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it it take 15 people 8 hours to make 100 items how long would it take 6 people working the same rate to make half as many items?
:
I think you are right;
Here's one way to do it:
:
Find out how many man-hrs per item
man-hrs = = = 1.2 man-hrs per item
:
Let x = hrs for 6 people to make 50 items
6x = 1.2(50)
:
6x = 60
x =
x = 10 hrs for 6 people to make 50 items
:
Put this all in one equation
x = *

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Use synthetic division to find the quotient and the remainder
(x^5+32)divided by (x+2)
:
:
Write it like this and proceed
....._____________________
-2|1 + 0 + 0 + 0 + 0 + 32
:
I got x^4 - 2x^3 + 4x^2 - 8x + 16, and remainder 0

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A polynomial in x has degree 3. The coefficient of x(with an exponent of 2) is 3 less than the coefficient of x(with an exponent of 3). The coefficient of x is three times the coefficient of x(with an exponent of 2). The remaining coefficient is 2 more than the coefficient of x(with an exponent of 3). The sum of the coefficients is -4. Find the polynomial.
:
Let a = coefficient of x^3
:
Write an expression for each statement:
;
"The coefficient of x^2 is 3 less than the coefficient of x^3"
(a-3)
:
"The coefficient of x is three times the coefficient of x^2."
3(a-3)
:
"The remaining coefficient is 2 more than the coefficient of x^3".
(a+2)
:
"The sum of the coefficients is -4."
a + (a-3) + 3(a-3) + (a+2) = -4
a + a - 3 + 3a - 9 + a + 2 = -4
6a - 10 = -4
6a = -4 + 10
6a = +6
a = +1
:
Using a = 1 we can write it:
(1)x^3 + (1-3)x^2 + 3(-2)x + (1+2)
:
y = x^3 - 2x^2 - 6x + 3; is the polynomial
:
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Let x = 10's digit
Let y = units digit
:
The original 2 digit number: (10x + y)
:
Write an equation for each statement:
"The units digit of a two-digit number is 1 less than 3 times the tens digit."
y = 3x - 1
:
"If the digits are reversed, the new number is 45 more than the original number."
10y + x = 45 + 10x + y
:
Find the number.
10y - y = 10x - x + 45
9y = 9x + 45
Simplify divide equation by 9
y = x + 5
;
Using the equation from the 1st statement, substitute (3x-1) for y; find x
3x - 1 = x + 5
3x - x = 5 + 1
2x = 6
x =
x = 3
:
Find y using y = x + 5
y = 3 + 5
y = 8
:
Our original number = 38
:
:
Check solution in statement:
"If the digits are reversed, the new number is 45 more than the original number."
83 = 45 + 38; confirms our solutions

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How do I solve using a system of two linear equations in two variables?
Barry is 8 years older than his sister. In 3 years, he will be twice as old as she will be then. How old is each now?
:
Let x = B's age now
Let y = S's age now
:
Write an equation for the statement,"Barry is 8 years older than his sister."
x = y + 8
or
x - y = 8
:
The same for this statement," In 3 years, he will be twice as old as she will be then.
(x+3) = 2(y+3)
x + 3 = 2y + 6
x = 2y + 6 - 3
x = 2y + 3
x - 2y = 3
:
The system, two equations two variables
x - y = 8
x - 2y = 3
------------ subtracting eliminates x
0 + y = 5 yrs is sister's age now
:
I'll let you find Barry's age

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Phil and Phyllis are siblings. Phyllis has twice as many brothers as she has sisters. Phil has the same number of brothers as sisters. How many girls and how many boys are in the family?
:
x = no. boys
y = no. girls
:
Phil: (x-1) bro, y sis
same number bro than sis:
(x-1) = y
x - y = 1
:
Phyl: x bro, (y-1) sis
Twice as many bro as sis:
x = 2(y-1)
x = 2y - 2
x - 2y = -2
;
Using these two equation for elimination
x - 2y = -2
x - y = 1
-------------subtracting eliminates x
0 - y = -3
y = 3 girls
:
We know x = 2y -2:
x = 2(3) - 2
x = 4 boys
:
Therefore:
Phil has 3 brothers, 3 sisters
Phyl has 4 brothers, 2 sisters

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A plane is flying the 3,458 mile trip from New York City to London has a 50 MPH tailwind. The flight's Point of No Return is the point at which the flight time required to return to New York is the same a the time required to continue to London. If the speed on the plane in still air is 360 MPH, how far is New York from the point of no return?
:
From the given information we can say:
360 + 50 = 410 mph speed (relative to the earth) with the wind
360 - 50 - 310 mph speed against the wind
:
Let d = dist to New York from point of no return
Then
(3458-d) = dist to London from point of no return
:
Write a time equation: Time = dist/speed
:
Return to N.Y time = Continue to London time
=
Cross multiply:
410d = 310(3458-d)
:
410d = 1071980 - 310d
:
410d + 310d = 1071980
:
720d = 1071980
d =
d = 1488.86 mi from N.Y.
:
:
Check solution by finding the distance to London and calculating the time to both:
3458 - 1488.86 = 1969.14 mi to London
:
Time to N.Y.: 1488.86/310 = 4.8 hrs
Time to Lond: 1969.14/410 = 4.8 hrs

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A rectangular lot measures 66m by 107m A sidewalk surrounds the lot The sidewalk is 2m wide what would the area of the sidewalk be
:
The overall dimensions including the surrounding sidewalk: 70 by 111 (add 4 m)
:
Overall area - lot area = sidewalk area
(70*111) - (66*107) =
7770 - 7072 = 708 sq/m is the sidewalk area

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Suppose the record shown played for 16.5 minutes on one side and the average of the radii of the grooves on the record was 3 and 3/4 inches. there are 33 1/3 revolutions per minute. Write an expression involving units that represents how many inches the needle passed through the grooves while the record was being played. then evaluate the expression.
:
This involves finding the average circumference, from the average radius (3.75")
Multiply this by the no. of minutes played at 33 1/3 rpm
:
D = 2*pi*r * time * record speed (rpm)
:
D = 2*pi*3.75 * 16.5 * 33.33
:
D = 25.562 * 16.5 * 33.33
:
D = 12,958 inches

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Mr. Graham invested some of his $18,000 in bonds and made a 5% profit and the rest in bonds that made a 12% profit. If the profit on the 12% bonds was $885 more than the profit on the 5% bonds, how much did Mr. Graham invest in the 5% bonds?
:
Let x = amt invested at 5%
:
It said the total invested was 18000, therefore:
(18000-x) = amt invested at 12%
:
5% profit + 885 = 12% profit
.05x + 885 = .12(18000-x)
:
.05x + 885 = 2160 - .12x
:
.05x + .12x = 2160 - 885
:
.17x = 1275
x =
x = $7500 invested at 5%
;
:
Check solution find the profit on each; 18000-7500 = 10500 invested at 12%
.12*10500 = 1260
.05*7500 = 375
----------------
difference = 885; confirms our solution

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Solve
x - = 5
Multiply equation by the common denominator, that would be 4*7 = 28
28(x) - 28* - 28* = 28(5)
Cancel the denominators:
28x - 7(3) - 4x = 140
:
28x - 4x - 21 = 140
:
24x = 140 + 21
:
24x = 161
x =
:
:
Check solution using decimals in calc; 161/24 ~ 6.71
6.71 - .75 - (6.71/7) = 5.00

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Solve for the indicated variable r+s/t=u.....solve for t
:
Assume you mean:
= u
Multiply both sides by t:
t* = t(u)
Cancel the t's on the left:
r + s = tu
:
Divide both sides by u and you have:
= t
:
:
If you meant the problem to be: r + = u, let me know, and I will edit this solution.

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Each day a man meets his wife at the commuter train station after work, and then she drives him home. She always arrives exactly on time to pick him up. One day he catches an earlier train and arrives at the station an hour early. He immediately begins walking home along the same route the wife drives. Eventually his wife sees him on her way to the station and drives him the rest of the way home. When they arrive home, the man notices that they arrived 20 minutes earlier than usual. How much time did the man spend walking?
:
Here's what I think. Let me know if I am wrong.
If she met him at the earlier time, he would be home 1 hr early. Since he is only 20 min early, he was walking for 40 min, before she picked him up

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Factor completely:
-2x to 3rd power + 16x to 2nd power - 24x
:
-2x^3 + 16x^2 - 24x
:
Factor out -2x and you have:
-2x(x^2 - 8x + 12)
Factors to
-2x(x - 6)(x - 2)

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You provide the manager with a quadratic equation that models the expected number of ticket sales for each day x. (x=1 is the day tickets go on sale).
Tickets = -0.2x^2+12x+11
:
The best way to do this, is to make the actual graph of the equation
it should look like this:
:

:
a. Does the graph of this equation open up or down? How did you determine this?
Look at it
:
b. Describe what happens to the tickets sales as time passes?
You could say the sales increase, reach a peak, then decrease to 0
:
c. Use the quadratic equation to determine the last day that tickets will be sold. (Note: Write your answer in terms of the number of days after ticket sales begin.)
Solve -0.2x^2+12x+11 = 0 using the quadratic formula:
The positive solution will be about approx x = 61. (60th day of ticket sales)
:
d. Will tickets peak or be at a low during the middle of the sale? How do you know?
Look at the graph
:
e. After how many days will the peak or low occur?
Look at the graph
Find the axis of symmetry: x = -b/(2a); a=-.2; b=12
:
f. How many tickets will be sold on the day when the peak or low occurs?
Find the vertex using the above x value
:
g. What is the point of the vertex? How does this number relate to your answers in parts e and f?
That is the max
:
h. How many solutions are there to the equation ? How do you know?
This equation has two solution, but only the positive solution makes sense
:
i. What do the solutions represent? Is there a solution that does not make sense? If so, in what ways does the solution not make sense?

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ratio 3:2:5.
ind 2 numbers such that their sum, their difference, and their product have the ratio 3:2:5.
:
Let the two numbers be x & y:
=
Cross multiply:
2(x+y) = 3(x-y)
2x + 2y = 3x - 3y
3y + 2y = 3x - 2x
5y = x
:
and
=
Cross multiply:
5(x-y) = 2xy
5x - 5y = 2xy
5x - 2xy = 5y
x(5-2y) = 5y
x =
Substitute 5y for x
5y =
Just looking at this we can assume
(5-2y) = 1
-2y = 1 - 5
-2y = -4
y = +2
:
Therefore:
x = 5(2)
x = 10
:
Testing the solution: x=10, y=2
=
and
=

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he sum of the base and height of a triangle is 25 cm. What is the maximum area of the triangle?
:
Let x = length of the base
Then
(25-x) = length of the height
:
Area = .5*L*B
Substituting of L and B
A = .5x(25-x)
A = 12.5x - .5x^2
As a quadratic equation find the axis of symmetry x = ;
a=-.5
b= 12.5
x =
x =
x = + 12.5 is base for max area
then
25 - 12.5 = 12.5 is the length for max area
:
A = .5 * 12.5 * 12.5
A = 78.25 sq cm is max area
:

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With the wind, a plane flew 1400 km in 4 hours. On the return trip, the pilot was forced to land after 1 h 30 min, having traveled only 450 km. Find the rate of the plane in still air and the rate of the wind.
:
Let x = speed of the plane in still air
Let y = speed of the wind
then
(x-y) = speed against the wind
(x+y) = speed with the wind
:
Write two distance equations; dist = time * speed:
;
4(x+y) = 1400
1.5(x-y) = 450
:
Simplify the 1st equation divide by 4
Simplify the 2nd equation divide by 1.5
:
x + y = 350
x - y = 300
-------------adding eliminates y
2x = 650
x =
x = 325 mph plane speed in still air
:
Use x + y = 350 to find y
325 + y = 325
y = 350 - 325
y = 25 mph wind speed.
:
:
Check solution in original 1st equation
4(325 + 25) =
4 * 350 = 1400, confirms our solutions

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he rectangular part of the field below is 140 yards long, and the diameter of each semicircle is 40 yards. Find the cost of reseeding the field at .70 per square yard.
:
Find the area of the two half circles, radius = 20
A =
A = 1256.637
:
Find the area of the rectangular portion:
A = 140 * 40
A = 5600
:
Total: 1256.637 + 5600 = 6856.637 sq yds
:
Cost: .7 * 6856.637 = $4799.65

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How do I solve this system of equations by using multiplication with the addition method?
:
1/5(x+2y) = -2
Multiply by 5 to get rid of the denominator
x + 2y = 5(-2)
x + 2y = -10
:
- =
Multiply equation by 4 to get rid of the denominator, resulting:
x - 2(3y) = 2(15)
x - 6y = 30
:
Multiply the 1st equation by -1 and add to the above equation
-x - 2y = +10
+x - 6y = +30
-----------------
0x - 8y = +40
y =
y = -5
:
Find x using x - 6y = 30:
x - 6(-5) = 30
x + 30 = 30
x = 0
;
:
Check solution using the 1st original equation:
1/5(x + 2y) = -2
1/5(0 + 2(-5) =
1/5 * -10 = -2

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seven times the reciprocal of a number plus 2/3 is 3. Find the number
:
Let the number = x
:
+ = 3
:
multiply equation by 3x to get rid of the denominators
3(7) + 2x = 3(3x)
21 + 2x = 9x
21 = 9x - 2x
21 = 7x
x =
x = 3
:
:
check solution:
+ = 3

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The director of a summer day camp estimates that 100 children will join if the camp fee is 250.00, but for each 20.00 decrease in the fee, ten more children will enroll.
Determine the linear equation that will represent the number of children who will enroll a a given fee.
:
Let x = cost per child
Let y = no. of children enrolled
:
From the information given, derive two set of x/y coordinates
100 kid at $250: x1=250, y1=100
and
110 kids at $230: x2=230, y2=110
:
Find the slope using the slope equation: m =
m = =
m =
:
Use the point/slope equation: y -y1 = m(x-x1)
:
y - 100 = (x - 250)
y - 100 = x + 125
y = x + 125 + 100
y = x + 225; is the linear equation
:
:
Graph the linear equation that represents the number of children who will enroll at a given fee.
:
Graph y = x + 225,
To graph this, use an x scale of 0 to +450, and y scale from 0 to 300
(we are not interested in any neg values here)
You only need two pair of coordinates to plot a linear (straight line) graph
However here are three sets: we will find y for x = 100, x = 200, x = 250
:
Substitute 100 for x, in the equation y = -.5x + 225 and find y
y = -.5(100) + 225
y = -50 + 225
y = +175
:
If we were making a table for x/y values, it would be:
x | y
-------
100|175
:
Substitute 200 for x and find y:
y = -.5(200) + 225
y = -100 + 225
y = +125
now our table is:
x | y
-------
100|175
200|125
:
Substitute 250 for x, find y
y = -.5(250) + 225
y = -125 + 225
y = +100
we have:
x | y
-------
100|175
200|125
250|100
0 |225, note this is the y intercept (the camp is free, 225 kids)
:
Plot these 4 points, join them with straight lines
:
it should look like this:

:
:
How many students will enroll if the camp fee is 180.00?
:
substitute 180 for x in our equation and find y
y = (180) + 225
6 = -90 + 225
y = +135 will enroll
;
:
How many students will enroll if the camp is free?
:
In this case x = 0
y = (0) + 225
y = 225 enrolled if it is free, you can see on the graph, that's the y intercept
:
:
did this make sense to you, any questions?

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need help with these 2 problems.Thanks for all your help.
write the equation of each root
:
1. 6, -2/3
Two factors
(x-6) = 0
and
(3x+2) = 0
FOIL
3x^2 - 18x + 2x - 12 = (3x^2 - 16x - 12) is the equation
:
2. -3/4, -8
Two factors:
(4x+3) = 0
and
(x+8) = 0
FOIL
4x^2 + 32x + 3x + 24 = (4x^2 + 35x +24) is the equation

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Which do you think pays more: 4.25% simple interest for 3 years or 4% interest compounded semi-annually for the same amount of time? Explain.
:
Here is the compound interest formula: A = P*
where
P = principal amt
A = resulting amt
r = interest rate (decimal)
t = time in years
n = no. of periods per year
:
We only need to use the multiplier which is:
:
For 4.25 % simple interest (compounded once a yr) for 3 yrs
= = 1.139955
:
For 4% compounded twice a year, for 3 yrs
= = 1.126162
:
The one with the highest multiplier pays more. (4.25%)
:
Did this help you understand this?

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Farmer Naz was carrying a basket of eggs to sell at the country market. As she was walking to the market, Mrs. Dropsy's cow startled the farmer and she dropped all of her eggs. Mrs. Dropsy felt guilty about this, so she insisted on paying for the eggs. She asked the farmer how many eggs she had, but she could not remember exactly. She did recall that when she divded the number of eggs by 2, 3, 4, or 5, there was only one egg left over, but when she took the eggs out in groups of 7, she completely emptied the basket. What was the least number of eggs in the basket?
:
3 * 4 * 5 = 60
:
Find the 1st integer solution to the equation:
;
y =
Starting with x = 1, it's apparent that the 1st one is x=5
(5 * 60) + 1 = 301 eggs; (43 groups of 7 eggs)

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A concert sold 1,068 tickets bringing in total revenue of $15,060. Three categories of tickets were sold. Type "A" sold for $10, type "B" sold for $15 and type "C" sold for $20. There were 4 times as many "A" tickets sold as there were "B" tickets. How many of each type of ticket generate
:
Total tickets equation:
A + B + C = 1068
:
Total Revenue equation
10A + 15B + 20C = 15060
:
"There were 4 times as many "A" tickets sold as there were "B" tickets."
A = 4B
:
Substitute 4B for A in both total equations
2B + B + C = 1068
3B + C = 1068
C = (1068-3B)
:
10(2B) + 15B + 20C = 15060
20B + 15B + 20C = 15060
35B + 20C = 15060
:
Substitute (1068-3B) for C in the above equation:
35B + 20(1068-3B) = 15060
35B + 21360 - 60B = 15060
35B - 60B = 15060 - 21360
-25B = -6300
B =
B = 252 ea $15 tickets
:
Using equation C = 1068 - 3B)
C = 1068 - 3(252)
C = 1068 - 756
C = 312 ea $20 tickets
:
Find A
A + 252 + 312 = 1068
A = 1068 - 564
A = 504 ea $10 tickets
:
:
You can check our solution:
10(504) + 15(252) + 20(312) =

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Tyrone is twice as old as his daughter. in six years Tyrone's age will be three times what his daughter's age was six years ago. how old are they at present?
:
The equation for the statements,"Tyrone is twice as old as his daughter."
t = 2d
:
and the equation for the statement:
"in six years Tyrone's age will be three time what his daughter's age was six years ago."
t + 6 = 3(d-6)
Simplify
t + 6 = 3d - 18
t = 3d - 18 - 6
t = 3d - 24
:
how old are they at present?
:
We know t = 2d so we have:
2d = 3d - 24
2d - 3d = -24
-d = -24
d = 24 is daughter's age
Then
2(24) = 48 is T's age
:
:
Check solutions in the statement:
in six years Tyrone's age will be three time what his daughter's age was six years ago."
48 + 6 = 3(24 - 6)
54 = 3(18); confirms our solution
:
That wasn't so hard, was it?

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What is the solution to two square root of three over square root of six minus two?
:
Multiply expression by the conjugate of the denominator
FOIL,the middle terms cancel in the denominator, so you eliminate the radical
* = =
:
2 will cancel into 2 and 4 giving us:

:
Factor 18 to reveal a perfect square, extract 3:
=

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A,B,and C can finish a job in six days. If B and C work together, the job will take 9 days; if A and C work together, the job will take 8 days. In how many days can each man working alone do the job?
:
Let the completed job = 1
write 3 equations
+ + = 1
:
+ = 1
:
+ = 1
:
Use elimination: mult the eq1 by 3, and eq2 by 2:
+ + = 3
+ + = 2
----------------------------------------- Subtraction eliminates B & C, Find A
= 1
A = 18 days by himself
:
Use elimination: mult the eq1 by 4, and eq3 by 3:
+ + = 4
+ + = 3
----------------------------------------- Subtraction eliminates A & C, Find B
24/B = 1
B = 24 days by himself
:
Use eq2, substitute 24 for B, find C
+ = 1
Multiply equation by 24C
9C + 24(9) = 24C
216 = 24C - 9C
216 = 15C
C =
C = 14.4 days by himself
;
Check solutions in eq1
+ + =
.333 + .250 + 4.17 = 1

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Let x = 10's digit of a two digit number
Let y = units digit
Let z = the resulting equal digits
:
Write an equation for each statement:
:
"the sum of a two digit number is 10."
x + y = 10
or
y = (10-x)
:
" if 18 be subtracted from it, the digits in the resulting number will be equal."
10x + y - 18 = 10z + z
10x + y = 11z + 18
:
Substitute (10-x) for y
10x + (10-x) = 11z + 18
10x - x = 11z + 18 - 10
9x = 11z + 8
:
The single digit integer values for x and z are limited to x=7 and z=5 in this equation:
x = 7, y = 3, z = 5
:
The number is 73
:
Check in the statement:
"if 18 be subtracted from it, the digits in the resulting number will be equal."
73 - 18 = 55

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f(x) = 2x-5 divided by x+4, find f-1(x).
:
f(x) = y
:
y =
solve for x:
Multiply both sides by (x+4)
2x - 5 = y(x+4)
2x - 5 = xy + 4y
2x = xy + 4y + 5
2x - xy = 4y + 5
x(2-y) = 4y + 5
x =

:
To find the inverse, swap x and y:
y =
f-1(x) =