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Lee can frame a cabin in 4 days less than Ron.
When they work together, they will do the job in 4 and 4/5 days.
How long would each of them take to frame the cabin
:
Change 4 to 4.8
:
Let t = time required by Lee to do the job
then
(t+4) = time required by Ron
:
Let the completed job = 1
:
+ = 1
Multiply by t(t+4) to get rid of the denominators, results:
4.8(t+4) + 4.8t = t(t+4)
:
4.8t + 19.2 + 4.8t = t^2 + 4t
:
9.6t + 19.2 = t^2 + 4t
:
0 = t^2 + 4t - 9.6t - 19.2
a quadratic equation
t^2 - 5.6t - 19.2 = 0
:
Use the quadratic formula:

In this problem: x=t; a=1; b=-5.6; c=-19.2

:

:

:

the positive solution

:
t =
t = 8 days, Lee working alone
and
8 + 4 = 12 days, Ron alone
:
:
Check solution in original work equation
4.8/8 + 4.8/12 =
.6 + .4 = 1

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3 girls, Laura, Tammy, and Jeri can wash the car, clean the pool,
and mow the lawn in 1 hour and 20 minutes.
If Jeri did all the work, she would take twice as long as Tammy and
2 hours longer than Laura.
How long would each girl take to do the job alone?
:
Use minutes for the time: 1 hr 20 min = 80 min; 2 hr = 120 min
:
let x = time required by Tammy alone
then
2x = Jeri's time alone
and
(2x-120) = Laura's time alone (2hrs less than Jeri)
:
let the completed job = 1 (includes all three tasks
:
+ + = 1
Multiply equation by 2x(2x-120) to clear the denominators
80(2(2x-120) + 80(2x-120) + 80(2x) = 2x(2x-120)
:
320x - 19200 + 160x - 9600 + 160x = 4x^2 - 240x
:
320x + 160x + 160x - 19200 - 9600 = 4x^2 - 240x
:
640x - 28800 = 4x^2 - 240x
:
0 = 4x^2 - 240x - 640x + 28800
:
0 = 4x^2 - 880x + 28800
Simplify, divide by 4
x^2 - 220x + 7200 = 0
Factor:
(x - 40)(x - 180) = 0
Two solutions,
x = 40; does not make sense
therefore
x = 180 min, (3 hrs) is Tammy's time alone
then
2*180 = 360 min, (6 hrs) is Jeri's time alone
and
360 - 120 = 240 min, (4 hrs) is Laura's time
;
;
Check solution using 1.33 hr working together
1.33/3 + 1.33/6 + 1.33/4
.44 + .22 + .33 = .99 ~ 1

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Trains A and B are traveling in the same direction on parallel tracks.
Train A is traveling at 60 mils per hour and train B is traveling at 70 mils per hour.
Train A passes a station at 5:15 A. M. if train B passes station at 5:27 A.M,
what time train b catch up train A?

:
from the given information, we know that train B is 12 min (.2 hr) behind
train A, when train A passes the station
:
The distance between the trains at this time: .2 * 70 = 14 mi
:
Let t = time required for train B to catch train a
:
write a distance equation: Dist = speed * time
:
Train A travel dist = Train b travel dist + 14 mi
70t = 60t + 14
70t - 60t = 14
10t = 14
t =
t = 1.4 hr or 1 hr 24 min
:
5:15 + 1:24 = 6:39 AM, B catches A
:
:
Check solution by finding the dist traveled by each train
70*1.4 = 98 mi
60*1.4 = 84 mi
----------------
differ = 14 mi

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: A fruit grower has 150 acres of land available to raise two crops, A and B.
It takes 1 day to trim an acre of crop A and 2 days to trim an acre of crop B,
and there are 240 days available each year for trimming.
It takes 0.3 days to pick an acre of crop A and 0.1 days to pick an acre of crop B,
and there are 30 days per year available for picking.
The profit is $140 per acre for crop A and $235 per acre for crop B.
Determine the optimal acreage for each fruit and the optimal profit.
(Show all your work including any graphs you used.)
:
Write an equation for each statement, rearrange for graphing:
:
" A fruit grower has 150 acres of land available to raise two crops, A and B."
A + B = 150
A = 150-B; (red)
:
It takes 1 day to trim an acre of crop A and 2 days to trim an acre of crop B,
and there are 240 days available each year for trimming."
1A + 2B = 240
A =< 240-2B; (green)
:
It takes 0.3 days to pick an acre of crop A and 0.1 days to pick an acre of crop B,
and there are 30 days per year available for picking."
.3A + .1B = 30
.3A = 30 - .1B
A =< 100 -B; (blue)
:
Graph these 3 equations, A on the y axis, B on the x axis

:
The corner points of the area of feasibility:
1. A=100,B=0
2. A=75, B=75
3. A=60,B=90
4. A=0, B=120
:
The profit is $140 per acre for crop A and $235 per acre for crop B.
Find the values of these points:
1. 140(100) + 235(0) = $14,000
2. 140(75) + 235(75) = $28,125
3. 140(60) + 235(90) = $29,550
4. 140(0) + 235(120) = $28,200
:
Plant 60 acres of Plant A, and 90 acres of Plant B, for max profit

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max is 24 years older than his son liam.
in two years liam will be half as old as max.
how old is liam now? and max?
i assumed it could be put like this but it has not been working for me.
m = l + 24
:
Both will be two years older, so you have
l+2 = .5(m+2)
l + 2 = .5m + 1
l = .5m + 1 - 2
l = .5m - 1
from the 1st equation substitute (l+24) for m
l = .5(l+24) - 1
l = .5l + 12 - 1
l - .5l = 11
.5l = 11
Multiply both sides by 2 to get 1l
l = 22
then
m = 22 + 24
m = 46
:
:
See if that is true in the statement:
"in two years liam will be half as old as max."
22 + 2 = .5(46 + 2)
:
Did this make sense to you now?

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If the price of copper is 65 cents/lb and the price of zinc is 30 cents/lb,
how many pounds of copper and zinc should be mixed to make 70 lb of brass
selling for 45 cents per pound?
:
Let x = amt of copper required
then
(70-x) = amt zinc required
:
65x + 30(70-x) = 45(70)
:
65x + 2100 - 30x = 3150
:
65x - 30x = 3150 - 2100
:
35x = 1050
x =
x = 30 lb of copper
and
70 - 30 = 40 lb of zinc
;
;
Check solution in original equation
65(30) + 30(40) = 45(70)
1950 + 1200 = 3150

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Together, Michelle, Sal, and kristen can clean and wax a car in 1 hr 20 min.
To complete the job alone, Michelle needs twice the time that Sal needs
and 2 hr more than kristen.
How long would it take each to clean and wax the car working alone?
:
Use minutes for the time: 1 hr 20 min = 80 min; 2 hr = 120 min
:
let x = time required by Sal alone
then
2x = Michelle's time alone
and
(2x-120) = Kristen's time alone
:
let the completed job = 1
:
+ + = 1
Multiply equation by 2x(2x-120) to clear the denominators
80(2(2x-120) + 80(2x-120) + 80(2x) = 2x(2x-120)
:
320x - 19200 + 160x - 9600 + 160x = 4x^2 - 240x
:
320x + 160x + 160x - 19200 - 9600 = 4x^2 - 240x
:
640x - 28800 = 4x^2 - 240x
:
0 = 4x^2 - 240x - 640x + 28800
:
0 = 4x^2 - 880x + 28800
Simplify, divide by 4
x^2 - 220x + 7200 = 0
Factor:
(x - 40)(x - 180) = 0
Two solutions,
x = 40; does not make sense
therefore
x = 180 min, (3 hrs) is Sal's time alone
then
2*180 = 360 min, (6 hrs) is Michelle's time alone
and
360 - 120 = 240 min, (4 hrs) is Kristen time
;
;
Check solution using 1.33 hr working together
1.33/3 + 1.33/6 + 1.33/4
.44 + .22 + .33 = .99 ~ 1

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A farmer sold 17 bushels of barley, and 13 bushels of wheat, for $31.55 ;
he received for the wheat 35 cents a bushel more than for the barley ;
what was the price of each per bushel ?
;
Let x = amt received for barley
then
(x+.35) = amt received for wheat
:
17x + 13(x+.35) = 31.55
:
17x + 13x + 4.55 = 31.55
:
30x = 31.55 - 4.55
x =
x = $.90 received for barley
and
.90 + .35 = $1.25 received for Wheat
:
:
You can check the solution on a calc
.9(17) + 1.25(13) =

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: Rose bought green mangoes at P50.00 a kilo and sold them at 4 for P35.00.
If she sold 20 kilos and there is an average of 7 mangoes per kilo,
how much did she earn?
:
Find the cost she paid for each Mango:
Find the no. of mangoes sold: 20 * 7 = 140 mangoes
Find the price she sold each Mango:
:
Let p = profit she made by buying and selling these Mangos
:
p = 140()
p = 140()
p = 140()
p = 5(45)
p = 225 P profit on 20 kilos of Mangos

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He sells a cereal mixture that is 60 parts oatmeal, 25 parts bran, and 15 parts wheat germ.
Each batch of cereal weights 80 kilograms.
Jake’s problem is that customers are requesting more bran in the mix.
They would like the batch he just prepared to have 30 parts bran out of the total 100 parts.
:
How many kilograms of the original mix of cereal should be removed and
replaced with bran so that his customers will be satisfied?
:
Determine what percentage is the Bran, in the original mixture
= = .15
:
Let x = amt of mixture to be removed, and amt of bran to be added
.15(80-x) + x = .30(80)
12 - .15x + x = 24
-.15x + x = 24 - 12
.85x = 12
x =
x = 14.1 kg removed, 14.1 kg of bran added

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The half-life of a substance is the time it takes for half of the substance
to remain after natural decay. Radioactive water (tritium) has a half-life
of 12.6 years. How long will it take for 85% of a sample to decay?
:
The half life formula:
A = Ao
where:
A = the resulting amt after t (yrs in this case)
Ao = initial amt
t = time (yrs)
h = half-life of substance (yrs)
:
Let initial amt: Ao = 1, then find A: 1.0 - .85 = .15
:
1*2^(-t/12.6) = .15
Find the log of both sides
.301 = -.8239
= -.8239
Multiply both sides by 12.6
-.301t = -.8239 * 12.6
:
-.301t = -10.381
t =
t = 34.49 yrs
:
:
Check solution on a calc: enter 2^(-34.49/12.6) = .1499 ~ .15

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I need to know how to set this problem up....
A silversmith combined pure silver that cost $5.20 per ounce with 50 oz of a
silver alloy that cost $ 2.80 per ounce. How many ounces of the pure silver
were used to make an alloy of silver costing $4.40 per ounce?
:
Treat it like a typical mixture problem
Let x = amt of pure silver required ($5.20)
Since the resulting total is 50 oz,
(50-x) = amt of $2.80 silver required
:
5.2x + 2.8(50-x) = 4.4(50)
:
Solve for x

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The measure of angle x is 18 more than twice the measure of angle y.
If angle x and y are complementary angles, find the measure of x
:
Write an equation for each statement
"The measure of angle x is 18 more than twice the measure of angle y."
x = 2y + 18
or
x - 2y = 18
:
"If angle x and y are complementary angles,"
x + y = 90
;
find the measure of x
:
Use elimination, subtract the 1st equation from the above equation:
x + y = 90
x - 2y = 18
---------------subtraction eliminates x, find y
3y = 72
y =
y = 24 degrees
:
Find x:
x = 2y + 18
x = 2(24) + 18
x = 48 + 18
x = 66 degrees
:
Check solution (are they complimentary?):
66 + 24 = 90

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Trains A and B are traveling in the same direction on parallel tracks.
Train A is traveling at 40 miles per hour and train B is traveling at 48 miles per hour.
Train A passes a station at 3:15 pm.
If train B passes the same station at 3:30 pm, at what time will train B catch up to train A?
:
from the given information, we know that train B is 15 min (1/4 hr) behind
train A, when train A passes the station
:
The distance between the trains at this time: * 48 = 12 mi
:
Let t = time required for train B to catch train a
:
write a distance equation: Dist = speed * time
:
Train B travel dist = Train A travel dist + 12 mi
48t = 40t + 12
48t - 40t = 12
8t = 12
t =
t = 1.5 hrs
:
3:15 + 1:30 = 4:45 pm, B catches A

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A Broadway theater has 500 seats, divided into orchestra, main, and balcony seating.
Orchestra seats sell for $40, main seats for $25, and balcony seats for $20.
If all the seats are sold, the gross revenue to the theatre is $13,150.
If all the main and balcony seats are sold, but only half the orchestra seats are sold,
the gross revenue is $11,150.
How many are there of each kind of seat?
:
Let x = Orchestra seats
Let y = Main seats
Let z = Balcony seats
:
Write an equation for each statement:
:
"A Broadway theater has 500 seats, divided into orchestra, main, and balcony seating."
x + y + z = 500
:
"Orchestra seats sell for $40, main seats for $25, and balcony seats for $20.
If all the seats are sold, the gross revenue to the theatre is $13,150."
40x + 25y + 20z = 13150
:
"but if only half the orchestra seats are sold, the gross revenue is $11,150."
40(.5x) + 25y + 20z = 11150
20x + 25y + 20z = 11150
:
use elimination on these two equations
40x + 25y + 20z = 13150
20x + 25y + 20z = 11150
-------------------------subtraction eliminates y and z
20x = 2000
x =
x = 100 orchestra seats
:
Using this value for x, write two equations with two unknowns.
100 + y + z = 500
y + z = 500 - 100
y + z = 400
and
40(100) + 25y + 20z = 13150
4000 + 25y + 20z = 13150
25y + 20z = 13150 - 4000
25y + 20z = 9150
:
Multiply the 1st equation by 20 and subtract from the above equation
25y + 20z = 9150
20y + 20z = 8000
-------------------subtraction eliminates z
5y = 1150
y =
y = 230 Main seats
:
Find the number of Balcony seats using the total seat equation:
100 + 230 + z = 500
z = 500 - 330
z = 170 Balcony seats
;
;
Check our solutions by finding the total revenue
40*100 = 4000
25*230 = 5750
20*170 = 3400
-------------
total = 13150; confirms our solutions
;
;
Did this make sense to you?

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1.) a plane left C and flew in the direction 270degrees for 3hrs. at 279mph
and landed at B.
a second plane flew from C in the direction 222degrees30minutes for 4hrs. and
landed at A.
If A is due south of B, find the distance between the two points and the average speed of the second plane
:
It will help to draw this out, a right triangle ABC, Angle B is a right angle
:
Change 222 deg, 30 min to 222.5 degrees
:
270 - 222.5 = 47.5 degrees is angle C:
:
279 * 3 = 837 mi is the distance C to B flown by the 1st plane
:
Let s = speed of the 2nd plane
then
4s = distance from C to A (also the hypotenuse of the triangle)
:
Use the cosine of 47.5 to find S:
Cos(47.5) = 837/4s
:
.67559 = 837/4s
:
.67559*4s = 837
:
2.70236s = 837
s =
s = 309.73 mph speed of the 2nd plane
:
Find the distance from C to A; (Traveled for 4 hrs at the above speed):
4 * 309.73 = 1238.9 miles from C to A
:
:
Find the distance from A to B (call it d) by using the Sine of 47.5
:
Sin(47.5) =
.73728 =
d = .73728 * 1238.9
d = 913.4 mi from A to B
:
:
Did all this make sense to you?

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a ski lift carries a skier up a slope at the rate of 800 feet per minute and
he returns from the top to the bottom on a path parallel to the lift at an
average rate of 2640 feet per minute.
How long is the lift if the round trip traveling time is 20 minutes?
:
Let d = the length of the lift
:
Write a time equation: Time =
:
Time up + time down = 20 min
+ = 20
Multiply equation by 211200:
211200* + 21120* = 211200*20
Cancel out the denominators:
264d + 80d = 4224000
:
344d = 4224000
d =
d = 12,279.07 feet
:
:
Check solution using this value in the time equation:
+ =
15.349 + 4.651 = 20.00

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Find 3 consecutive even numbers where the product of the smaller two numbers is 64 less than the square of the largest number.
:
"3 consecutive even numbers": x, (x+2), (x+4)
:
" where the product of the smaller two numbers is 64 less than the square of the largest number"
x *(x+2) = (x+4)^2 - 64
:
x^2 + 2x = x^2 + 8x + 16 - 64
:
x^2 - x^2 + 2x - 8x = 16 - 64
:
-6x = -48
x =
x = +8
:
8, 10, 12 are the numbers
:
:
Check solution in the statement:
"the product of the smaller two numbers is 64 less than the square of the largest"
8 * 10 = 144 - 64

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In a triangle, the perimeter is 121 feet, the shortest side is 15 feet longer
than one-half of the longest side, the second longest side is 4 feet
shorter than the longest side. What is the length of each side?
:
Three sides: a, b, c; (c is the shortest side)
:
"the perimeter is 121 feet, "
a + b + c = 121
:
"the shortest side is 15 feet longer than one-half of the longest side,"
c = .5a + 15
:
" the second longest side is 4 feet shorter than the longest side."
b = a - 4
:
Using the perimeter equation, substitute for b and c; find a
a + (a-4) + (.5a+15) = 121
a + a + .5a - 4 + 15 = 121
2.5a + 11 = 121
2.5a = 121 - 11
2.5a = 110
a =
a = 44 ft is the longest side
:
you can find b and c using the given equations

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When the digits of a two-digit number are reversed, the new number is 9 more
than the original number, and the sum of the digits of the original number
is 11. What is the original number?
:
Two digits:
x = the 10's digit
y = units digit
:
The original number; 10x + y
:
"When the digits of a two-digit number are reversed, the new number is 9 more
than the original number,"
10y + x = 10x + y + 9
:
10y - y = 10x - x + 9
:
9y = 9x + 9
Simplify, divide by 9
y = x + 1
:
"the sum of the digits of the original number is 11."
x + y = 11
Replace y with (x+1)
x + x + 1 = 11
2x = 10
x = 5
then
y = 5 + 1
y = 6
:
56 = the original number
:
Check solution in the statement:
"when the digits of a two-digit number are reversed, the new number is
9 more than the original number,"
65 = 56 + 9

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A rectangular piece of plywood is trimmed to make a square by cutting a 4-cm
strip off the top and a 2-cm strip off one side.
If the area of the original piece is 74 cm2 greater than the area of the
square, find the dimensions of the rectangle.
:
Let L = length of the rectangle
Let W = width of the rectangle
then draw a diagram and it will be apparent that:
(W-2) = side of the square
and
(L-4) = side of the square
Therefore
W - 2 = L - 4
W = L - 4 + 2
W = (L-2)
:
"If the area of the original piece is 74 cm2 greater than the area of the square, find the dimensions of the rectangle."
L * W = 74
Replace W with (L-2)
L*(L-2) = 74
L^2 - 2L - 74 = 0
We have to use the quadratic equation to solve this:

In this problem: x=L; a=1; b=-2; c=-74

:

:

:

Positive solution

:

L = 9.66 cm is the length of the rectangle
then
W = 9.66 - 2 = 7.66 cm is the width of the rectangle
;
:
Check solution by finding the area using these values:
9.66 * 7.66 = 73.9975 ~ 74; confirms our solution

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A retailer spent $48 to purchase a number of special mugs. Two of them were broken in the store, but by selling each of the remaining mugs for $3 above the original cost per mug, she made a total profit of $22. Need to solve for the number of mugs originally purchased? Denoted by n.
:
Price paid for each mug =
:
No. of mugs sold = (n-2)
:
Selling price = + 3
;
Total revenue = 48 + 22 = $70:
:
#sold * Price sold = total revenue
(n-2) * [ + 3] = 70
FOIL
48 + 3n - 96/n - 6 = 70
3n - 96/n + 42 - 70 = 0
3n - 96/n - 28 = 0
Multiply equation by n, forming a quadratic equation:
3n^2 - 28n - 96 = 0
Factor
(3n +8)(n-12) = 0
Positive solution
n = 12 mugs originally bought
:
Check
Cost: 48/12 = $4
Sold: 10 * 7 = $70 (cost + profit)

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The square root of 4a divided by square root of 3b.
:

:
Multiply by (any number over itself is one)
*
:
Gets rid of the radical in the denominator
=
Factor inside the radical to reveal perfect square

Extract the square root of 4, which is 2

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What is the formula?

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Find a quadratic funtion in the form y=ax^2+bx+c
That passes through the points (-3,-13),(-1,-1),(2,32)
:
Write an equation for each set of coordinates:
:
x = -3; y = -13
9a - 3b + c = -13
:
x = -1; y = -1
a - b + c = -1
:
x = 2; y = 32
4a + 2b + c = 32
:
Subtract the 2nd equation from the 1st equation
9a - 3b + c = -13
a - b + c = -1
----------------eliminates c
8a - 2b = -12
:
Subtract the 2nd equation from the 3rd equation
4a + 2b + c = 32
a - b + c = -1
-----------------eliminates c
3a + 3b = 33
Simplify,divide by 3
a + b = 11
:
multiply above equation by 2, add to
8a - 2b = -12
2a + 2b = 22
-------------eliminates b
10a = 10
a = 1
:
Find b using a + b = 11
1 + b = 11
b = 11 - 1
b = 10
;
find c using a - b + c = -1
1 - 10 + c = -1
-9 + c = -1
c = -1 + 9
c = 8
:
The equation: y = x^2 + 10x + 8
:
Prove this to yourself by substituting the given values for x and finding y

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Half the supplement of an angle is 12 degrees less than twice the complement of the angle.
Find the measure of the angle.
:
Let the angle be x:
:
Write a equation for the statement:
"Half the supplement of an angle is 12 degrees less than twice the complement of the angle."
(180-x) = 2(90-x) - 12
Multiply both sides by 2 to get rid of the fraction, results:
180 - x = 4(90-x) - 24
:
180 - x = 360 - 4x - 24
-x + 4x = 360 - 24 - 180
3x = 156
x =
x = 52 degrees is the angle
;
:
Check solution in the original equation:
(180-52) = 2(90-52) - 12
(128) = 2(38) - 12
64 = 76 - 12

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3 is a zero of f(x)=2x^3-3x^2-11x+6. Use synthetic division to find the zeros of f.
....._________________
3 |2 - 3 - 11 + 6
..............6 + 9 - 6
......------------------
......2 + 3 - 2 + 0
;
so we have
2x^2 + 3x - 2 = 0
Factors to
(2x - 1)(x + 2) = 0
Two solutions
2x = 1
x =
and
x = -2
:
:
The zeros: -2, +, +3

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The weight M of an object on Mars varies directly as its weight E on Earth.
A person who weighs 95 lbs on Earth weighs 38 lbs on Mars.
How much would a 100 lb person weight on Mars?
:
Since it just varies directly, we can use a ratio equation.
m = weight on Mars
=
Cross multiply
95m = 38*100
m =
m = 40 lb on Mars

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The moving sidewalk at Chicago's airport moves 1.8 ft/sec.
Walking on the moving sidewalk, Joe travels 105 ft forward in the time it takes
to travel 51 ft. in the opposite direction.
How fast would Joe be walking on a non-moving sidewalk?
:
Let s = speed on a non-moving sidewalk
:
(s+1.8) = speed moving with it
(s-1.8) = speed moving against it
:
Write a time equation: Time = dist/speed
:
=
Cross multiply
105(s-1.8) = 51(s+1.8)
105s - 189 = 51s + 91.8
105s - 51s = 91.8 + 189
54s = 280.8
s =
s = 5.2 ft/sec on an ordinary sidewalk
:
:
Check solution by find that the times are equal
105/(5.2+1.8) = 15 sec
51/(5.2-1.8) = 15 sec

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Find the cube root of 64, multiply by 4

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The arithmetic mean of 6 numbers is 17.
If two numbers are added, the new set of numbers will have an arithmetic mean of 19.
What are the two numbers if their difference is 4?
:
Mean is the same as average.
:
Let the two number be x & y
:
Write an equation for each statement:
;
"If two numbers are added, the new set of numbers will have an arithmetic mean of 19."
= 19
= 19
Multiply both side by 8, results
x + y + 102 = 152
x + y = 152 - 102
x + y = 50
:
"What are the two numbers if their difference is 4?
x - y = 4
x = (y+4)
:
Substitute (y+4) for x in the 1st equation, find y
(y + 4) + y = 50
:
2y = 50 - 4
y =
y = 23 is one number
then
x - 23 = 4
x = 27 is the other number
;
:
Check solution
=
= 19
:
:
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