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the problem is:

ten times the sum of 30 and 15 is the same as the sum of a number and 12, what is the number?

let x be equal to the number.
10 times the sum of (30 + 15) is equal to 10 * 45 which is equal to 450.
this is the same as the sum of a number and 12.
sum of a number and 12 is equal to x + 12 because we just assigned x to be equal to the number.
your equation becomes
450 = x + 12
subtract 12 from both sides of this equation to get:
450 - 12 = x which becomes:
438 = x
your number should be 438 if we did this right.

to confirm, assume your number is 438 and go back to the original problem and and see if it holds true.

the original problem statement is:

ten times the sum of 30 and 15 is the same as the sum of a number and 12, what is the number?

ten times the sum of 30 and 15 is still equal to 450.
the sum of 438 and 12 is equal to 450.
they are both equal to 450 so the problem statement holds true when the number is assumed to be 438.

looks like you got it right on the second try.

ten times the sum of 30 and 15 means you take the sum of 30 and 15 first and then multiply that by ten.

the key is the statement "the sum of"
that implies the sum has to be done before the multiplication.

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i presume that cm equals centimeters and m equals meters.

you need to know the conversion factors first of all.

1 meter is equal to 10 centimeters

this means that 1 centimeters is equal to 1/10 = .1 meter.

1 hour is equal to 60 minutes
1 minute is equal to 60 seconds
1 hour is therefore equal to 60 * 60 = 3600 seconds.

this means that 1 second is equal to 1/3600 = .0002777778 of an hour.


you start with 360,000 centimeters per hour.

since 1 meters is equal to 10 centimeters, then you need to divide centimeters by 10 in order to get equivalent meters.


360,000 centimeters / 10 = 36,000 meters.

your 360,000 centimeters is equivalent to 36,000 meters.

since 1 hour is equal to 3600 seconds, you need to multiply 1 hour by 3600 to get 3600 seconds.

your 1 hour has been 3600 seconds.

360,000 centimeters per hour becomes 36,000 meters per 3600 seconds.

to find the number of meters per second you need to divide 36,000 by 3600 to get 10 meters per second.

you 360,000 centimeters per hour becomes 36,000 meters per 3600 seconds which becomes 10 meters per second.

that should be your answer.

360,000 centimeters per hour is equivalent to 10 meters per second.

to see if this is good, try to get back to centimeters per hour from this equation.

since 1 meter is equal to 10 centimeters, multiply 10 by 10 to get 100 centimeters per second.

10 meters per second becomes equivalent to 100 centimeters per second.

since there are 3600 seconds in an hour, multiply 100 centimeters per second by 3600 to get 360,000 centimeters per hour.

the conversion have been done successfully.

360,000 centimeters per hour is equivalent to 10 meters per second.

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the celsius temperature shows the freezing point at 0 degrees and the boiling point at 100 degrees.

the fahrenheit temperature shows the freezing point at 32 degrees and the boiling point at 212 degrees.

at the freezing point, the fahrenheit temperature is equal to the celsius temperature + 32 degrees.

if we let x equal to celsius temperature and we let y equal the fahrenheit temperature, and we assume a linear equation is suffiicient to model the relationship, then our formula will be:

y = mx + 32

y is the fahrenheit temperature and x is the celsius temperature.

when x is equal to 0, y is equal to 32 degrees.

that's the fahrenheit temperature when the celsius temperature is 0 degrees.

when x = 100, we know that y is equal to 212 degrees.

y = mx + b is the slope intercept form of a straight line.
m is the slope.
b is the y intercept.

we already found b = 32 because we know that when x is equal to 0, y is equal to 32.

that led to y = mx + 32

we now need to find m (the slope).

the slope is determined by the following equation:

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

(x1,y1) is one point on the line of the equation.
(x2,y2) is the other point on the line of the equation.

our first point is equal to (x1,y1) which is equal to (0,32).
that's the point where celsius (x) is equal to 0 and fahrenheit (y) is equal to 32 degrees.

our second point is equal to (x2,y2) which is equal to (100,212).
that's the point where celsius (x) is equal to 100 and fahrenheit (y) is equal to 212 degrees.

the slope is therefore equal to (y2 - y1) / (x2 - x1) which is equal to (212 - 32) / (100 - 0) which is equal to 180 / 100 which is equal to 18 / 10 which is equal to 9 / 5.

our slope is equal to 9/5.
our y intercept is equal to 32.

out equation is y = 9/5 * x + 32.

if you know the celsius temperature, then use this formula to find the equivalent fahrenheit formula.

if you know the the fahrenheit temperature, then the formula to find the celsius temperature is derived from this formula as follows:

start with y = 9/5 * x + 32
subtract 32 from both sides of this equation to get:
y - 32 = 9/5 * x
multiply both sides of this equation by (5/9) to get:
5/9 * (y - 32) = x
this is the equation to convert from fahrenheit to celsius.

to convert from celsius to fahrenheit, the equation is:
y = 9/5 * x + 32
to convert from fahrenheit to celsius, the equation is:
x = 5/9 * (y - 32)

to test this out, assume the celsius is 0 degrees.
to convert to fahrenheit, the formula becomes:
y = 9/5 * 0 + 32 which results in y = 32
to convert 32 degrees fahrenheit to celsius, the formula becomes:
x = 5/9 * (y - 32) which results in x = 5/9 * 0 which results in x = 0.

on the other end, assume the celsius is 212 degrees.
to convert to fahrenheit, the formula becomes:
y = 9/5 * 100 + 32 which results in y = 180 + 32 which results in y = 212.
to convert 22 degrees fahrenheit to celsius, the formula becomes:
x = 5/9 * (212 - 32) which results in x = 5/9 * 180) which results in x = 100.

the formula works at the extremes that we used to derive the formula in the first place.

it will work in between as well.

bear in mind that the celsius to fahrenheit conversion is not necessarily a straight line function, so the estimates in between will be close to the actual temperature but not exactly on.

for example:

assume celsius is 50 degrees.
fahrenheit is equal to 9/5 * 50 + 32 which is equal to 122 degrees.
the actual fahrenheit temperature will be near 122 degrees but not necessarily right at 122 degrees.
the formula will give you an approximation that is close but will not necessarily be right on.

as far as the equation you showed up top in your problem, i don't understand what that has to do with celsius or fahrenheit.

the equation is:

x+10y=2, for y

it appears you want to solve for y.

here's how you would do that:

x + 10y = 2
subtract x from both sides of the equation to get:
10y = -x + 2
divide both sides of the equation by 10 to get:
y = (1/10) * -x + (1/10) * 2
this is equivalent to:
y = -(1/10)x + (1/5)

your original equation is x + 10y = 2
after you solved for y, your final equation is y = (-1/10)x + (1/5)

take any value for x and solve for y in the second equation.
use those values for x and y in the first equation and you'll see that it becomes a true equation which will confirm the final equation as being a good one.

for example:

i assumed x was equal to 32.
i chose this number at random.
i substituted for the value of x in the second equation to get;
y = (-1/10) * 32 + (1/5) to get y = -3
i substituted for x and y in the first equation to get:
x + 10 y = 2 becomes 32 + (10 * -3) = 2 which becomes:
32 - 30 = 2 which becomes 2 = 2
this equation is true so the value of 32 for x and -3 for y is good.
the equation is true means that the values of x and y that we used are solutions to the equation.

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your son is 20 years old on august 31.
you are 41 on february 5th.
let's try to simplify this and assume you were both born on the same day.
today you are 41
today he is 20
our assumption is that in x years your son will be half your age.
the formula for this would be:
(x + 20) = 1/2 (x + 41)
we need to solve for x.
multiply both sides of this equation by 2 to get rid of the fractionh.
you get:
2 * (x + 20) = (x + 41)
simplify this equation to get:
2x + 40 = x + 41
subtract x from both sides of this equation to get:
x + 40 = 41
subtract 40 from both sides of this equation to get:
x = 1
your son will be 1/2 your age in 1 year.
he will be 21
you will be 42
now we get to birthdays.
his birthday is august 31.
your birthday is february 5.
he will be 21 on august 31 of 2012
you will be 42 on february 5 of 2012.
for a period of time, you will be 42 and he will still be 20.
on august 31 of 2012 he will be 21 and be half your age.
this will continue until february 5 of 2013 when you become 43 and he will still be 21.
so, from august 31 of 2012 until february 5 of 2013, your son will be half your age.
this assumes the smallest increment you want to get down to is years.
if you want to find the exact month or the exact day when your son will be half your age then the formula needs to be revised to account for that.
i won't get into that unless you ask, but the exact day and hour and minute and second can be determined by use of the same formula modified for the increments involved.

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presumably these are all part of the same line.

you have the line of af equal to ab + bc + cd + de + ef

the line would look like this:

         ---------------------------------------------------
         a         b         c         d         e         f

you are given that the line segment ac is equal to df.
you are also given that the line segment bc is equal to ef

that would look like this:

         ---------------------------------------------------------
         a         b         c               d         e         f



         ---------------------               ---------------------
         a                   c     equals    d                   f


                   -----------                         -----------
                   b         c         equals          e         f

you are given that ac = df
you know that ac = ab + bc
you know that df = de + ef

this means that ab + bc = de + ef by substitution of equals.

you are given that bc = ef

this means that you can substitute bc for ef or you can substitute ef for bc in any equation.

the equation of interest is:

ab + bc = de + ef

substitute bc for ef to get:

ab + bc = de + bc

subtract bc from both sides of this equation to get:

ab = de

you can do this because of the basic algebraic property that says:

if a + c = b + c then a = b

this follows from the fact that if you subtract the same quantity from both sides of an equality, then the equality is preserved.

you are left with ab = de which is what you wanted to prove so you are done.

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you have:
(3r-2) / r = 7/8
multiply both sides of this equation by r to get:
(3r-2) = r * (7/8)
multiply both sides of this equation by 8 to get:
8 * (3r - 2) = r * 7
simplify this equation to get:
24r - 16 = 7r
add 16 to both sides of this equation and subtract 7r from both sides of this equation to get:
24r - 7r = 16
simplify to get:
17r = 16
divide both sides of this equation by 17 to get:
r = 16/17
substitute 16/17 for r in your original equation to see if the equation holds true.

your original equation is:
(3r-2) / r = 7/8
substitute for r to get:
(3 * (16/17) - 2) / (16/17) = 7/8
multiply both sides of this equation by (16/17) to get:
(3 * (16/17) - 2) = (7/8) * (16/17)
simplify this equation to get:
(48/17 - 2) = (112/136)
simplify further to get:
(48/17 - 34/17) = (112/136)
simplify further to get:
(14/17) = (112/136)
112/136 is equivalent to 14/17 so you get:
(14/17) = (14/17) which is true, confirming the value of (16/17) for r is good.

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your equation is:
9x + 3 = 9x + b
subtract 9x from both sides of this equation to get:
b = 3
in order for this equation to have a solution, the value of b has to be equal to 3.
any other value of b will result in no solution.

if you set b equal to 3, the equation reduces to 3 = 3 which is true, confirming the value of 3 for b is a solution.

if you set b to any other value, like 2, the equation reduces to 3 = 2 which is false, confirming the value of 2 for b is not a solution.

try any value for b other than 3 and you will see that the equation does not have a solution.

the fact that you have 9x + 3 = 9x + b should be your clue that b has to be equal to 3.

the 9x terms will always be equal regardless of the value of x.
that means the other term in each expression has to also be equal.
that can only happen when b = 3.

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one side of the white board is x + y inches.
the other side of the white board is x^2 - 3xy + 7y^2 inches.
the area of the whiteboard is equal to length * width.
let W equal the width and set W equal to x + y inches.
let L equal the length and set L equal to x^2 - 3xy + 7y^2 inches.
let A equal the area which is equal to L * W.
you get:
A = L * W = (x + y) * (x^2 - 3xy + 7y^2) square inches.
you need to multiply these factors together.
(x + y) * (x^2 - 3xy + 7y^2) will be equal to:
(x * (x^2 - 3xy + 7y^2)) + (y * (x^2 - 3xy + 7y^2)) which becomes equal to:
(x^3 - 3x^2y + 7xy^2) + (yx^2 - 3xy^2 + 7y^3)
remove the parentheses and reorder the variables in each term so that the x variable is first and the y variable is second so you can see the like terms easier.
you get:
x^3 - 3x^2y + 7xy^2 + x^2y - 3xy^2 + 7y^3
now reorder the terms so that the like terms are next to each other.
you get:
x^3 [- 3x^2y + x^2y] [+ 7xy^2 - 3xy^2] + 7y^3
i enclosed the like terms in brackets so you can see them better.
you would not ordinarily do that.
i was careful to include the operation signs so you can see whether they needed to be added together or subtracted from each other.
now combine like terms to get:
x^3 -2x^2y + 4xy^2 + 7y^3
that's your area.
it is expressed in square inches.
you would say:
A = x^3 - 2x^2y + 4xy^2 + 7y^3 square inches.
if you want to confirm that you did the calculations correctly, then the easiest way is to provide some random values for x and y and use them to solve the original equation and the final equation after simplification.
the original equation is:
A = (x + y) * (x^2 - 3xy + 7y^2) square inches.
the final equation is:
A = x^3 - 2x^2y + 4xy^2 + 7y^3 square inches.
i did the check using x = 2 and y = 3.
i got an answer of 245 both times so i'm reasonably confident in did the calculations and the simplifications correctly.

the multiplication uses the distributive property of mathematics.
(a + b) * (c + d) would be equal to:
a * (c + d) + b * (c + d)
a + b) * (c + d + e) would be equal to:
a * (c + d + e) + b * (c + d + e)

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p(heads) = .5
p(tails) = .5
since the probability of success and failure are the same, then the probability for each occurrence is the same.
that probability is .5*.5*.5 = .125
what differs is the ways in which each occurrence can happen.
3 heads can only happen in 1 way.
3 tails can only happen in 1 way.
1 head and 2 tails can happen in 3 ways.
they are:
htt
tht
tth
2 heads and 1 tail can happen in 3 ways.
they are:
hht
hth
thh
the formula for determining the ways they can happen is the combination formula.
that formula is
nCx = n! / (x! * (n-x)!))
you are drawing 3 coins and you want exactly 1 head.
the formula for that becomes:
n = 3
x = 1
nCx = 3! / (1! * 2!) = 3! / 2! = (3 * 2!) / 2! = 3
3! means 3 * 2 * 1
2! means 2 * 1
1! means 1
the total probability of all possible occurrences has to equal to 1.
for your 3 coins, here's how the probabilities would be calculated.
probability of 0 heads would be:
3C0 * (.5)^3 = 3! / (0! * 3!) * .125 = 3! / 3! * .125 = 1 * .125 = .125
0! is always equal to 1
1! is always equal to 1
probability of 1 head would be:
3C1 * (.5)^3 = 3! / (1! * 2!) * .125 = 3! / 2! * .125 = 3 * .125 = .375
probability of 2 heads would be:
3C2 * (.5)^2 = 3! / (2! * 1!) * .125 = 3! / 2! * .125 = 3 * .125 = .375
probability of 3 heads would be:
3C3 * (.5)^2 = 3! / (3! * 0!) * .125 = 3! / 3! * .125 = 1 * .125 = .125
your total probability is:
.125 + .125 + .375 + .375 = .250 + .750 = 1.0
this is what it should be so we calculated correctly.

the general formula is:
nCx * p^x * q^(n-x)
p is the probability of success
q is the probability of failure.

in your experiment, the probability of success and the probability of failure were the same so it didn't matter.

in other case, they will be different.

suppose the coin was weighted so the probability of heads was .7 and the probability of tails was .3

the same problem would give you the following probabilities.
probability of exactly 0 heads = 1 * .7^0 * .3^3 = .027
probability of exactly 1 head = 3 * .7^1 * .3^2 = .189
probability of exactly 2 heads = 3 * .7^2 * .3^1 = .441
probability of exactly 3 heads = 1 * .7^3 = .343

the total probability is still equal to 1.
the number of ways each occurrence can happen is the same.
the probability of each occurrence, however, is different.

the answer to your questions from the data for equal probabilities is:

If I drop three coins on a table, what is the probability of finding
1. exactly 2 heads
exactly 2 heads would be .375
2. at most 1 head
at most 1 head would be exactly 0 and exactly 1 which would be .125 + .375 = .5
3. at least 2 heads
at least 2 heads would be exactly 2 and exactly 3 which would be .375 and .125 = .5
4. at least 1 head
at least 1 head would be exactly 1 and exactly 2 and exactly 3 which would be .375 + .375 + .125 = .875
not that this is also 1 minus exactly 0 which would be equal to 1 - .125 = .875.

.125 = 1/8
.375 = 3/8

i think you tallied the possibilities correctly but may not have tallied the probabilities correctly.

doing it the way you did it, i got:
HHH
HHT
HTH
THH
HTT
THT
TTH
TTT
this part seems to be ok.
looking at this set of data, however, i get the following.
probability of exactly 0 heads is 1/8 (TTT)
probability of exactly 1 head is 3/8 (HTT, THT, TTH)
probability of exactly 2 heads is 3/8 (HHT, HTH, THH)
probability of exactly 3 heads is 1/8 (HHH)

the answers to the questions are:
1. exactly 2 heads = 3/8
2. at most 1 head = 4/8
3. at least 2 heads = 4/8
4. at least 1 head = 7/8

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ANSWER TO PROBLEM NUMBER 1

1. If n(A) = 8, then how many proper subsets does A have?

n(A) = 8 says that the set A has 8 elements.

Here's a definition of elements of a set from Wikipedia.

Writing A = {1, 2, 3, 4 },means that the elements of the set A are the numbers 1, 2, 3 and 4. Sets of elements of A, for example {1, 2}, are subsets of A.
Sets can themselves be elements. For example consider the set B = {1, 2, {3, 4}}. The elements of B are not 1, 2, 3, and 4. Rather, there are only three elements of B, namely the numbers 1 and 2, and the set {3, 4}.
The elements of a set can be anything. For example, C = { red, green, blue }, is the set whose elements are the colors red, green and blue.

Based on this definition, then n(A) = 8 means that the set A consists of 8 subsets.

These would need to be proper subsets since the set itself is not counted as an element of the set.

it was difficult to find an answer that specifically addressed this question so the answer i gave you is pieced together from what i was able to find on the net.

the main source of the information that helped answer the question was wikipedia.

i believe it's true based on the definitions of elements of a set and the excerpt above about how to count elements of a set.

they did not specifically address proper or improper subsets but it is clear that the set itself is not counted as an element of the set, so all elements of the set have to be proper subsets.

an element can be one element or it can be a subset.

an element itself is also considered a subset.

here's the link from wikipedia that clued me in to the fact that a subset is considered an element of a set. you count the set as an element and you do not count the elements within that set.

http://en.wikipedia.org/wiki/Element_%28mathematics%29

the answer to your question is that n(A) tells you the number of proper subsets that are contain in the set A.

this is to the best of my knowledge based on what i learned from wikipedia.

ANSWER TO PROBLEM 2

2. At a local college, a survey was taken to determine where students studied on campus. Of 180 students surveyed, it was determined that:
89 studied in the library
76 studied in the student center
55 studied in both places
Of those interviewed,
a. how many studied in only the library?
b. how many studied in only the student center?
c. how many did not study in either location?

to answer this question, you need to clean up the data set.
that means taking out elements that are in multiple sets and creating a new set that contains the multiples onlyh.

you have 89 studied in the library and 55 studied in both.
remove 55 from 89 to get 34 studied in the library only.

you have 76 studied in the student center and 55 studied in both.
remove 55 from 76 to get 21 studied in the student center only.

add 55 that studied in both.

you have:
34 studied in the library only.
21 studied in the student center only.
55 studied in both.
that's a total of 110
since there were 180 students surveyed, then 180 - 110 = 70 of them studied in neither the library nor the student center.

note that if you did not remove the students that studied both from each category, then the total of students that studied in the library or the student center or both would be equal to 89 + 76 - 55 = 110.

the 55 would have had to be subtracted from the total in order to remove double counting of the 55 who studied in both and were included in both.

by removing them from each category up front, the double counting was avoided and you came up with the same result.

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i was not able to come up with an example on how the score is actually calculated, but i was able to find a link to a place that might be able to help you.
http://www.schools.nsw.edu.au/learning/k-6assessments/shsplacement/ss_marksexplain.php

this assessment includes scores for:
reading
writing
math
general ability

i think you only had 3 mentions (english, math, general ability).

never the less, i think i'm in the right place.

i sent them an email requesting an example of how the calculation would be performed.

you should probably contact them as well for information if you can't find it any other way.

send me an email through algebra.com so if i do get some information from them, i can relay it back to you.

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if the interest is compounded monthly (normal bank practice), then $720.00 would be equal to $847.2093772in 3 years.
if the interest is compounded annually, then $720.00 would be equal to $844.1213401 in 3 years.
if the interest was compounded daily, then $720.00 would be equal to $847.4908701 in 3 years.

if the interest was compounded continuously, then $720.00 would be equal to $847.50

continuous compounding is given by the formula:
F = P * e^(i*y)

annual, monthly, daily compounding is given by the formula:

F = P * (1+(i/c)^(y*c)

P is the present amount
F is the future amount
i is the annual interest rate
y is the number of year
c is the compounding period.
e is the scientific constant of 2.718281828

example:

for monthly compounding, the variables become:

P = $725.00
F = F
i = .0525
y = 3
c = 12

the formula of F = P * (1+(i/c))^(y*c) becomes:

F = $724.00 * (1 + (.0525/12))^(*3*12) which becomes:

F = $724.00 * (1.004375)^36 which becomes:

F = $847.2093772

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you are looking at:
sqrt(x) * sqrt(2x) * sqrt(10x^5) i believe.
you want to multiply these all together if i understand you correctly in order to simplify the expression.
in general, sqrt(a) times sqrt(b) equals sqrt(a * b)
that means that:
sqrt(x) * sqrt(2x) * sqrt(10x^5) = sqrt(x * 2x * 10x^5) which becomes:
sqrt(20x^7)
this is because x * 2 * x * 10 * x^5 is equivalent to:
x * x * x^5 * 2 * 10 which is equivalent to:
x^2 * x^5 * 20 which is equivalent to:
x^7 * 20 which is equivalent to:
20 * x^7 which can be shown as:
20x^7
the ability to make x * x * 2 * x = 2 * x * x * x is the law of commutativity.
this law says that a * b is the same as b * a
the ability to make x * x = x^2 is the algebraic law of exponents that says that a^b * a^c is equal to a^(b + c)
you need to simplify this and you should be done.
sqrt(20x^7) is equal to sqrt(4*5*x^2*x^2*x^2*x)
you can bring out the 4 from under the square root sign because the square root of 4 is 2.
the expression becomes 2 * sqrt(5*x^2*x^2*x^2*x)
you can bring out 3 of the x^2 terms from under the square root sing because the square root of x^2 is equal to x.
the expression becomes 2 * x * x * x * sqrt(5x)
since x * x * x = x^3, then your expression becomes:
2 * x^3 * sqrt(x)
that should be as simplified as it gets.
the final expression is 2x^3sqrt(x) which looks like this:

if you want to show that the final expression is equivalent to the original expression, then just pick a random value for x and solve both the original expression and the final expression.
if they give you the same answer then you did good.
the original expression is:
sqrt(x) * sqrt(2x) * sqrt(10x^5)
the final expression is:
2 * x^3 * sqrt(x)
we'll pick a random value for x of 7 and use the calculator to solve the original expression and the final expression.l
the original expression becomes 4058.430731
the final expression becomes 4058.430731
i get the same answer so the simplification was done correctly.
here's a couple of references that might help you to understand this better.
http://www.bymath.com/studyguide/ari/ari4.html
http://www.mathsisfun.com/algebra/exponent-laws.html
http://www.school-for-champions.com/algebra/properties_add_multiply.htm
http://www.wtamu.edu/academic/anns/mps/math/mathlab/int_algebra/int_alg_tut39_simrad.htm

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A = L * W
this means area equals length times width.
if you double the length and you double the width, the formula becomes:
A = 2 * L * 2 * W which is the same as:
A = 2 * 2 * L * W which is the same as:
A = 4 * L * W
if you double the length and the width, you are quadrupling the area.
this is because 2 * 2 = 4.
an example:
L = 2 and W = 4
Area = 2 * 4 = 8
double the length and double the width to get:
L = 4 and W = 8
Area = 4 * 8 = 32
32 is 4 times as large as 8.
you doubled the length and the width and you quadrupled the area.
this is because 2 * 2 = 4
if you tripled the length and doubled the width, then the area would be 6 times as large because 2 * 3 = 6
an example:
L = 2 and W = 4
Area = 2 * 4 = 8
double the length and triple the width to get:
L = 4 and W = 12
Area = 4 * 12 = 48
48 is 6 times as big as 8.
what you did was:
L * W = Area
you doubled the length and tripled the width to get:
2 * L * 3 * W = Area
this becomes:
2 * 3 * L * W = Area which becomes:
6 * L * W = Area
the area becomes 6 times as large because 2 * 3 = 6.

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the slope intercept form of the equation of a straight line is y = mx + b
m is the slope
b is the y intercept
the point you are given is:
(x,y) = (2,2)
the equation you are given is:
y = (-1/5)x + 5
the slope is (-1/5)
the y intercept is 5
the line perpendicular to this line will have a slope that is a negative reciprocal of the slope of this line.
if the slope is a/b, then the negative reciprocal of the slope is -b/a
since our slope is (-1/5), then the negative reciprocal is (5/1) which is equal to 5.
the slope of our line is 5 and it passes through the point (2,2)
the equation of our line will be y = mx + b
we replace the m with 5 and we get:
y = 5x + b
to find b, we replace y with 2 and x with 2 and solve for b.
this is because the point (x,y) = (2,2) which means that x = 2 and y = 2 at that point.
we get:
2 = 5*2 + b which becomes:
2 = 10 + b
subtract 10 from both sides of this equation to get:
-8 = b which becomes:
b = -8
our equation becomes:
y = 5x - 8
a graph of both equations will show that our new equation is perpendicular to the original equation and will pass through the point (2,2).
i have created a horizontal line at y = 2 and a vertical line at x = 2 in order to show you that our new line passes through that point.

for your additional information, the 2 lines will intersect at the point where the two equations are equal to each other.
that point will be solved for below.
the 2 equations are:
y = (-1/5)x + 5
y = 5x - 8
we set both these equations equal to each other to get:
(-1/5)x + 5 = 5x - 8
add 8 to both sides of this equation to get:
(-1/5)x + 13 = 5x
add (1/5)x to both sides of this equation to get:
13 = 5x + (1/5)x which becomes:
13 = 26/5x
that makes x = (5*13)/26 = 5/2)
this is the same as x = 2.5
we'll solve for y in both equations.
the answers should be the same.
y = -(1/5)x + 5 becomes y = -(1/5)*2.5 + 5 which becomes y = 4.5
y = 5x - 8 becomes y = 5*2.5 - 8 which becomes y = 4.5
we have our point of intersection of the 2 lines at x = 2.5 and y = 4.5
this can be expressed as (x,y) = 2.5,4.5)
the graph is shown below with the vertical and horizontal lines removed so you can see the intersection point easier.
i also made the x and y axis symmetrical so you can see that the lines are really perpendicular to each other easier.

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rate * time = units of work
the unit of work is equal to 1 which is the house.
john takes 30 minutes to clean the house.
if time is in minutes, then the equation for john is:
rate * 30 = 1
divide both sides of this equation by 30 and you get:
rate = 1/30
this means that john can clean 1/30 of the house in 1 minute.
when john's wife helps him, they both can clean the house in 18 minutes.
rate * time = units of work.
the unit of work is still the house so it is equal to 1
the rate of work is the combined rate of john and his wife.
the time is takes is now 18 minutes.
the formula of rate * time becomes:
(john's rate + john's wife's rate) * 18 = 1
we know john's rate because we just solved for it.
we'll make john's wife's rate = x
our formula becomes:
(1/30 + x) * 18 = 1
we multiply this out to get:
18/30 + 18x = 1
we subtract 18/30 from both sides of this equation to get:
18x = 1 - 18/30
we simplify this to get:
18x = 12/30
we divide both sides of this equation by 18 to get:
x = (12/30) / 18 = 1/45
this means that john's wife's rate is equal to 1/45.
john's rate is 1/30 of the house in one minute.
john's wife's rate is 1/45 of the house in one minute.
i believe we will see that it will take john's wife 45 minutes to clean the hours by herself.
the formula is rate * time = units of work.
the units of work is still equal to 1.
the wife's rate is equal to 1/45.
the formula becomes:
1/45 * time = 1
multiply both sides of this equation by 45 to get:
time = 45 minutes.
when they work together, their rates are additive.
the formula becomes:
(1/30 + 1/45) * time = 1
this simplifies to:
(3/90 + 2/90) * time = 1 which simplifies further to:
5/90 * time = 1 which simplifies further to:
1/18 * time = 1
multiply both sides of this equation by 18 to get:
time = 18
the number work out ok so we have the answer.
the answer is:
john's wife takes 45 minutes to clean the house by herself.

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area of rectangular swimming pool is equal to 3x^2 + 11x + 6 square feet.
one side of the pool is equal to 3x + 2 feet.
the formula for the area of a rectangle is L * W = A
L is the length
W is the width.
A is the area.
you have A = 3x^2 + 11x + 6
you have L = 3x + 2
if A equals L * W, then:
W = A / L
this would be:
(3x^2 + 11x + 6) / (3x + 2)
we can do the division, or we can determine that this is a quadratic equation whose factors will be (3x + 2) * (x + something).
since 6 is equal to 2 * 3, that something is probably 3.
let's try the factors of (3x + 2) * (x + 3)
we multiply those factors together to get:
3x^2 + 9x + 2x + 6
combine like terms to get 3x^2 + 11x + 6
since this is the quadratic equation that represents the area of the pool, then we have found the width of the pool.
A = L * W = (3x+2) * (x+3)
L = 3x + 2 feet
W = x + 3 feet
we were given the length of the pool as 3x + 2 feet.
the other side of the pool is the width of the pool which is equal to x + 3 feet
if you multiply L * W you will get the area of the pool which is equal to 3x^2 + 11x + 6 square feet

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your given solution is or
the equation is ax^2 + bx + c = 0
if the solutions to the equation are:


then the factors of the equation are:

if you multiply these factors together you will get the equation.
the following picture shows you the mutiplication of the factors.

your equation is:

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y = x^2 makes y a function of x.
the independent variable is x and the dependent variable is y.
x^2 = y makes x^2 a function of y.
the independent variable is y and the dependent variable is x^2.
if we let z = x^2, then:
x^2 = y becomes z = y
z is considered a function of y because it depends on the value of y.
you would solve this like you would solve any equation.
if y = 5, then z = 5
since z = x^2, this means that x^2 = 5.
suppose you had:
x^2 = y-3
x^2 is still a function of y because it depends on the value of y.
if y = 5, x^2 = y - 3 = 5 - 3 = 2
you get x^2 = 2 when y = 5
i believe this to be true, but i won't swear to it.
there may be some rule that says you can't do this, but i haven't seen it.
in practice, you can set anything equal to anything else.
the dependent variable is a function of the independent variable.
the equation determines the value of the dependent variable based on the value of the independent variable.
x^2 is a function of y meets that definition.
the value of x^2 is dependent on the value of y.
the right side of that equation doesn't have to be just y.
it can be any expression involving y.
x^2 = y^3 + 5y - 3 would be an example.
x^2 is a function of y.
when y = 3, x^3 = 27 + 15 - 3 which becomes:
x^2 = 39

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your equation is:
sin (2x) + cos(60) = 0
since cos(60) = 1/2, this equation becomes:
sin (2x) + 1/2 = 0
subtract 1/2 from both sides of this equation to get:
sin (2x) = -(1/2)
the sin is negative in the third and fourth quadrants only.
solve the angle as if it was in the first quadrant.
you get sin(2x) = 1/2
this means that 2x = sin^-1(1/2) which becomes 2x = 30 degrees.
your answer would be 2x = 30 degrees if the angle was in the first quadrant.
the equivalent angle in the third quadrant would be 180 + 2x which would make the angle 210 degrees.
the equivalent in the fourth quadrant would be 360 - 2x which would make the angle 330 degrees.
your original equation becomes:
sin(210) + cos(60) = 0
sin(330) + cos(60) = 0
solve both of these equations using your calculator and you'll see that they are both true.
a picture of what we just did is shown below.

we first solved for the reference angle.
that was the angle in the first quadrant.
we got a positive value because we were in the first quadrant.
we then determined that our value had to be negative.
we had to know that the sine is negative in the third and fourth quadrant only.
we then had to find an equivalent angle in the third and fourth quadrant.
the formula to derive that is as follows:
the reference angle is always in the first quadrant.
if the angle is in the second quadrant, then the angle is equal to 180 minus the reference angle.
if the angle is in the third quadrant, then the angle is equal to 180 plus the reference angle.
if the angle is in the fourth quadrant, then the angle is equal to 360 minus the reference angle.
we found the angle in the third and fourth quadrant because the sine is negative in the third and fourth quadrant.

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a circular sector would be an arc.
if you looked at it from above, it would look something like a slice of a round pie that was cut from the outside of the pie to the center, making a wedge.
the radius is equal to 1.875 cm
the central angle is 165.58 degrees.
the outside arc of the cam would be equal to 165.58/360 * 2 * pi * r
r is the radius of the circle.
this becomes:
165.58 / 360 * 2 * pi * 1.875 which becomes 5.418592829 cm.
you also have the pieces that go from the ends of the arc to the center.
those pieces are 1.875 cm apiece.
the perimeter is therefor equal to the length of the arc plus the pieces that go from the ends of the arc to the center.
this would be equal to 5.418592829 + 2 * 1.875 which equals 9.168592829 cm.
here's a reference you can look at regarding area and perimeter of a circular section.
http://www.efunda.com/math/areas/CircleSectionGen.cfm

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the sum of 2 acute angles is 110.
the supplement of the smaller angle is 15 times the complement of the larger angle.
what is the measure of the 2 acute angles.
let the smaller angle be equal to x.
let the larger angle be equal to y.
you have:
x + y = 110
the supplement of the smaller angle is 180 - x.
the complement of the larger angle is 90 - y.
since the supplement of the smaller angle is 15 times the complement of the larger angle, you also have:
(180-x) = 15 * (90-y)
the 2 equations that you have to solve simultaneously are:
x + y = 110 (first equation)
(180-x) = 15 * (90-y) (second equation)
we'll solve by substitution.
from the first equation, you can solve for y to get:
y = 110 - x
substitute for y in the second equation to get:
(180-x) = 15 * (90 - (110-x))
remove parentheses to get:
180 - x = 15 * (-20 + x) which becomes:
180 - x = -300 + 15x
add 300 to both sides of this equation and add x to both sides of this equation to get:
180 + 300 = 15x + x
combine like terms to get:
480 = 16x
divide both sides of this equation by 16 to get:
x = 30
your original equations are:
x + y = 110 (first equation)
(180-x) = 15 * (90-y) (second equation)
substitute for x in the first equation to get:
x + y = 110 becomes:
30 + y = 110
solve for y to get:
y = 80
your values for x and y are:
x = 30
y = 80
to see if these values are good, substitute for x and y in the second equation to get:
(180-x) = 15 * (90-y) becomes:
(180-30) = 15 * (90-80) which becomes:
150 = 15 * (10) which becomes:
150 = 150
this confirms the values for x and y are good.
x is the smaller angle.
y is the larger angle.
the answer to your question is:
the smaller angle is equal to 30 degrees.
the larger angle is equal to 80 degrees.

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this one's a bear, but i think i have it.
you are given that:
sum of an infinite series is 12.
each term in the series is equal to 3 times the sum of the infinite series starting with the term that comes after it.
we'll let your series be represented by:
a1, a2, a3, a4, .................................a[infinity].
the sum of the series starting with a1 is equal to:
a1/(1-r) = 12
the sum of the series starting with a2 is equal to:
a2/(1-r)
since the sum of the infinite series starting with a1 is equal to (the sum of the infinite series starting with a2 plus a1), then we can derive the formula of:
a2/(1-r) + a1 = 12
we now have:
a1/(1-r) = 12 (equation 1)
and we have:
a2/(1-r) + a1 = 12 (equation 2)
we also know that each term in the series is equal to 3 times the sum of the infinite series starting with the next term in the series.
this means that:
a1 = 3 * (a2/(1-r)
we can substitute for a1 in equation 2 to get:
a2/(1-r) + 3a2/(1-r) = 12 (equation 4)
we now have:
a1/(1-r) = 12 (equation 1)
a2/(1-r) + 3a2/(1-r) = 12 (equation 4)
since they are both equal to 12, we can make the expressions on the left side of equation 1 and equation 4 equal to each other to get:
a1/(1-r) = a2/(1-r) + 3a2/(1-4)
if we multiply both sides of this equation by (1-r), we get:
a1 = a2 + 3a2 which becomes:
a1 = 4a2
if we solve this equation for a2, then we get:
a2 = (1/4) * a1
this means that our common ratio is equal to (1/4).
if our common ratio is equal to (1/4), then the equation of:
a1/(1-r) = 12 becomes:
a1/(1 - (1/4)) = 12 which becomes:
a1/(3/4) = 12 which becomes:
a1 = (3/4) * 12 which becomes:
a1 = 9
we now have:
a1 = 9
r = (1/4)
because a2 = a1*r, we can derive a2 to get:
a2 = (9/4)
we can derive the infinite sum starting with (9/4) by using the formula:
(9/4)/ ( 1-r) = S
since r = (1/4), this formula becomes:
(9/4) / (3/4) = S
Solving this formulas for S gets us:
S = 3
the infinite sum of the geometric series starting with a2 is equal to 3.
a1 should be equal to 3 times this and it is.
we have:
a1 = 9
r = (1/4)
the question is:
what is the first term of the sequence?
the answer is:
the first term of the sequence is 9.

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the formula for the sum of the interior angles of a polygon is equal to (n-2)*180.
the polygon has 12 sides, so the sum of the interior angles of the polygon is equal to (12-2)*180 = 10*180 = 1800 degrees.
there are 12 angles in the polygon and it is assumed to be a regular polygon (equal angles and equal sides), so each interior angles will be equal to 1800 / 12 = 150 degrees.
an interior angle and a vertex go together.
each vertex contains an interior angle that is formed by the sides connecting to that vertex.
the derivation of that is as follows:
a straight angle is an angle of 180 degrees.
the formula stated above of (n-2) * 180 could be re-worded to say (n-2) * a straight angle.
the formula depends on the fact that the sum of the angles of a triangle is equal to 180 degrees.
accept that and you can then derive the formula from that.
look at the picture below and then read the comments below it.

the sum of the angles in a triangle is equal to 180 degrees.
each polygon can be divided into the same number of triangles as the sum of the sides of the triangle.
each of these triangles has a vertex at the center of the polygon.
a rectangle has 4 sides and can be divided into 4 triangles.
a hexagon has 6 sides and can be divided into 6 triangle.
the sum of the angles of the triangle within the polygon is therefore equal to the number of sides * 180.
if we want to get the sum of the interior angles of the polygon, we have to subtract the sum of the central angles of the polygon.
the sum of the central angles of the polygon will always be equal to 360 degrees because the number of degrees in a circle is equal to 360 degrees.
we therefore have to subtract 360 from the sum of the angles of the triangles that form the polygon to get the sum of the interior angles of the polygon.
this means that our formula becomes:
n * 180 - 360.
now, 360 is equal to 2 * 180, so our formula becomes:
n * 180 - 2 * 180.
if we factor out the 180, then we are left with (n-2) * 180.
that's how the formula is derived.
when you have a dodecagon, then it has 12 sides.
the formula then becomes:
(12 - 2) * 180 which becomes 10 * 180.
here's a link that names the polygons.
http://mathcentral.uregina.ca/qq/database/QQ.09.96/rosa1.html
the names are usually derived from the greek names for the number of sides.

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1. x years ago, Paul was 20 years old, how old is he now?
let y equal the age that paul is now.
your equation becomes:
y - x = 20
add x to both sides of the equation to get:
y = x + 20
in order to determine what y is, you need to know what x is.
the equation establishes the relationship between y and x, but cannot tell you the value of x.
y depends on x, so x is needed before you can determine y.
for example:
if x = 20, then y = 20 + 20 = 40
if x = 10, then y = 10 + 20 = 30
y will always be 20 more than x but you have no idea of the value of y unless you know the value of x.

2.the sum of a & b, diminished by one-third of c
this would not be an equation.
it would be an expression.
you can make it into an equation by setting it equal to something.
we'll start with the expression
sum of a & b is expressed by (a + b)
if you diminish it by one third of c, then you are subtracting 1/3 of c from it.
the expression becomes:
(a + b) - (1/3)*c
if you want to turn the expression into an equation, then set it equal to something, like the variable y.
you then get:
y = (a + b) - (1/3)*c
example:
let a = 20 and let b = 30 and let c = 120
the sum of a & b is equal to 20 + 30 = 50
if you diminish it by one third of c, then you are diminishing it by 40 because 40 is equal to one third of 120.
if you diminish it by 40, then you are subtracting 40 from it, so you will get 50 - 40 = 10.

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you don't have enough information to determine the length of the block or the area of the block.

the area of the block will differ depending on the length of the block and the width of the block.

the length of the block requires you to know the width of the block before determining what it is.

here's some examples:

72 is the perimeter.
you are given that.

the perimeter is determined by the equation 2L + 2W = perimeter.

you have 2L + 2W = 72

if you let W = 8, then 2W = 16 and the formula becomes:
2L + 16 = 72
subtract 16 from both sides of this equation to get:
2L = 56
divide this equation by 2 to get:
L = 28

your dimensions are:
L = 28
W = 8

the area is L * W which equals 28 * 8 which equals 224 square meters.

now, assume W = 16.
this means that 2W = 32.
this means that 2L + 2W = 72 becomes 2L + 32 = 72
subtract 32 from both sides of that equation to get:
2L = 40
divide both sides of that equation to get
L = 20

your dimensions are:

L = 20
W = 16

the area is now L * W = 20 * 16 = 320 square units.

Given that the perimeter is 72 meters, the length is dependent on the width and the area is dependent on the length times the width.

without knowing the width, you can have many possible lengths depending on the dimension of the width that you choose.

here's a table that shows you the possible lengths and widths with a perimeter of 72.

these are all integers in multiples of 4.
if you assume real numbers that could include fractional parts of a whole number, then the list can be endless.

Perimeter
	Length	Width	Area
72	4	68	272
72	8	64	512
72	12	60	720
72	16	56	896
72	20	52	1040
72	24	48	1152
72	28	44	1232
72	32	40	1280
72	36	36	1296
72	40	32	1280
72	44	28	1232
72	48	24	1152
72	52	20	1040
72	56	16	896
72	60	12	720
72	64	8	512
72	68	4	272

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y = (2x - 2) - 5
remove the parentheses to get:
y = 2x - 2 - 5
combine like terms to get:
y = 2x - 7

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this is a quadratic equation in standard form.
the standard form of a quadratic equation is ax^2 + bx + c = 0
set your equation to 0 and it becomes:
(1/2)x^2 + 6x + 7 = 0
this means that:
a = (1/2)
b = 6
c = 7
the formula for finding the x value of the vertex is:
x = -b/2a
substitute for b and a in this formula to get:
x = -6 / (2 * (1/2)) which becomes:
x = -6 / 1 which becomes:
x = -6
to find the value of y for the vertex, then substitute -6 in the equation to get:
y = (1/2)x^2 + 6x + 7 becomes:
y = (1/2)*(-6)^2 + 6*(-6) + 7 which becomes:
y = (1/2)*36 - 36 + 7 which becomes:
y = 18 - 36 + 7 which becomes:
y = -11
the vertex should be at the point (x,y) = (-6,-11)
a graph of your equation looks like this:

i drew a horizonal line at y = -11 and a vertical line at x = -6 to show your where the vertex would be.
it would be at the intersection of the vertical and horizontal line.
the vertical line is not perfectly vertical, but it should be vertical enough for you to get the idea.

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her total bill was $207.80
she had x months at a rate of $17.10
she had y months at a rate of $18.40
you have 2 equations.
x + y = 12
17.10*x + 18.40*y = 207.8
you need to solve these 2 equations simultaneously to get your answer.
we'll solve this equation by elimination.
your equations are:
x + y = 12 (equation 1)
17.10x + 18.40y = 207.8 (equation 2)
multiply both sides of equation 1 by 17.10 to get:
17.10x + 17.10y = 205.2 (equation 3)
17.10x + 18.40y = 207.8 (equation 2)
subtract equation 3 from equation 2 to get:
1.3y = 2.6
divide both sides of this equation by 1.3 to get:
y = 2
since x + y = 12, this means that:
x = 10
y = 2
substitute these values for x and y in equation 2 to get:
17.10*10 + 18.40*2 = 207.8 which becomes:
171.0 + 36.8 = 207.8 which becomes:
207.8 = 207.8
this confirms the values for x and y are good.
your answer is:
she had 10 months at $17.10 and 2 months at $18.40
the rate increase took effect in November.

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colony of bacteria are modeled by n(t) = 1000 * e^(0.00145 * t).
n is measured in bacteria per milliliter.
t is in minutes.
evaluate n(0) and interpret results.
estimate how long it takes for n to double.

n(0) would be equal to 1000 * e^(0.00145 * 0) which would be equal to 1000 * e^0 which would be equal to 1000 * 1 which would be equal to 1000 bacteria per milliliter.

this is because e^0 = 1.
actually, anything to the 0 power is equal to 1 except 0.
0^0 cannot be determined.
that's not your problem though.
in your problem, e^0 = 1 and the formula of:
n(0) = 1000 * e^(0.00145 * t) becomes:
n(0) = 1000 * e^(0.00145 * 0) which becomes:
n(0) = 1000 * e^0 which becomes:
n(0) = 1000 * 1 which becomes:
n(0) = 1000
since n is expressed in number of bacteria per milliliter then:
n(0) = 1000 bacteria per milliliter.

your formula of:
n(0) = 1000 * e^(0.00145 * 0) becomes:
1000 = 1000 * e^(0.00145 * 0)
divide both sides of this equation by 1000 and you get:
1 = e^(0.00145*0)
this becomes:
1 = e^0 which becomes:
1 = 1
this confirms the calculation was good.

in order for the bacteria to double, it would have to go from 1000 to 2000.
you have n(t) = 1000 * e^(0.00145 * t)
you want to know the value of t when n(t) is equal to 2000.
your formula becomes:
2000 = 1000 * e^(0.00145 * t)
divide both sides of this equation by 1000 to get:
2 = e^(0.00145 * t)
take the natural log of both sides of this equation to get:
ln(2) = ln(e^(0.00145 * t))
since, in general, log(a^x) = x * log(a), then your equation of:
ln(2) = ln(e^(0.00145 * t)) becomes:
ln(2) = 0.00145 * t * ln(e)
since ln(e) is equal to 1, then your this equation becomes:
ln(2) = 0.00145 * t
divide both sides of this equation by 0.00145 and you get:
ln(2) / 0.00145 = t
use your calculator to solve for t to get:
t = 478.l0325383
if we did this right, it should take 478.10325383 minutes for the bacteria to double.
we test this by substituting that value for t in the equation of:
2000 = 1000 * e^(0.00145 * t) to get:
2000 = 1000 * e^(0.00145 * 478.10325383) which becomes:
2000 = 1000 * e^(.693147181) which becomes:
2000 = 2000
the formula works, so the answer is that:
the bacteria will double in 478.10325383 minutes.

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(8x)^(2/3) is equal to ((8x)^(1/3))^2
you get the cube root of (8x) and then you square it.
in radical form, this would be shown as:

(8x)^(2/3) is also equal to ((8x)^2)^(1/3)
you square (8x) and then you take the cube root of it.
in radical form, this would be shown as:

you'll get the same answer either way.
assume x = 27
the first form gets the cube root of 27 which is equal to 3 and then squares that to get 9.
the second form squares 27 to get 729 and then takes the cube root of that to get 9.