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If Steven can mix 20 drinks in 5 minutes, Sue can mix 20 drinks in 10 minutes, and Jack can mix 20 drinks in 15 minutes, how much time will it take all 3 of them working together to mix the 20 drinks?
:
The completed job is the mixing of 20 drinks.
Let the completed job = 1
:
Let t = time required when working together:
:
+ + = 1
:
Multiply the equation by 30 to get rid of those annoying denominators:
30* + 30* + 30* = 30(1)
:
Cancel them denominators and you have:
6t + 3t + 2t = 30
11t = 30
t =
t = 2.73 minutes for all three to mix 20 drinks
:
Convert the decimal portion to seconds: .73 * 60 = 43.6 seconds
2 min 43.6 sec

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Write an expression for each phrase
:
Let x = H's age now
;
One-half of Heather's age two years from now
(x+2)
:
plus one-third of her age three years ago
+ (x-3)
is twenty years. How old is she now?
:
Put it all together, ("is" usually means "="
(x+2) + (x-3) = 20
:
Get rid of those annoying denominators, multiply equation 6, resulting in:
3(x+2) + 2(x-3) = 6(20)
:
3x + 6 + 2x - 6 = 120
5x = 120
x = 120/5
x = 24 yrs is H's age now
:
check it out in the statement
(24+2) + (24-3) = 20
(26) + (21) = 20
13 + 7 = 20 confirms our answer

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graph each of the following inequalities on the cartesian plane
2x + y <_ -5
:
First put it in the general "y="
y <= -2x - 5
:
You can choose the value for x to find y, A good start is x = -3
y = -2(-3) -5
y = +6 - 5
y = +1
:
Then x = +3
y = -2(3) - 5
y = -6 - 5
y = -11
:
This gives y a rather high value, perhaps x = +2 would be better for graphing
y = -2(2) - 5
y = -4 - 5
y = -9
:
From what we have done we can make a table
x | y
--------
-3 |+1
0 |-5; I always include x = 0 because it's so easy to know the value of y
+2 |-9
+3 |-11
:
If you plot this on +/-10 x-y graph:

:
Note that x=3 will not be included on our 10 by 10 by graph, choose values that
will keep it on a 10 by 10 graph
:
Remembering that this is an <= inequaltity, shade the area at or below the line on this graph.
:
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An art dealer sold two artworks for $1520 thereby making a profit of 25% on the first work and 10% on the other, whereas if he had approached any exhibition he would have sold them together for $1535 with a profit of 10% on the first and 25% on the other artwork. Find the actual cost of each artwork.
:
Let x = cost of the 1st artwork, Let y = cost of 2nd artwork
:
Art dealer: Retail price 1st piece = 1.25(x); Retail price of 2nd = 1.10(y)
1.25x + 1.10y = 1520
:
Exhibition: Retail price 1st piece = 1.10(x); Retail price of 2nd = 1.25(y)
1.10x + 1.25y = 1535
:
Use two equation in the elimination method:
multiply the 1st equation by 100, and the 2nd equation by 88,*resulting in:
125x + 110y = 152000
96.8x +110y = 135080
----------------------subtracting eliminates y
28.2x + 0y = 16920
x = 16920/28.2
x = $600 is the cost of the 1st piece
:
2nd piece cost using the 1st equation:
1.25(600) + 1.1y = 1520
750 + 1.1y - 1520
1.1y = 1520 - 750
1.1y = 770
y = 770/1.1
y = $700 is the cost of the 2nd piece:
:
:
Check our solutions in the 2nd equation:
1.1(600) + 1.25(700) =
660 + 875 = 1535 as given
:
Did this make sense to you? *In case you wondered how I came up 88 as multiplier
in the 2nd equation. I multiplied the 1st equation by 100 to get rid of the
decimal. Then divided 1.25 into 110 and saw it came out to an even 88.

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Write the equation of the line passing through each of the given pairs of points. Write your result in slope-intercept form, where possible.
(-1, 3) and (4,-2)
:
First find the slope using the slope (m) equation: m =
:
In your problems we will assign the coordinates as follows:
x1 = -1; y1 = 3; x2 = 4; y2 =-2
:
Substitute these in the slope equation:
m = = ; minus a minus = plus
:
m = = -1 is the slope
:
Next use the point/slope equation which is y - y1 - m(x - x1)
We know x1 y1 and m, substitute these in the equation
y - 3 = -1(x - (-1)
y - 3 = -1(x + 1)
y - 3= -x - 1
y = -x - 1 + 3
y = -x + 2 is the slope intercept form
:
You can confirm this by substituting the given values for x and see if the equal the give values for y
:
Note that the intercept (which is the y intercept) always occurs when x = 0
Therefore in this equation the y intercept is +2
:
The x intercept occurs when y = 0, use the equation to find that
-x + 2 = 0
-x = -2
x = +2; x has to positive, multiply equation -1 if necessary
:
If you graphed this, it's even more apparent.
:

:
How about this? Did it shed some light on this stuff

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A company needs to create an open topped box to ships it's product. The box is going to be made by cutting congruent squares from each corner of a piece of cardboard, and then folding the cardboard to create a container of maximum volume. What is the size of each square that is removed from the corners, what are the dimensions of the box, and what is the volume of the box with the largest area? The piece of cardboard has a length of 65 inches and a width of 56 inches.
:
Draw a rough diagram of the rectangular piece of cardboard, label it 65 by 56
Let the side of the small squares = x
It will be apparent that the base dimensions of the box will = (65-2x by (56-2x)
The height will = x
:
let y = volume of the box:
y = x * (65-2x) * (56 - 2x)
y = x(4x^2 - 242x + 3640)
y = 4x^3 - 242x^2 + 3640x
:
The easiest way to determine max volume is graph this equation

:
Note that max volume seems to occur when x = 10 inches
:
To answer the questions then:
The size of the squares are 10" by 10"
The volume:
(65-20)*(56-20) * 10
45 * 36 * 10 = 16200 cu inches
:
You can confirm this by finding the volume using x = 11 and x = 9
The volume will be less.
:
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Two pipes can be used to fill a pool. Working together the two pipes can fill the pool in 4 hours. The larger pipe can fill the pool in 4 hours less time than the smaller pipe can alone. Find the time to the nearest tenth of an hour it takes for the smaller pipe working alone to fill the pool.
:
Let t = time required for the smaller pipe to fill the pool alone
then
(t-4) = time required for the larger pipe
:
Let the filled pool = 1
:
+ = 1
:
Multiply equation by t(t-4) to get rid of the denominators:
t(t-4)* + t(t-4)* = t(t-4)*1
:
Cancel out the denominators and you have:
4(t-4) + 4t = t(t-4)
:
4t - 16 + 4t = t^2 - 4t
:
8t - 16 = t^2 - 4t
:
Arrange as a quadratic equation:
0 = t^2 - 4t - 8t + 16
:
t^2 - 12t + 16 = 0
:
WE have to solve this using the quadratic equation: a=1; b=-12; c=16
:

:

:

:

:

t = 1.528, obviously this is not the solutions
:
Solution 2:

:

:
t = 10.472 ~ 10.5 hrs required by the small pipe to fill the pool
:
:
Check solution:
4/10.5 + 4/6.5 =
.4 + .6 = 1

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Ellen and Kate raced on their bike to the library after school. They both left school at 3p.m. and biked along the same path. Ellen rode at a speed of 12 mph and Kate rode at a speed of 9 mph. Ellen got to the library 15 minutes before Kate. How long did it take Ellen to get to the library and what time did Ellen get to the library.
:
Let t = time required (in hrs) for E to get to the library
Then
(t + .25) = time required for K to get to the library (changed 15 min to hrs)
:
We know they both traveled the same distance. Make a distance equation
Dist = speed * time
:
E's dist = K's dist
12t = 9(t + .25)
12t = 9t + 2.25
12t - 9t = 2.25
3t = 2.25
t = 2.25/3
t = .75 hr or 45 min
:
E got to the library at 3:45
:
:
Confirm out solution by finding that they did, in fact, travel the same distance
K's time .75 + .25 = 1 hr
9 * 1 = 9 mi
12 * .75 = 9 mi
;
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Jane invested some amount at the rate of 12% simple interest and some other amount at the rate of 10% simple interest. She received yearly interest of $130. Randy also invested in the same scheme, but he interchanged the amounts invested, and received $4 more as interest. How much amount did each of them invest at different rates?
:
Let x = amt J invested at 12% and amt R invested at 10%
:
Let y = amt J invested at 10% and amt R invested at 12%
:
J's interest equation: .12x + .10y = 130
:
R's interest equation: .10x + .12y = 134
:
Multiply J's equation by 1.2 and subtract R's equation:
.144x + .12y = 156
.10x + .12y = 134
------------------subtracting eliminates y
.044x + 0y = 22
x = 22/.044
x = $500 invested at 12% by J and 10% by R
:
Find y using .10x + .12y = 134; (.10(500) = $50)
50 + .12y = 134
.12y = 134 - 50
.12y = 84
y = 84/.12
y = $700 invested at 10% by J and 12% by R
:
Check J's total interest
.12(500) + .10(700) =
60 + 70 = 130
:
Check R's total interest
.12(700) + .10(500) =
84 + 50 = 134
:
:
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The length of a rectangle is three times the width. If the width were increased by 3 m while the length remained the same, the new rectangle would have an area of 30 m square. Find the length and width of the original rectangle.
:
Let x = width
And
3x = length
:
New rectangle
(x+3) = new width
and
3x = unchanged length
then the new area
3x(x+3) = 30
:
3x^2 + 9x = 30
:
3x^2 + 9x - 30 = 0; a quadratic equation
:
Simplify, divide equation by 3:
x^2 + 3x - 10
:
Factors to:
(x + 5)(x - 2) = 0
Positive solution:
x = 2 is the original width
3(2) = 6; length
:
Check solution by finding the area
6(2+3) = 30
:
:
64. Two cars left an intersection at the same time. One traveled west, and the other traveled 14 mi less, but to the south. How far apart were they then, if the distance between them was 16 mi more than the distance traveled south?
:
Let x = dist traveled by the westbound car
then
(x-14) = dist traveled by the southbound car
:
It said,
"the distance between them was 16 mi more than the distance traveled south?"
From this we can say the distance between them, (which is the hypotenuse), is:
(x-14) + 16 = (x+2)
:
Using pythag a^2 + b^2 = c^2
:
x^2 + (x-14)^2 = (x+2)^2
:
x^2 + x^2 - 28x + 196 = x^2 + 4x + 4; FOILed (x-14)^2 and (x+2)^2
:
x^2 + x^2 - x^2 - 28x - 4x + 196 - 4 = 0, group like terms on the left
:
x^2 - 32x + 192 = 0; a quadratic equation
:
With a little diligence we can factor this:
(x - 8)(x - 24) = 0
:
Our two solutions
x = +8 and x = + 24
:
Obviously x = 24 is the solution since one car when 14 mi less.
:
The distance apart (the hypotenuse): 24 + 2 = 26 mi
:
Check our solution using a^2 + b^2 = c^2; a=24, b=10 (14 mi less) and c = 26
:
24^2 + 10^2 = 26^2
576 + 100 = 676; confirms our solution.
:
I tried to explain things as we went along here. Did it make sense to you? Any questions?

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This same problem came up a few weeks ago, this is what I submitted then, perhaps it will help you.
:
A small firm produces both AM and AM/FM car radios. The AM radios take 15 h to produce, and the AM/FM radios take 20 h. The number of production hours is limited to 300 h per week. The plant's capacity is limited to a total of 18 radios per week, and existing orders require that at least 4 AM radios and at least 3 AM/FM radios be produced per week. Write a system of inequalities representing this situation. Then, draw a graph of the feasible region given these conditions, in which X is the number of AM radios and Y is the number of AM/FM radios.
:
Let x = number of AM radios; let y = number of AM/FM radios

The production hour constraint:
15x + 20y =< 300
Arrange in general form to plot the graph
20y =< 300 - 15x
y =< 300/20 - (15/20)x
y =< 15 - .75x, use this to plot the production hr graph (Purple line)
:
Plant's capacity constraint:
x + y =< 18
y =< 18 - x, plot this graph also (Green line)
:
Min AM radio production constraint:
x => 4
This will be a vertical line going thru the x axis at +4, (I can't draw this in)
:
Min AM/FM radio production constraint
y => 3
This will be a horizontal line going thru the y axis at +3, (Dark blue line)
:
Here is the graph as presented by these equations except for x => 4,
:

:
The feasibility region:
1. At or below the purple or green line, whichever is lowest
2. At or above the horizontal line
3. At or to the right of the vertical line which you have to draw in at x = +4
:
How about this, did it all make some sense to you?

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:
subtract 9 from both sides and you have the difference of squares:

:
Which factors to:
(2z + 3)(2z - 3) = 0
:
2z = -3
z =
and
2z = +3
z =

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+
Which is:
-
:
Common denominator would be
:

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Jay's favoritebook is 182 pages less than three times Mitchel's favorite book. If both books have the same number of pages, how many pages are in Jay's favorite book?
:
It says both books have the same number of pages
Both books = x pages
:
"Jay's book is 182 pages less than three times Mitchel's book."
x = 3x - 182
x - 3x = -182
-2x = -182
x = -182/-2
x = +91 pages
:
:
Check:
91 = 3(91) - 182
91 = 273 - 182

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An investment of $5000 is made at an annual simple interest rate of 5.3%. How much additional money must be invested at an annual simple interest rate of 8.5% so that the total interest earned is 6.5% of the total investment?
:
Let x = additional amt invested at 8.5%
Therefore:
The resulting amt = (x+5000)
:
The interest equation:
.053(5000) + .085x = .065(x + 5000)
:
Can you solve this now?

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: Show the steps that you would take to solve the following algebraically:
4- X\X-2= -2\X-2
Assume you mean:
4 - =
:
Multiply equation by (x-2) to get rid of the denominators:
4(x-2) - (x-2)* = (x-2)*
:
Cancel out the denominators and you have
4(x-2) - x = -2
:
4x - 8 - x = -2
:
4x - x = -2 + 8
:
3x = +6
:
x = 2
:
:
b) What potential solution did you obtain? Explain why this is this not a solution.
:
Substituting 2 for x in the original equation gives you denominators of 0

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A lidless box is constructed from a square piece of tin by cutting a 10cm
square from each corner and bending up the sides for the box. If the volume of
the box is 1200cm cubed, find the dimensions to the nearest tenth of a centimetre.
:
Let x = length of the sides of the base
:
If you draw a rough diagram, showing the 10cm squares on the corners it will be
apparent that the height of the box will be 10 cm:
:
We know the Area of the base =
:
Base area = = 120 sq cm
:
x =
:
x = 2* or 10.95445 cm
:
The dimensions to the nearest tenth: 11 by 11 by 10
:
Check solution: * * 10 = 1200

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Are you serious?

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A real estate office handles 50 apartment units. When the rent is $540 per month, all units are occupied. However, for each $30 increase in rent, one unit becomes vacant. Each occupied unit requires an average of $18 per month for service and repairs. What rent should be charged to obtain the maximum profit?
:
Let x = no. of $30 increases
and
Also x = no. of vacant apartments
:
Y = Revenue
y = price per apt * no. of apts
:
y = (540+30x) * (50-x); FOIL this
:
y = 27000 - 540x + 1500x - 30x^2
:
y = -30x^2 + 960x + 27000; arranged as a quadratic equation (a parabola)
:
The axis of symmetry will be the value of x for max profit
x =
x =
x = 16; no. of increases of $30 and 16 apts vacant
:
Use x = 16 to find the rent for max profit:
540 + (30*16) = $1020 per month
:
Find the actual profit by subtracting the service fee from the rental of 34 apts:
(50-16)*(1020 - 18) =
34 * 1002 = $34,068
:
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A small company produces both standard and deluxe playhouses. The standard playhouses take 12 hours of labor to produce, and the deluxe playhouses take 20 hours. The labor available is limited to 800 hours per week, and the total production capacity is 50 items per week. Existing orders require the company to produce at least 10 standard playhouses and 15 deluxe playhouses per week. Write a system of inequalities representing this situation, where x is the number of standard playhouses and y is the number of deluxe playhouses. Then graph the system of inequalities.
:
Let x = standard p.h; y = deluxe p.h
:
The labor equation:
12x + 20y =< 800
:
put equation in the general (y=) form to plot the graph:
20y =< 800 - 12x
y =< 800/20 - (12/20)x
y =< 40 - .6x
:
:
The production capacity equation:
x + y =< 50
Put this in the "y=" form also
y <= 50 -x
:
Existing order constraints
x => 10
and
y => 15
:
Plot these using the equation givens.
y = 40 - .6x; (purple line)
y = 50 - x; (green line)
y = 15; Note that y = 15 is a horizontal line going thru y = 15; black line
x = 10 is a vertical line going thru x = 10, I am unable to put this in
:
I assume you know how to make up an x/y table and plot a graph, if you can't,
let me known and we will go thru the graphing routine
:
Here is the graph:

:
When you draw your graph (or print this one) draw a vertical line thru x = 10
This will complete the area of feasibility which is:
:
1.At or below, the purple line or the green line whichever is lower.
2.At or above the black horizontal line
3.At or to the right of the vertical line which you will add to the graph
:
Does this all make sense to you?

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The difference of two numbers is 63. The second is 9 less than 4 times the first. What are the two numbers?
:
Let x = a number:
:
Just write an equation/expression for what it says:
"The difference of two numbers is 63"
x and (x + 63)
:
"The second is 9 less than 4 times the first."
(x + 63) = 4x - 9
:
What are the two numbers?
x + 63 = 4x - 9
:
63 + 9 = 4x - x
:
72 = 3x
:
x = 72/3
:
x = 24
:
Our two numbers are 24 and 87
:
Check our answers in the statement:
"The second is 9 less than 4 times the first."
87 = 4(24) - 9
87 = 96 - 9; confirms our solution

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Use elimination to solve:
6x - y/4 = -9
3x + 2y = -30
:
Get rid of the denominator in the 1st equation by multiplying it by 4
4(6x) - y = 4(-9)
24x - y = -36
Now multiply this equation by 2 and add to the 2nd equation:
:
48x - 2y = -72
3x + 2y = -30
-----------------adding eliminates y; find x
51x + 0y = -102
x = -102/51
x = -2
:
Find y using the 2nd equation:
3(-2) + 2y = -30
-6 + 2y = -30
2y = -30 + 6
2y = -24
y = -24/2
y = -12
:
:
Check solutions using the 1st equation
6x - y/4 = - 9
6(-2) - (-12/4) = -9
-12 + 3 = -9; minus a minus is a plus

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You've answered the question, you don't have to go anywhere from there.
:
You found the axis of symmetry: x = 6/5 or 1.2 seconds
You found the vertex to be 24 ft
:
It took 1.2 sec to get to your friend and it was then 24 ft high

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A home-based company produces both hand-knitted scarves and sweaters. The scarves take 2 hours of labor to produce, and the sweaters take 14 hours. The labor available is limited to 40 hours per week, and the total production capacity is 5 items per week. Write a system of inequalities representing this situation, where x is the number of scarves and y is the number of sweaters. Then graph the system of inequalities.
:
x = no. of scarves; y = no. of sweaters:
:
The labor (in hours) inequality:
2x + 14y =< 40
Arrange in the general (y=) form so we can graph it:
14y =< 40 - 2x
y =< (40/14) - (2/14)x
y =< 20/7 - (1/7)x
:
The production inequality:
x + y = 5
y = 5 - x
:
Plot these two inequalities (I assume you know how to plot a graph, if not you
can email me at ankor@dixie-net.com and I will add the procedure to do that.)
:
your graph should look like this:

:
The area of feasibility would be at or below either line, whichever is lower
It's assumed that x and y => 0

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-9- radical 108, over 3(negative nine minus a square root of 108 over 3)
My answer is the square root of 3 times sqr root of 4, is this correct?
:
I don't think so; I would do it this way.
:
Assume you mean:

:
Factor inside the radical to reveal a perfect square (9)

:
Extract the sqrt of 9

:
Which is:
-
:
cancel the 3's and

:
Factor inside the radical to reveal another perfect square:

:

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simplest radical form: radical 50x^3y^5
;

:
Factor it inside the radical to show the perfect squares

:
Extract those perfect squares

:
Did you follow what we did here OK?

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One day, the temperature at 9:00 A.M. was 42°F, and by 2:00 P.M. the temperature was 57°F. What was the hourly rate of temperature change?
:
you can translate this to: 57-42 = 15 degree change in 5 hrs
Therefore
= 3 degrees per hour

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A cement manufacturer buys limestone for $3 per pound and a special shale mix for $5 per pound. The two ingredients are mixed into a 100-pound compound that cost $4.25 per pound. How much limestone and how much of the shale mix should be added into the compound?
:
Let x = amt of limestone
Then
(100-x) = amt of shale
:
3x + 5(100-x) = 4.25(100)
:
3x + 500 - 5x = 425
:
3x - 5x = 425 - 500
:
-2x = -75
:
x = -75/-2
:
x = 37.5 lb of limestone required
and
100 - 37.5 = 62.5 lb of shale

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Just multiply: .85 * 8882

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If a rock is thrown upward with an initial velocity of 64 ft/second from the top of a 25-foot building.
:
:
1) write the hight (s) equation using this information.
:
This will be a quadratic equation: s = -16t^2 + 64t + 25
S is the height after t seconds
:
The three terms of the above equation represent:
-16t^2: the force of gravity, negative because it's pulling down
64t: the velocity of the rock thrown upward, positive because it is going up
25: the height of the building where all this takes place
:
2) how high is the rock after one second?
Substitute 1 for t in the equation and find s
s = -16(1^2) + 64(1) + 25
s = -16 + 64 + 25
s = 73 ft after 1 second
:
3) after how many seconds will the graph reach maximum height?
This will occur at the axis of symmetry: Remember x = -b/(2a)
t = -64/2(-16)
t = -64/-32
t = + 2 seconds
:
4) what is the maximum height?
This occurs when t = 2; substitute 2 for t in the equation, find s
s = -16(2^2) + 64(2)+25
s = -16(4) + 128 + 25
s = -64 + 128 + 25
s = +89 ft is the maximum
:
A graphical presentation would be:

Y axis is the height and x axis is the time in seconds

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pointing to a man in a photograph,margie said,"his mother's only dughter is my mother".how is margie related to that man.
:
I am usually not too good at these kind of problems, however, no one seems to be jumping in on this;
:
Here's what I think
"His mother's only daughter is my mother" means it has to be the brother of her mother. Therefore she's his niece.
: