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Let m - Martin's age now; Let g = Gary's age now
:
Write an equation for each statement:
"If Martin was four times as old as Gary 8 years ago"
(m-8) = 4(g-8)
m - 8 = 4g - 32
m - 4g = -32 + 8
m - 4g = -24
:
" and if Martin will be twice as old as Gary 8 years hence,"
(m+8) = 2(g+8)
m + 8 = 2g + 16
m - 2g = 16 - 8
m - 2g = 8
:
Subtract the 1st equation from the above equation:
m - 2g = 8
m - 4g = -24
--------------subtraction eliminates m
0 + 2g = +32
g =
g = 16 yrs Gary's age now
:
Use the equation; m - 2g = 8 to find m
m - 2(16) = 8
m - 32 = 8
m = 8 + 32
m = 40 yrs is Martin's age now
:
:
Check solution in the statement:
"If Martin was four times as old as Gary 8 years ago and if Martin will be twice as old as Gary 8 years hence, find their ages."
40 - 8 = 4(16-8)
and
40 + 8 = 2(16+8)

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Henry sold 120 magazine subscriptions in x+2 days. If he sold at the same rate for another week, then how many magazines did he sell in the extra week?
:
The no. of mags sold per day:

:
No. of mags sold in 7 days (the extra week):
7* =
:
:
You can prove this to yourself, let x = 2
= 30 mags per day
In one week: 7 * 30 = 210
:
Using our solution:
= 210

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A new automobile engine can run on a mixture of gasoline and a substitute fuel. If gas costs $1.50 per gallon and the substitute fuel costs 40 cents per gallon, what percent of a mixture must be substitute fuel to bring the cost down to $1 per gallon?
:
Let x = percent of Substitute fuel
:
.40x + 1.50(100-x) = 100
:
.4x + 150 - 1.5x = 100
.4x - 1.5x = 100 - 150
-1.1x = -50
x =
x = 45.45 % of substitute fuel
:
Check:
1.50 gas: 100 - 45.45 = 54.55
.40(45.45) + 1.50(54.55) =
18.18 + 81.825 = 100.00
:
I am not absolutely sure about this. Let me know if I screwed this one up. Carl

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Find all values of x satisfying the given conditions.
:
y1 = 10(2x-1), y2 = 2x+1, and y1 is 14 more than 8 times y2.
:
Use some logic here, it says:
y1 = 8(y2) + 14
:
From the 1st two given equations:
we can substitute 10(2x-1) for y1 and 2x+1 for y2, in the above; therefore:
:
10(2x-1) = 8(2x+1) + 14
Multiply the terms inside the brackets, and solve for x
20x - 10 = 16x + 8 + 14
20x - 10 = 16x + 22
Some basic algebra:
20x - 16x = 22 + 10
4x = 32
x =
x = 8
:
:
y1= 1/x, y2= 1/2x, y3= 1/x-1, and the sum of 3 times y1 and 4 times y2 is the product of 4 and y3(corrected this typo).
:
Write an equation for the statement:
"the sum of 3 times y1 and 4 times y2 is the product of 4 and y3"
3(y1) + 4(y2) = 4(y3)
:
Substitute: (1/x) for y1; (1/2x) for y2; and (1/(x-1)) for y3

which is:

2 will cancel into 4 so we have:

we can add like fractions

Cross multiply and find x:
4x = 5(x-1)
4x = 5x - 5
4x -5x = -5
-x = -5
x = 5
:
:
did this make sense to you now?

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The 3 consecutive integers: x, x+1, x+2
:
find three consecutive integers such that the sum of their squares is 77
x^2 + (x+1)^2 + (x+2)^2 = 77
FOIL
x^2 + (x^2 + 2x + 1) + (x^2 + 4x + 4) = 77
:
x^2 + x^2 + 2x + 1 + x^2 + 4x + 4 = 77;
Group like terms
x^2 + x^2 + x^2 + 2x + 4x + 1 + 4 = 77
:
3x^2 + 6x + 5 = 77
:
3x^2 + 6x + 5 - 77 = 0
:
3x^2 + 6x - 72 = 0
Simplify divide by 3
x^2 + 2x - 24 = 0
Factor
(x+6)(x-4) = 0
Two solutions:
x = -6 and x = +4
:
Our consecutive numbers:
-6, -5, -4
and
4, 5, 6

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48.A rectangular plot of ground is 120 yards long and 80 yards wide.To double the area of the plot, while retaining the rectangular shape, strips of equal width will be added at one end and along one side. Find the width of the strip?
:
See #154457
:
49.A motorboat takes 4 hours to travel 30 miles downstream and then 18 miles upstream on a river whose current flows at the rate of 3 miles per hour. How fast can the boat travel in still water?
:
Let s = boat speed in still water
then
(s+3) = speed downstream
and
(s-3) = speed upstream
:
Write a time equation: Time = dist/speed
:
Time upstream + Time downstream = 4 hrs
+ = 4
Multiply the equation by the common denominator, (s+4)(s-4); results:
30(s-3) + 18(s+3) = 4(s+3)(s-3)
:
30s - 90 + 18s + 54 = 4(s^2 - 9)
:
48s - 36 = 4s^2 - 36
arrange as a quadratic equation:
4s^2 - 48s -36 + 36 = 0
4s^2 - 48s = 0
4s(s - 12) = 0
s = 12 mph boat in still water
;
Check solution; speed upstream 9 mph, downstream 15 mph; confirm the time:
(30/15) = 2
(18/9) = 2
-------------
total = 4 hrs
:
:
50.The floor of a basement game room is covered by 480 square asphalt tiles of a certain size. if 3 inches were added to the length of a side of each tile, the floor could be covered by 270 tiles. find the length of a side of the tile on the floor?
:
Let x = the side dimension of the original tile
then
(x+3) = the side dimension of the projected tile
:
That areas remain the same so we have:
480x^2 = 270(x+3) ^2
FOIL the right
480x^2 = 270(x^2 + 6x + 9)
:
480x^2 = 270x^2 + 1620x + 2430
Arrange as a quadratic on the left
480x^2 - 270x^2 - 1620x - 2430 = 0
:
210x^2 - 1620x - 2430 = 0
Simplify, divide equation by 30:
7x^2 - 54x - 81 = 0
Factor
(7x+9)(x-9) = 0
Positive solution:
x = 9 in, the side of the original tile
:
:
Check solution by finding the area of each scenario:
480(9^2) = 38880
270(12^2) = 38880

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draw a graph of an equation that contains two x-intercepts, -2 and 3.
:
Write the equation for this; we know x=-2 and x=3
Therefore the factors can be (x+2)(x-3) = 0
FOIL these and you have:
x^2 - 3x + 2x - 6 = 0
x^2 - x - 6 = 0 is the equation
:
Plot y = x^2 - x - 6, for x = -4 to x = +5, it should look like this:

Note the x intercepts are x=-2 and x=+3

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An inlet pipe can fill a tank in 1 hour; a drain pipe can empty it in 3/2 hours. The inlet pipe was opened at 6:00 a.m. to fill the tank. At 6:20 a.m. someone noticed that the drain has been left open and so he closes it immediately. How long will it take the first pipe to finish filling.
:
Let's deal in minutes here:
Inlet time = 60 min
Drain time = 90 min
:
let t = time (in minutes) required to finish filling the tank.
:
Let the full tank = 1:
:
Total on time for the inlet pipe = (t+20)
On time for the drain = 20 min
:
- = 1
Multiply equation by 180 to get rid of the denominators, results:
3(t+20) - 2(20) = 180
:
3t + 60 - 40 = 180
3t + 20 = 180
3t = 180 - 20
3t = 160
t =
t = 53.333 min or 53 min 20 sec to finish filling the tank
:
:
Check with calc (inlet time: 53.33 + 20 = 73.33)
(73.33/60) - (20/90) = 1

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Jan invested $1200 at a certain simple interest rate and $2200 at a rate 3% higher. Her annual earnings were $253. Find the two interest rates if she earned $121 more on the larger investment than on the smaller.
:
You don,t need the last statement:
:
Let x = interest rate (in decimal form) of the $1200 investment
Then
(x+.03) = interest rate of the $2200 investment
:
1200x + 2200(x+.03) = 253
:
1200x + 2200x + 66 = 253
:
3400x = 243 - 66
:
3400x = 187
x =
x = .055, 5.5% int rate on the $1200
then
.055 +.03 = 8.5% int rate on the $2200
:
Check:
.055 * 1200 = 66
.085 * 2200 = 187
------------------
total int: = 253
:
:
187 - 66 = 121, proves the last statement is true, but we did not need it

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Write an equation for each statement.
:
"the sum of the ages of Ira and Sofia is 26.
I + S = 26
or
I = (26-S)
:
" the difference between 4 times Sofia's age and twice Ira's age is 8."
4S - 2I = 8
;
Substitute (26-S) for I in the above equation, find S;
4S - 2(26-S) = 8
4S - 52 + 2S = 8
4S + 2S = 8 + 52
6S = 60
S = 10
Then
I = 26 - 10
I = 16
;
:
Check solution using the statement:
"the difference between 4 times Sofia's age and twice Ira's age is 8."
4(10) - 2(16) = 8

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1.a rectangular plot of ground is 120 yards long and 80 yards wide.To double the area of the plot, while retaining the rectangular shape, strips of equal width will be added at one end and along one side. Find the width of the strip?
:
Let x = width of the strip
:
Dimensions of new rectangle will be (x+120) by (x+80)
Old rectangle area = 120*80 = 9600
:
Write an area equation for the new rectangle
(x+120)*(x+80) = 2(9600)
FOIL
x^2 + 80x + 120x + 9600 = 19200
:
x^2 + 200x + 9600 - 19200 = 0
:
x^2 + 200x - 9600 = 0
Factors to:
(x+240)(x-40) = 0
Positive solution
x = +40 yds is the width of the strip that doubles the area
;
;
Check solution:
(40+120) * (40+80) = 19200 which is double the original area

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a florist sold 80 stalks of lilies and threw away 1/7 of the remaining stalks of lilies.the florist has 2/5 of the lilies left. how many stalks of lilies were left?
;
This is the algebra forum, so algebra is all we know about.
:
Let x = Original no. of lilies
Total - sold - threw away = lilies left(given as 2/5 of the total)
x - 80 - (x-80) = x
:
Multiply equation by 35 to get rid of the denominators
35x - 35(80) - 5(x-80) = 7(2x)
:
35x - 2800 - 5x + 400 = 14x
:
35x - 5x - 14x - 2800 + 400 = 0
:
16x - 2400 = 0
:
16x = 2400
x =
x = 150 lilies originally
:
* 150 = 60 left
:
Check:
150 - 80 = 70 not sold
*70 = 10 threw away
60 remaineth, which is 2/5 of 150

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I need to find the surface area of a Christmas tree I made for my summer project. The cone shape is 22 inches tall and has a diameter of 7.5 inches. There is also a base for the cone shape. I also have a cylinder shape for the tree trunk which is 3 inches in diameter and 5 inches in height. Please show me the calculations. I am using 2TTR(R+H) for both shapes but I do not think it is right and then I am not sure how to do the base of the cone.
:
First just find the surface area of the cone
The formula for that: SA = +
:
In your problem r = 3.75 (half the diameter),
:
l is the slanted distance of the cone,
that will be the hypotenuse of a triangle formed by r and h
l =
l = 22.3173 inches
;
SA = +
SA = 44.1786 + 262.9195
SA = 307.0981 sq inches
:
Add the trunk:
Trunk =
Trunk = 47.1239
:
Total; 307.0981 + 47.1239 = 354.222 sq/in (this includes the circular surface area of the bottom of the trunk)

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Mrs. Shieh was 24 years old when Ingred was born. In three years the sum of their ages will be 68 years. How old is each now?
:
We can say that S is 24 yrs older than I, therefore
S = (I+24)
:
An equation for the statement:
"In three years the sum of their ages will be 68 years."
(S+3) + (I+3) = 68
S + I + 6 = 68
S + I = 68-6
S + I = 62
:
How old is each now?
:
Substitute (I+24) for S in the above equation
(I+24) + I = 62
2I = 62 - 24
I =
I = 19 yrs
then
19+24 = 43 yrs is S
:
:
Check solution in the statement:
"In three years the sum of their ages will be 68 years."
(43+3) + (19+3) = 68

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mechanic has found that a car with a 16-quart radiator has a 40% antifreeze mixture in the radiator. The mechanic has 70% antifreeze solution on hand. How much of the 40% solution would the mechanic have to replace with the 70% solution to get the solution in the radiator up to 50%.
:
Let x = amt of 40% to be removed, and the amt of 70% to be added
:
.40(16-x) + .70x = .50(16)
;
6.5 - .4x + .7x = 8
:
-.4x + .7x = 8 - 6.4
:
.3x = 1.6
x =
x = 5 quarts of 40% removed
x = 5 quarts of 70% to be added
;
:
Check solution on a calc (16 - 5.33 = 10.67):
.4(10.67) + .7(5.33) = .5(16)
4.268 + 3.731 = 7.999 ~ 8, confirms our solution

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one number is two less than a second number. If you take one-half of the first number and increase it by the second number, the result is at least 41. Find the least possible value for the second number.
:
Write an equation for each statement:
"one number is two less than a second number.
x = "one number"
and
(x+2) = "a second number"
;
" If you take one-half of the first number and increase it by the second number, the result is at least 41."
.5x + (x+2) => 41
1.5x + 2 = 41
1.5x = 41 - 2
1.5x = 39
x =
x => 26
:
Find the least possible value for the second number.
:
Least value: 26 + 2 = 28
:
Check solution in statement
"one-half of the first number and increase it by the second number, the result is at least 41.
.5(26) + 28 = 41

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An employer has a daily payroll of $1225 when employing some workers at $80 per day and others at $85 per day. When the number of $80 workers is increased by 50% and the number of $85 workers is decreased by 1/5, the new daily payroll is $1540. How many were originally employed at each rate?
:
Let x = no. of $80 workers (originally)
Let y = no. of $85 workers
:
$1225 payroll equation
80x + 85y = 1225
:
$1540 payroll equation; (50% increase = 1.5; and 1/5 decrease = .8
80(1.5x) + 85(.8y) = 1540
120x + 68y = 1540
:
Multiply the 1st equation by 1.5 and subtract the above equation:
120x + 127.5y = 1837.5
120x + 68.0y = 1540
------------------------subtraction eliminates x, find y:
0x + 59.5y = 297.5
y =
y = 5 ea $85 workers originally
:
Find x using the 1st equation:
80x + 85(5) = 1225
80x + 425 = 1225
80x = 1225 - 425
x =
x = 10 ea $80 workers originally
:
:
Check solution in equation: 120x + 68y = 1540
120(10) + 68(5) = 1540, confirms our solutions

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Write an equation for the line.
Containing x-intercept of 5 and y-intercept of -2.
:
Using the slope/intercept form: y = mx + b;
:
The y intercept occurs when x = 0, therefore
y = mx - 2
:
The x intercept occurs when y = 0, therefore
m(5) - 2 = 0
5m = + 2
m =
:
Therefore the equation is:
y = x - 2

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6x + 7 = 20/x
;
Multiply both sides by x to get rid of the denominator
x(6x + 7) = 20
:
Mult terms inside the brackets:
6x^2 + 7x = 20
;
Subtract 20 from both sides to reveal our old friend, the quadratic equation
6x^2 + 7x - 20 = 0
:
With diligent effort we can factor this to:
(2x + 5)(3x - 4) = 0
Two solutions
2x = -5
x =
x = -2.5
and
3x = 4
x =
;
:
Check the 1st solution in the original equation
6(-2.5) + 7 =
-15 + 7 = -8
;
Check the 2nd solution
6* + 7 =
8 + 7 = 15

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Simplify and write with positive exponents 9x ( x^-3) / 18 x^15.
:
Use the reciprocal to get rid of the neg exponent:
/
Same as:
*
:
Cancel 9x into 18x^15, multiply denominators:
* =

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Sue can shovel snow from her driveway in 65min. Bill can do the job in 50min. How long would it take sue and Bill to shovel the same driveway together?
:
Let = time required, working together (in minutes)
:
Let the completed job = 1 (Provided they each have a shovel)
:
Each will do a fraction of the job which will add up to 1:
:
+ = 1
;
Multiply equation by a common denominator 65*50; resulting in:
50t + 65t = 65*50
;
115t = 3250
t =
t = 28.26 minutes working together
:
:
Check solution using a calc: enter
(28.26/65) + (28.26/50) = .99997 ~ 1

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/
Invert the dividing fraction and multiply
*
Cancel like terms and you have
which is

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Elena bicycles 5km/h faster then Carlos. In the same time it takes Carlos to bicycle 39km, Elena can bicycle 54km. How fast does each bicyclist travel?
:
Let s = C's speed
then
(s+5) = E's speed
:
it says the times are equal: write a time equation: time = dist/speed
:
=
cross multiply
54s = 39(s+5)
:
54s = 39s + 195
:
54s - 39s = 195
:
15s = 195
s =
s = 13 km/hr is C's speed
then
13 + 5 = 18 km/hr is E's speed
:
:
Check solutions by finding the times
54/18 = 3 hr
39/13 = 3 hr

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Y varies directly as X. When X is 33, Y is 165. Find Y, when X is 14.
;
We know there is a constant (k) involved here:
y = k*x
:
When y = 165 and x = 33, we can write it:
33k = 165
k =
k = 5
:
When x = 14
y = 5(14)
y = 70
:
;
Check it with a calc using proportion:
165/70 = 33/14

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A company determines that its total cost is given by the function
C(x) = 2.5x^2 - 125x + 10,000 where x is the number of units produced and C(x) is in dollars.
:
A. How many units must be produced to minimize cost?
:
In a quadratic equation, we can find the axis of symmetry to find at what value of x, c is minimum:
:
Axis of symmetry: x =
In this equation a=2.5; b+ -125
:
x = =
x = 25 units will give minimum cost
;
;
B. What is the minimum cost?
:
Find min cost by substituting 25 for x in the given equation:
C(x) = 2.5(25^2) - 125(25) + 10000
C(x) = 2.5(625) - 3125 + 10000
C(x) = 1562.5 - 3125 + 10000
C(x) = $8437.50 is the minimum cost

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An object is projected upward with an initial velocity of 251 feet per second from a height of 32 feet. During what time period will its height exceed 91 feet?
:
-16t^2 + 251t + 32 = 91
:
-16t^2 + 251t + 32 - 91 = 0
:
-16t^2 + 251t - 59 = 0
:
Using the quadratic formula;
the period will be between .24 sec and 15.45 sec it's height will be above 91 ft

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you have thrown a baseball straight upward from a height of about 6 feet. A friend has used a stop watch to record the time the ball is in the air. If it takes approximately 6.5 seconds before the ball strikes the ground, explain how you can find the ball's initial velocity.
;
Using the equation -16t^2 + vt + c = h
Where:
h = height of the ball at t sec
t = time in seconds
v = velocity (upward)
-16t^2 = gravity (downward)
c = initial height
:
In this problem t = 6.5 sec. c = 6 ft; h = 0 (ground)
:
-16(6.5^2) + 6.5v + 6 = 0
:
-16(42.25) + 6.5v + 6 = 0
:
-676 + 6.5v + 6 = 0
:
6.5v - 670 = 0
v =
v = 103.1 ft/sec is the velocity

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An employer has a daily payroll of $1225 when employing some workers at $80 per day and others at $85 per day. When the number of $80 workers is increased by 50% and the number of $85 workers is decreased by 1/5, the new daily payroll is $1540. How many were originally employed at each rate?
:
Let x = no. of $80 workers (originally)
Let y = no. of $85 workers
:
$1225 payroll equation
80x + 85y = 1225
:
$1540 payroll equation; (50% increase = 1.5; and 1/5 decrease = .8
80(1.5x) + 85(.8y) = 1540
120x + 68y = 1540
:
Multiply the 1st equation by 1.5 and subtract the above equation:
120x + 127.5y = 1837.5
120x + 68.0y = 1540
------------------------subtraction eliminates x, find y:
0x + 59.5y = 297.5
y =
y = 5 ea $85 workers originally
:
Find x using the 1st equation:
80x + 85(5) = 1225
80x + 425 = 1225
80x = 1225 - 425
x =
x = 10 ea $80 workers originally
:
:
Check solution in equation: 120x + 68y = 1540
120(10) + 68(5) = 1540, confirms our solutions

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(sqr of 12xy)cubed times (x times sqr of xsquared y)=
:
Assume your mean:
=
We can write this:

Which is:
= = ; factor to reveal the perfect squares
Extract the perfect squares
=

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two-thirds of the diameter of a circle that is 9/16 across
:
* = =

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Joe has a collection of nickels and dimes that is worth $5.65. If the number of dimes were doubled and the number of nickels were increased by 8, the value of the coins would be $10.45. How many dimes does he have?
:
Write an equation for each statement:
:
"Joe has a collection of nickels and dimes that is worth $5.65."
.05n + .10d = 5.65
:
" If the number of dimes were doubled and the number of nickels were increased by 8, the value of the coins would be $10.45. "
.05(n+8) + .10(2d) = 10.45
simplify
.05n + .4 + .20d = 10.45
.05n + .20d = 10.45 - .40
.05n + .20d = 10.05
:
Now subtract the 1st equation from the above equation and eliminate n, find d
: