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multiply 60 * 90 to get 3600

this is divisible by 6 and 9 because I created it using factors that contained 6 and 9

3600 / 6 = 600
3600 / 9 = 400

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let m = number of minutes spent on math.
let e = number of minutes spent on english.
let s = number of minutes spent on social studies.

e = s (spend as much time on english as on social studies)
m = 2*e (spend twice much time on math as on english)

since e = s, we can say m = 2*e or m = 2*s or m = e+s. they would all mean the same thing.

let t = total number of minutes allison can study.

2 hours = 120 minutes
3 hours = 180 minutes

120 < t < 180 (number of minutes allows is more than 2 hours * 60 and less than 3 hours * 60)

we want to find the minimum e.
we want to find the maximum s.
we want to find the maximum m.

our constraints are:
t = m + s + e
120 < t < 180
s = e
m = 2*e = 2*s = s+e

we have 2e = m
we have s = e

we have 120 < t < 180 which means that 120 < e + s + m < 180

since m = 2e and s = e, this equation becomes:

120 < e + e + 2e < 180 which becomes:

120 < 4e < 180

we divide all sides of this equation by 4 to get:

30 < e < 45

the minimum e is > 30
the maximum e is < 45

since m = 2e, then:

the minimum m is > 60
the maximum m is < 90

the minimum and maximum s is the same as the minimum and maximum e.

let's see how this works.

if m and e and s are minimized they would have to be > 60 + 30 + 30 = 120 minutes which is the minimum total time allowed.

if m and e and s are maximized they would have to be < 90 + 45 + 45 = 180 minutes which is the maximum amount of time allowed.

the questions were:

a. What is the minimum number of minutes she can spend on english homework? b. What is the maximum number of minutes she can spend on social studies? c. What is the maximum number of minutes she can devote to math?

the answer to a is that she can spend a minimum of 30 minutes on english.
the answer to b is that she can spend a maximum of 45 minutes on social studies.

the answer to c is that she can spend a maximum of 90 minutes on math.

assume 30 minutes on english.
this means 30 minutes on social studies.
this means 60 minutes on math.
total time spent is 120 minutes.

assume 45 minutes on social studies.
this means 45 minutes on english.
this means 90 minutes on math.
total time spent is 180 minutes.

technically the problem states > 120 minutes and < 180 minutes.
minimum of 30 minutes or maximum of 45 minutes for english would violate this becdause then the problem should have been stated as >= 120 minutes and <= 180 minutes.

this is picking hairs but mathematically you could not state a minimum of 30 minutes. you would have to state a minimum of more than 30 minutes. likewise for the rest.

hopefully this is not an issue but i'm just mentioning it in case it comes up.

since there is a fixed relationship between m and e and s, a minimum of any one of them automatically implies a minimum of the others and a maximum of any one of them automatically implies a maximum of the others.

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(+9) + (-8) is the same as (+9) - (+8).

You would subtract.

the answer is 9 - 8 = 1

It's the operator combined with the sign that determines what you do.

The sign is within the parentheses. It tells you whether the number is negative or positive.

the operator is outside the parentheses. It tells you whether you will be adding or subtracting.

adding a negative number is the same as subtracting a positive number.

+ (-9) is the same as - (+9)

when you combine the operator with the sign, the rules are:

++ means add
+- means subtract
-+ means subtract
-- means add

the first sign is the operator.

the second sign is the sign of the operand (the number you are working on).

check the following website out.

It might help clarify it.

Adding and Subtracting Positive and Negative Numbers

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set x^2 + 2x - 8 = 0

this would be the point at which the equations crosses the x-axis.

factor x^2 + 2x - 8 to get
(x+4) * (x-2) = 0

solving this equation will make both of these terms equal to 0.

example:

divide both sides of the equation by (x-2) and you get (x+4) = 0
divide both sides of the equation by (x+4) and you get (x-2) = 0

x+4 = 0 results in x = -4
x-2 = 0 results in x = 2

that would be selection (1)

a graph of the equation is shown below.



as can be seen, the graph of the equation x^2 + 2x - 8 crosses the x-axis at x = 2 and x = -4

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Not sure what your equation is.
|y| >= 2 has a solution.

The solution is y >= 2 or y <= -2

|x| <= 1 also has a solution.

The solution is -1 <= x <= 1

|y| >= 2*|x| also has a solution.

The solution is y >= 2*|x| or y <= -2*|x|

I could go further but it would get more complicated and you might not need to do so.

In order to test the solutions out, you need to put in test cases where the value of y is within limits and the value of y is out of limits to see if the equations hold true or not.

Let's take |y| >= 2

Let y = -3, -2, 0, 1, 2, 3
We get |y| equal to 3, 2, 0, 1, 2, 3
|y| >= 2 when y = -3, -2, 2, 3 which is consistent with our requirements that y >= 2 or y <= -2.

Let's take |x| <= 1

Let x = -2, -1, 0, 1, 2
Then |x| = 2, 1, 0, 1, 2
|x| <= 1 when x = -1, 0, 1 which is consistent with our requirements that -1 <= x <= 1

Let's take |y| >= 2*|x|

Let |x| = 5
This means that 2*|x| = 10

Let's take y = -15, -10, 0, 10, 15
Then |y| = 15, 10, 0, 10, 15
|y| >= 10 when y = -15, -10, 10, 15 which is consistent with out requirements that y <= -10 or y >= 10

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equation is c/6 = -21
multiply both sides of equation by 6 to get:
c = 6*(-21)
c = -126

That's what it looks like to me.

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This looks like the vertex form of the equation for a circle.

The general form of that should be:

(x-h)^2 + (y-k)^2 = r^2

where:

(h,k) is the (x,y) coordinates of the center of the circle.

Since -h = +7, then h must be -7.

Since -k = -5, then k must be 5.

Center of your circle is at x,y) = (-7,5)

To graph your circle, we need to solve for y.

Your equation is:

(x + 7)^2 + (y - 5)^2 = 16

Subtract (x+7)^2 from both sides to get:

(y-5)^2 = 16 - (x+7)^2)

Take the square root of both sides of the equation to get:

y-5 = +/-

Add 5 to both sides of the equation to get:

y = 5 +/-

graph of your equation looks like.....

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80 bins holding 20 tons each would hold 1600 tons.

If you didn't know that, you would solve this puzzle as follows:

Let x = number of bins holding 20 tons.

Let y = number of bins holding 25 tons.

Total number of bins is 80 so x + y = 80 is one of your equations.

Total tons required to be held is 1600 tons, so 20*x + 25*y = 1600

You would solve both these equations simultaneously.

this means that the same value for x and y would solve both equations at the same time.

Your two equations are:

x + y = 80
20x + 25y = 1600

You can solve by substitution or you can solve by elimination.

We'll do elimination.

You want one of the variables to drop out of the two equations so you are left with one equation and one variable.

You do this by making the same variable in each equation have same coefficient.

You can then add or subtract the two equations to eliminate that variable and be left with one equation with one unknown.

Here's how we do it.

Your equations are again.

x + y = 80
20x + 25y = 1600

Multiply both sides of the first equation by 25. This allows the coefficient of y in the first equation to be the same as the coefficdient of y in the second equation and it doesn't change the equality because you are doing the same thing on both sides of the equal sign.

You get:

25x + 25y = 2000
20x + 25y = 1600

Subtract the second equation from the first equation to get:

5x = 400

Divide both sides of this equation by 5 to get:

x = 80

Use the value of x to solve for y in one of your equations.

Use x + y = 80.

You get:

y = 0

You have:

x = 80
y = 0

Replace x and y in both original equations to see if this solves both equations.

First equation is x + y = 80 and x = 80 and y = 0 works ok.

Second equation is 20x+ 25y = 1600 and x = 80 and y = 0 works ok because 20*80 = 1600.

Your answer is 80 bins of 20 tons will handle 1600 tons.

Your assumption in the equality equation of x + y = 80 was that all bins had to be used.

That's why your answer used all 80 bins.

Suppose you only wanted to use 70 bins.

Your equation of total number of bins used would have been x + y = 70

that changes the equation.

The two simultaneous equations would have been.

x + y = 70
20x + 25y = 1600

You can also solve this by substitution.

Solve for x in the first equation to get x = 70 - y

Substitute for x in the second equation to get:

20*(70-y) + 25y = 1600
remove partntheses to get:
1400 - 20y + 25y = 1600
subtract 1400 from both sides simplify and combine both sides to get:
5y = 200
divide both sides by 5 to get:
y = 40
Since x + y = 70, then x = 30

30 * 20 + 40 * 25 = 600 + 1000 = 1600

The answer to your problem, however, was 80 bins because that's the number that you wanted to use.

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Average iss $9.83.

Normal distribution.

Standard deviation is $4.58.

Probability that a randomly selected employee earns less than $5.43 is.....

.168352

Click on the following hyperlink to go to a tool that helps you calculate these things.

It's called a z-table calculator.

Z Table Normal Distribution Table Calculator

Enter 9.83 into the mean input box.

Enter 4.58 into the standard deviation input box.

Enter 5.43 in the below input box.

Select Below

Your answer pops out underneath all the selections in blue.

This is the top chart at that website. Bottom chart is for other analyses.

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slope-intercept form of the equation of a line is y = mx = b where m is the slope and b is the y-intercept.

equation of 2x + 2y = 6 is the standard form of the equation of a line.

That standard form is ax + by = c

You need to convert that equation to the slope intercept form first.

2x + 2y = 6 is the original equation.
subtract 2x from both sides of the equation to get:
2y = -2x + 6
divide both sides of the equation by 2 to get:
y = -x + 3

your original equation in slope-intercept form is:

y = -x + 3

The slope of your original equation is -1.

Your equation will have the same slope.

take the general slope-intercept form of:

y = mx + b

and replace m with -1 to get:

y = -x + b

now take your points of (x,y) = (.9,.3) and replace x and y in this equation with them.

you get:

y = -x + b becomes:
.3 = -.9 + b
add .9 from both sides of this equation to get:
.3 + .9 = b
simplify to get:
b = 1.2

replace b in the slope-intercept form of your equation to get:

y = -x + 1.2

That should be your equation.

Graph the original equation and your equation to show as follows:



The horizontal line at y = .9 shows you that this occurs on your parallel line at around x = .3

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absolute value of a number is always positive.

If the number in the absolute value sign is negative, you multiply it by -1 to make it positive.

If the number in the absolute value sign is positive, you leave it that way.

|-6| = 6
|3| = 3

6 is greater than 3.

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I believe it would be .75 * .80

75% of the time he would survive the first stage.

80% of that 75% of the time he would survive the second stage.

.8*.75 = .6 which is 60%.

Looks like answer B.

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25 sets cost $150 apiece.

markup of 37.5% would be 150 + .375*150 = 150 +56.25 = 206.25

he needs to collect 25 * 206.25 to get the 37.5% markup on all the tv sets he sells.

25 * 206.25 means he needs to collect 5156.25

25 * 150 means the total cost to him was 3750

5156.25 / 3750 = 1.375 - 1 = .375 * 100% = 37.5%

He sold 20 at $265 apiece meaning he collected 20 * 265 = 5300.

He needs to collect 5156.25 minus 5300 to exactly equal his 37.5% markup.

5156.25 minus 5300 = -143.75 / 5 = -28.75.

This means he can give the remaining 5 sets away plus give each person an additional $28.75 in order to realize his original markup.

His cost for the set is $150.
He gives it away plus an addition $28.75.

His markdown is $150 plus $28.75 equals $178.75 / $150.00 = 1.11916666667 * the original price equals 111.9166666667%.

Here's how that would work.

Original cost is $150 * 25 = $3750.
Needs to make ($150 + .375*150) * 25 = 206.25 * 25 = $5156.25
Sold 20 at $265 = $5300.00
Gave away 5 plus $28.75 = -$143.75

Add $5300 and -$143.75 to get $5156.25

Markdown on the remaining 5 is 111.9166666667%

He made too much money on the original 20.

He can give the remaining sets away and still realize more than he wanted to make in the first place.

Formula to use would be:

(20 * 265) + (5 * x) = 25 * 150 * 1.375

This becomes:

5300 + 5x = 5156.25

This becomes:

5x = 5156.25 - 5300 = 143.75

This becomes:

x = - $28.75

Markdown is Sale price minus cost, divided by cost.
Markdown is therefore -$28.75 - $150 = -$178.75 / $150 = - -1.19166666667 * 100% equals -111.9166666667 percent.

Numbers look a little odd but that's the way it works out.

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rate times time = distance

distance = 5040

time = 6 hours

rate * 6 = 5040

rate = 5040/6 = 840 kmph (kilometers per hour)

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area of a circle is equal to
r equals diameter / 2

Circle 1 has a radius of 8/2 = 4 inches.
Circle 2 has a radius of 6/2 = 3 inches.
Circle 3 has a radius of 2/2 = 1 inch.

Area of circle 1 equals = 16*

Area of circle 2 equals = 9*

Area of circle 3 equals = 1*

Differences in the areas is (16-9-1)* equals 6* equals 18.84955592 square inches.

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Area equals bh/2

If you double each dimension, you would probably get an area 4 times as large.

Try 6-8-10

base and height will be 3 and 4 in the original triangle to get an area of 12/2 = 6.

base and height will be 6 and 8 in the new triangle to get an area of 48/2 = 24

24/6 = 4 meaning the new triangle area is 4 times as large as the original triangle area.

Your answer should be C.

Your new triangle should also be proportionate to the original triangle.

6-8-10 is exactly double 3-4-5 in the same proportions.

Answer B would give you the same area, but the proportions would be off.

3-4-5 and 4-12-13 do not have the same porportions for all of the dimensions.
4/3 not equal to 2
12/4 not equal to 2
13/5 not equal to 2

Answer C keeps the proportions the same

6/3 = 2
8/4 = 2
10/5 = 2

Keeping the sides in porportions means the angles will stay the same meaning you have the same triangle only larger. The triangle would be similar.

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is the same as .

Since is the same as , then the answer is 4/3 because equals 4 and equals 3.

4/3 = 1.3333333

If you use your calculator to solve , you will get 1.3333333

If you use your calculator to solve , you will also get 1.3333333

If you use your calculator to solve , you will also get 1.3333333

If you use your calculator to solve , you will also get 1.333333

They are all equivalent.

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In all these equations, let y = f(x).

To find the inverse function, solve for x and then replace y with x and x with y.

alternatively you can replace y with x and x with y first, and then solve for y.

Either way you'll get the same answer.

selection A
y = x+3
Solve for x to get:
x = y-3
Replace x with y and y with x to get:
y = x-3 which is the inverse function of y = x+3

Selection B
y = x/2
Solve for x to get:
x = 2y
Replace x with y and y with x to get:
y = 2x which is the inverse function of y = x/2

Selection C
y = -x-2
Solve for x to get:
x = -y-2
Replace x with y and y with x to get:
y = -x-2 which is the inverse function of y = -x-2

Selection D
y= sqrt(x+2)
Solve for x as follows:
Square both sides to get:
y^2 = x+2
Subtract 2 from both sides to get:
y^2-2 = x which is the same as:
x = y^2-2
Replace x with y and y with x to get:
y = x^2-2 which is the inverse function of y = sqrt(x+2)

Selection C inverse equation is the same as the original equation.


graph of this equation is shown below:



The line y = -x-2 is a reflection about the line y = x which is a definition of inverse function.

Take any point (x,y) on the line y = -x-2 above the line y = x. The opposite point (y,x) on the line y = -x-2 below the line will be the same distance from the line y = x.

example:

let x = -2
then y = -(-2)-2 = 2-2 = 0
your coordinate point is (-2,0).

now let x = 0
then y = 0-2 = -2
your coordinate point is (0,-2)

the point (-2,0) is a reflection of the point (0,-2) about the line y = x

to prove that the distance between these points and the line y = x is the same, we need to find the point of intersection between these two lines.

the point of intersection with the line y = x would be (-1,-1) as shown on the graph.

The distance between the point (-2,0) and (-1,-1) is given by the equation:

which equals

The distance between the point (0,-2) and (-1,-1) is given by the equation:

which equals

The point (-2,0) is a reflection of the point (0,-2) about the line y = x which is a requirement of inverse functions.

Selection C is your answer.

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60 = 2002
66 = 2003
70 = 2004
76 = 2005
78 = 2006
85 = 2007

rate of increase for each year would be the value in that year minus the value in the previous year divided by the value in the previous year.

rate of increase in 2003 over 2002 = 6/60 = .1
rate of increase in 2004 over 2003 = 4/66 = .06
rate of increase in 2005 over 2004 = 6/70 = .085
rate of increase in 2006 over 2005 = 2/76 = .026
rate of increase in 2007 over 2006 = 7/78 = .089

percent equals 1005 * rate.

biggest increase percent is 2003 over 2002 = 10%

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The amount of a investment of P dollars are t years at simple interest rate r is given by a=p + prt
1) rewrite this formula by factoring out the GCF on the right side.
2) find a if $8300 is invested for 3 years at a simple interest rate of 15%.

your formula is:

a = p + p*r*t

where:
r is the interest rate
t is the number of years

you factor out the p on the right hand side of the equation to get:

a = p * (1 + r*t)

if you invest $8300 for 3 years at 15% interest per year, then this formula would become:

a = 8300 * (1 + (.15*3))

the interest rate of 15% has to be converted to a rate before you do anything with it.

rate = percent divided by 100%.

15% / 100% = .15 which is the interest rate per year.

your answer would be:

a = 8300 * (1 + (.15*3))

this would become:

a = 8300 * (1 + .45)

this would become:

a = 8300 * 1.45

this would result in:

a = 12035

15% of 8300 = .15 * 8300 = 1245 * 3 = 3735 + 8300 = 12035 confirming the answer is correct.

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A tank holds 800 gallons of water. The gauge indicates that the tank is ¾ full; then 100 gallons are drained from the tank to fill a pool 7/8 full. Next, 1/3 of a tank of water is added back in. What fraction of the tank is filled with water?

Let T represent a full tank = 800 gallons of water.
Let P represent a full pool = ???

3/4 * T - 100 = 7/8 * P

This formula says that three quarters of the tank minus 100 gallons equals 7/8 of the capacity of the pool.

You are adding 1/3 of a tank of water back in to the tank ?????

This problem apparently doesn't have anything to do with the pool.

Here's why I think that way.

Your tank is 3/4 full meaning it have 800*3/4 = 600 gallons in it.

you drain 100 gallons from it so you are left with 500 gallons in the tank.

you add 1/3 of a tank full of water back into it, so you are adding 800/3 = 266 and 2/3 gallons into it.

500 + 266 and 2/3 = 766 and 2/3 gallons / 800 = .9583333333 percent full.

in fractions, you would have (2300/3) / (2400/3) which is the same as 2300/2400.

2300/2400 is the same as 23/24 = .95833333333 so your fraction of the tank that is full would be 23/24.

The pool had nothing to do with it other than to add extraneous information as far as I could see.

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easiest thing to do normally is to multiply both sides of the equation by a factor that will result in the removal of the fractions.

example:

(5/8)*x + (3/15) * y = 70

multiply both sides of this equation by (8*15) to get:

(5/8)*8*15*x + (3/15)*8*15*y = 70*8*15

this results in:

5*15*x + 3*8*y = 70*8*15

simplify and then solve.

another way is to get a common denominator so you can combine the fractions.

this should result in the same equation above.

original equation is again:

(5/8)*x + (3/15) * y = 70

this equation is the same as:

5*x/8 + 3*y/15 = 70

multiply numerator and denominator of the first fraction by 15/15 and multiply the numerator and denominator of the second fraction by 8/8 to get:

(5*x*15)/(8*15) + (3*y*8)/(8*15) = 70

this becomes:

(5*x*15 + 3*y*8) / (8*15) = 70

multiply both sides of this equation by 8*15 to get:

5*15*x + 3*8*y = 70*8*15

I prefer to multiply both sides of the equation by a common factor to get rid of the denominators but you can do it either way.

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A rectangular painting with a width of X centimeters has an area of x^2 + 50x square centimeters. Find a binomial that represents the length.

Area equals length * width.

you have the area equal to x^2 + 50x

If you let x = width, and y = length, then you get:

area = x*y = x^2 + 50x

divide both sides of this equation by x to get:

y = (x^2 + 50x) / y = x+50

your answer should be:

y = x+50

if that's true, then x * (x+50) = x*y = area of the painting.

x * (x+50) = x^2 + 50x so answer should be good.

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1/2 b*h = 240
b = h+8

formula becomes:
1/2 * (h+8)*h = 240
multiply both sides of this equation by 2 to get:
(h+8) * h = 480
remove parentheses to get:
h^2 + 8h = 480
subtract 480 from both sides of the equation to get:
h^2 + 8h - 480 = 0
Solve this equation using the quadratic formula to get:
h = 18.27105745
b = h+8 = 26.27105745

1/2*b*h = 240 which proves the values for b and h are good.

I believe they are asking you for the other leg, which is b.

I don't know what else they would be asking for.

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Let x = length and let y = width.

Your 2 equations are:

2x + 2y = 500
xy = 21800

Solvle for y in both equations to get:

y = 250-x
y = 21800/x

Graph these equations to get what is shown below:



The graphs of these 2 equations do not intersect, therefore you do not have a solution that is common to both.

If you solve these two equations simultaneously by substitution, it will lead to an equation of x^2 - 250x + 21800 = 0.

Solve this using the quadratic equation and you will wind up with imaginary roots.

This means there is no solution for x that is real that will satisfy both equations, I believe.

The graph of the two equations bears this out.

I'm not 100% sure that I'm right but it sure looks like that.

I did a test, however, to see what happens if I do get an intersection on the graph.

I doubled the perimeter to 1000.

Your 2 equations would become:

2x + 2y = 1000
xy = 21800

Solvle for y in both equations to get:

y = 500-x
y = 21800/x

Graph these equations to get what is shown below:



You do get an intersection now which says there is a solution to this revised equation.

Solving it algebraically using the quadratic formula does provide real roots to the equation this time.

We have a quadratic equation of x^2 - 500x + 21800 = 0 which yields roots of:

x = 48.25758998
and:
x = 451.74241

When x is 48.25758998, y = 451.74241

When x is 451.74241, y = 48.25758998

2 * x + 2 * y = 1000 which is the perimeter of the revised equation.

x*y = 48... * 451... = 21800 which is the area of the revised equation.


My original analysis is proved correct because when I do get an intersection ojn the graph, I have real solutions, which means that there is no solution to the equations you posed originally. Those are:

2x + 2y = 500
x*y = 21800

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you were on the right track.
-----
you let x = number of houses with floor plan 1
you let y = number of houses with floor plan 2
-----
you had x + y = 38
you had 175000*x + 200000*y = 7200000
-----
you need to solve both these equations simultaneously.
-----
you can do it by substitution.
-----
take the first equation and solve for x or y.
either one will do.
I used x = 38-y
-----
substitute for x in the second equation to get:
175000 * (38-y) + 200000 * y = 7200000
this expands to be:
175000 * 38 - 175000 * y + 200000 * y = 7200000
this simplifies to:
6650000 + 25000y = 7200000
subtract 6650000 from both sides of this equation to get:
25000y = 550000
divide both sides by 25000 to get:
y = 22
that makes x = 16 because 22 + 16 = 38
-----

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looking at the graph, it looks like you have potential solutions when x >= 5+(2/3) and when x >= 13+(1/2)
-----
between x = 5+(2/3) and x = 13+(1/2) your y value would have to be >= 5 and <= 3x-12.
-----
from x = 13+(1/2) to infinity your y value would have to be >= 5 and <= x+15.
-----
hopefully that's the answer you are looking for.
solving this without looking at a graph is not something I would look forward to doing.
-----
It would be best to solve the equalities first and then get the intersection points.
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you have 3 equations.
solving for equalities, you would get:
y = 5
y = 3x-12
y = x+15
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solving for the intersection points you would get:
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y = 5 intersect with y = 3x-12 at x = 5+(2/3)
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y = 5 intersects with y = x+15 at x = -10
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y = 3x-12 intersects with y = x+15 at x = 13+(1/2)
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now that you have the intersection points, you can try to find where y has to fit in those intervals.
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the intervals are:
x = -10
x = 5+(2/3)
x = 13+(1/2)
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you can forget x = -10 because x has to be > 2 (one of the equations shown)
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when x = 2, you have 3 solutions of equality.
y = 5
y = 3x-12 = 3*2-12 = 6-12 = -6
y = x+15 = 3+15 = 18
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y can be greater than 5 and smaller than 18 at the same time, but it cannot be smaller than -6 and greater than 5 at the same time so it looks like the interval between 2 and 5+(2/3) is not going to be good. The graph shows that easily, but without the graph, you might need to plot more points to see it. Logiclly it would make sense since 3x-12 is below y = 5 and intersect with y = 5 at x = 5+(2/3) which means it must have been below it up to that point if it started below it at x = 2.
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you would do the same thing with the other intervals between the intersection points.
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as I said before, looking at the graph make the job so much easier. you still have to logically deduce what's going on but seeing what it looks like on the graph is much better than not.
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you can see a picture of the graph with comments by clicking on the following hyperlink.
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Picture of Graph with Comments

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your equation is 6y = 18
solving for y, you get y = 18/6 = 3
this is true for all values of x.
a graph of this equation is shown below:

your input / output table would look like this:
x / y
-5 3
-4 3
-3 3
-2 3
-1 3
0 3
1 3
2 3
3 3
4 3
5 3
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tan (15) = .267949192 by the calculator.
tan (15) = 0.267949192431123 by excel spreadsheet function.
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I don't know if it just stops there or if excel doesn't have the capability to extend the fraction any further.
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For it to be exact, it would have to be the result of the ratio of two integers.
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I would be at a total loss to determine what integers would result in that fraction.
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my guess is that it's probably irrational and so the fraction would go on endlessly without any repeating patterns to indicate that even if it's an endless fraction, it was the result of the ratio of two integers.
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but that's just a guess.
I really don't know.
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amount of aspirin in the bloodstream is 1/2 of the original amount every 20 minutes.
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formula is:
y = 200*(.5)^(.05x)
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graph of this equation looks like this:

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this equations looks like it models it pretty well.
y is the amount of aspirin left in the bloodstream.
x is the number of minutes.
the horizontal lines are at 200,100,50,25 to show you the amount of aspirin left in the bloodstream after 0,20,40,60 minutes.
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It does not appear that the bloodstream will ever be completely free of aspirin based on this formula.
.5^(.05x) will never be equal to 0.
It will become very small, but never zero.
the calculator might show 0, but that's because the calculator can only go so far in the number of decimal places it can show, even with scientific notation.
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My calculator shows a result of 0 when 200 is multiplied by .5^333
It shows 2.285974783 * 10^-98 when 200 is multiplied by .5^332
Since .05*x = 333 when the calculator give a result of 0, this means that x = 20*333 = 6660 minutes
That's 4.625 days for the body to be completely free of aspirin enough so that my calculator can't determine how much is still left.
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best thing to do with this is determine a point at which the level of aspirin in the blood is less than a certain amount.
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that can be calculated.
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assume that amount is 1 milligram.
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formula would be 1 = 200*(.5)^(.05x)
divide both sides of this formula by 200 to get:
1/200 = (.5)^(.05x)
take the log of both sides of this equation to get:
log(1/200) = (.05x)*log(.5)
divide both sides of this equation by log(.5) to get:
log(1/200)/log(.5) = (.05x)
solve for (.05x) to get:
(.05x = 7.64385619
solve for x to get:
x = 152.8771238 minutes
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this means that it will take 152.8771238 minutes for the aspirin in the bloodstream to be less than 1 milligrams.
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the following graph should show that about as clear as can get with the tools available.

I added the horizontal line at y = 1 to show you the cutoff point better. It's at somewhere around x = 153 which is just about right.

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perfect score is 10
11 scores with:
mean of 8
mode of 7
median of 8
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total of 11 scores with a median of 8 means that a maximum of 5 scores are below 8 and 5 scores are above 8. this will occur when there is only 1 number 8 in the distribution. we will assume only one number 8 in the distribution because that would allow for more number 10's if possible. we need at least one though, because we have 11 numbers and that means that one of the numbers has to be the median in order to get 5 on each side plus the 1 in the middle equal to 11 total.
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mode of 7 means that 7 is the score that has occurred with the most frequency.
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integer scores means that there are no fractions to worry about.
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we need:
1 number 8
a minimum of 2 number 7's
5 numbers above 8
5 numbers below 8
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let's start off with 5 number 10's.
that would be the maximum number of 10's without any other constraints except the fact that the median is 8 meaning the number of units above 8 had to be 5 at most.
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5 number 10's means that we would have to have 6 number 7's because 7 is the mode which is the number that occurs most frequently. we can't have 6 number 7's because the total of numbers below 8 has to be 5 and 6 is greater than 5.
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so 5 number 10's is no good.
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can we have 4 number 10's?
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this means we must have at least 5 number 7's and number 7 would then be the most frequent number.
if we have 4 number 10's, we would need 1 more number above 8 to make 5 numbers above 8. we'll pick the lowest number which is 9.
so far we have:
4 number 10's equals a total of 40
1 number 9 equals a total of 9
1 number 8 equals a total of 8
5 number 7's equals a total of 35
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we have a total of 11 numbers with a grand total of 92
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we have exceeded the grand total of 88 so 4 number 10's is not possible given all the constraints.
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can we have 3 number 10's?
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3 number 10's means we have to have a minimum of 4 number 7's to keep number 7 the most frequently used number in the set.
we also need 2 more numbers above 8 which means we have to choose 9 because 9 is the smallest number above 8.
so far we have:
3 number 10's equals a total of 30
2 number 9's equals a total of 18
1 number 8 equals a total of 8
4 number 7's equals a total of 28
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this give us a total of 10 numbers with a grand total of 84
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we need 1 more number that equals 4.
adding 1 number 4 give us a total of 11 numbers with a grand total of 88.
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the maximum number of 10's we can have is 3.
since 10 is a perfect score, then:
the maximum number of perfect scores we can have is 3.
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