateral triangle.
3 * 2 * 3 is equal to the perimeter of the equilateral triangle.
that perimeter is equal to 18.
the diagram is shown below:

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based on the question, i'm assuming you want to know how many miles per hour is 124 kilometers per hour.
the conversion factor would be 1 kilometer = 0.621371192 miles
multiply 124 by .621371192 and you get 77.05002781 miles per hour.

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i'm assuming that your sister took 2000 out of her credit card and gave it to you as a loan.
then she had to pay off the 2000 at 22% annual interest rate per year.
the calculations would be as follows:
annual interest rate percent = 22%
annual interest rate = .22
monthly interest rate = .22/12 = .01833333333
in fractional form that would be equal to 22/1200.
assuming no other activity in the account, your balance at the end of each month would be as shown below:

month	interest	payment	    balance
0	0	        0	    $2,000.00
1	$36.67	        200	    $1,836.67
2	$33.67	        100	    $1,770.34
3	$32.46	        100	    $1,702.80
4	$31.22	        100	    $1,634.01
5	$29.96	        100	    $1,563.97

month 0 is when the loan is made.
month 1 is the end of the first month.
month 2 is the end of the second month.
etc.
you pay interest on the unpaid balance at the end of each month.
then you deduct the payment.
what is left is the remaining balance.
this same process occurs each month.
at the end of the fifth month, you have $1,563.97 remaining balance.
that's the amount of the loan that still remains to be paid.
that's the amount you owe your sister.
you already paid her a total of $600.00
the amount of the loan that was paid off was $2000.00 - $1,563.97 = $436.03
the difference between $600 and $436.03 was the interest that was paid off with the money you gave her.
pay her $1,563.97 and you should be square.

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the law of transitivity means this:
if a implies b and b implies c then a implies c.
the statement above boils down to:
if i tell you the time, then my life will be miserable.
this assumes the chain of logic follows through to the end.
the chain is as follows:
if i tell you the time then we start chatting (a implies b)
if we start chatting, then we'll meet at the local pub (b implies c)
if we meet at the local pub then we'll discuss my family (c implies d)
if we discuss my family then you'll find out that my daughter is available (d implies e)
if you find out that my daughter is available then you'll want to marry her (e implies f)
if you want to marry her then my life will be miserable. (f implies g).
the law is:
if a implies b and b implies c, then a implies c.
we now have a implies c
the law then states:
if a implies c and c implies d, then a implies d.
we now have a implies d.
the law then states:
if a implies d and d implies e, than a implies e.
we now have a implies e.
the law then states:
if a implies e and e implies f, then a implies f.
we now have a implies f.
the law then states:
if a implies f and f implies g, then a implies g.
we now have a implies g.
this means that:
if i tell you the time then i will be miserable by the law of transitivity.

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let x = the number.
the quotient of nine and the number becomes 9/x.
increasing by half of the number becomes + x/2.
put them together to get:
9/x + x/2
here's a reference you might find helpful.
http://www.purplemath.com/modules/translat.htm

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300 degrees is in the fourth quadrant.
the equivalent angle in the first quadrant would be 360 - 300 = 60 degrees.
this is a common angle that you can find the exact trigonometric functions for.
they are:
sin(60) = sqrt(3)/2
cos(60) = 1/2
tan(60) = sqrt(3)
cot(60) = sqrt(3)/3
sec(60) = 2
csc(60) = 2*sqrt(3)/3
that's the equivalent angle in the first quadrant where all trigonometric functions are positive.
when the angle is in the fourth quadrant, the signs of the trigonometric functions are as follows:
sine = negative
cosine = positive
tangent = negative
cotangent = negative
secant = positive
cosecant = negative
based on that, then:
sin(300) = - sqrt(3)/2
cos(300) = + 1/2
tan(300) = - sqrt(3)
cot(300) = - sqrt(3)/3
sec(300) = + 2
csc(300) = - 2*sqrt(3)/3
note that:
cotangent = 1 / tangent
secant = 1 / cosine
cosecant = 1 / sine