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A rectangle is three times as long as it is wide.
The perimeter is 40 feet.
Find the width.
:
Let x = the width
then
3x = the length
:
The perimeter
2L = 2W = 40
so we have:
2(3x) + 2x = 40
6x + 2x = 40
8x = 40
x =
x = 5 ft is the width
;
;
Check:
2(15) + 2(5) = 40

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If 3ax+b=c, then what does x equals
:
2ax + b = c
Subtract b from both sides
2ax = c - b
Divide both sides by 2a
x =

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An airplane travels 260 miles per hour into a 50 mile per hour head wind.
On the return trip the wind has slowed to 25 miles per hour and the plane averages 335 miles per hour.
How fast would the plane have gone with no wind on either leg?
:
Let s = speed of plane in still air
then given that:
(s-50) = 260
s = 260 + 50
s = 310 mph in still air
and return trip
(s+25) = 335
s + 25 = 335
s = 335 - 25
s = 310 mph, as expected

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the distance between two cities is southwest ten miles. to drive, you must travel 8 miles east.
how many miles north will you have to drive?
:
This is a right triangle problem; use a^2 + b^2 = c^2
where
a = 8 (east)
c = 10 southwest
find b (north)
8^2 + b^2 = 10^2
64 + b^2 = 100
b^2 = 100 - 64
b^2 = 36
b =
b = 6 mi north

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The ages of Sean and Bryan now total 49 years.
Bryan’s age now is twice the age that Sean’s was when Bryan’s age was the same as Sean’s is now.
How many years old is Bryan?
:
Write an equation for each statement:
:
"The ages of Sean and Bryan now total 49 years."
s + b = 49
s = (49-b)
:
"Bryan’s age now is twice the age that Sean’s was when Bryan’s age was the same as Sean’s is now."
b = 2(s - (b - s))
b = 2(s + s - b)
b = 2(2s - b)
b = 4s - 2b
b + 2b = 4s
3b = 4s
Substitute (49-b) for s
3b = 4(49-b)
3b = 196 - 4b
3b + 4b = 196
7b = 196
b =
b = 28 yrs is Bryan's age
then
49 - 28 = 21 yrs is Sean's age
:
Let see if that works in the statement:
There is a 7 yr difference in their ages
"Bryan’s age now is twice the age that Sean’s was when Bryan’s age was the same as Sean’s is now."
28 = 2(21 - 7)
28 = 2(14)

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x + y = 8
y = x - 3
:
Rearrange the 2nd equation, add to the 1st equation
x + y = 8
-x+ y = -3
-------------adding eliminates x find y
0 + 2y = 5
y =
y = 2.5
:
Find x, the original 2nd equation
2.5 = x - 3
2.5 + 3 = x
5.5 = x
:
Ordered pair: 5.5, 2.5
:
Check solution in original 1st equation
x + y = 8
5.5 + 2.5 = 8

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The units of a two digit number is 1 more than twice the tens digit. If the digits are reversed, the new number is 36 more than the original number. Find the number.
:
Let x = the 10's digit
Let y = the units
then
10x + y = the number
:
"The units of a two digit number is 1 more than twice the tens digit."
y = 2x + 1
:
"If the digits are reversed, the new number is 36 more than the original number."
10y + x = 10x + y + 36
10y - y = 10x - x + 36
9y = 9x + 36
Simplify, divide by 9
y = x + 4
Substitute (2x+1) for y
2x + 1 = x + 4
2x + x = 4 - 1
x = 3
then
y = 2(3) + 1
y = 7
:
37 is the number
:
:
Check solution in the statement:
"If the digits are reversed, the new number is 36 more than the original number. "
73 = 37 + 36

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a paint company wants to make a 40% solution from a 25% solution and 50% solution of a chemical.
how many gallons of the 25% and 50% mixture do they need to make 100 gallons of the 40% mixture?
:
Let x = amt of 50% solution
then
(100-x) = amt of 25% solution
:
.50x + .25(100-x) = .40(100)
.50x + 25 - .25x = 40
.50x - .25x = 40 - 25
.25x = 15
x =
x = 60 gal of 50% solution
and
100 - 60 = 40 gal of 25% solution
:
:
Check
.5(60) + .25(40) = .4(100)
30 + 10 = 40

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Two cyclist start biking from a trail's start 3 hours apart,
The second cyclist travels @ 10mph and starts 3 hours after the first cyclist who is traveling @ 6mph.
How much time will pass before the second cyclist catches up with the first from the time the second cyclist started biking?
:
Let t = travel time of the 2nd cyclist
then
(t+3) = travel time of the 1st cyclist
:
When the 2nd catches the 1st, they will have traveled the same distance
:
Write a distance equation: dist = speed * time
:
2nd cyclist dist = 1st cyclist dist
10t = 6(t+3)
10t = 6t + 18
10t - 6t = 18
4t = 18
t =
t = 4.5 hrs for the 2nd cyclist to catch the 1st
:
:
Check solution (1st cyclist travel time = 7.5 hr)
10 * 4.5 = 45 mi
6 * 7.5 = 45 mi

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How much sugar (g) do I need to make a 20% solution, total volume = 1 gallon?
:
1 gal of water weighs
:
If the solution is 20% sugar by weight, take the weight of one gallon of water and multiply by 20%:
One gallon of water weighs 8.34 lbs
0.2 * 8.34 = 1.668 lbs.

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Let their present ages by a & n
:
Write an equation for each statement:
:
two years ago, Anton was three times as old as his sister, Nathalie.
a - 2 = 3(n-2)
a - 2 = 3n - 6
a = 3n - 6 + 2
a = 3n - 4
:
In 4 years, he will be only twice his sister's age.
a + 4 = 2(n + 4)
a + 4 = 2n + 8
a = 2n + 8 - 4
a = 2n + 4
:
Find their present ages?
Replace a with the 1st equation, find n
3n - 4 = 2n + 4
3n - 2n = 4 + 4
n = 8
:
I'll let you find a

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How many different prime numbers are factors of 1988?
Prime factors, 2 * 2 * 7 * 71: so B.3

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just add em up and divide by 4:
=

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Find two consecutive positive numbers
x, (x+1)
:
such that the product of the sum and difference of the numbers plus 8 is the sum of their squares.
(x + (x+1))*((x+1)-x) + 8 = x^2 + (x+1)^2
:
((2x+1) * 1) + 8 = x^2 + (x^2 + 2x + 1)
:
2x + 9 = 2x^2 + 2x + 1
:
9 - 1 = 2x^2 + 2x - 2x
:
8 = 2x^2
:
4 = x^2
:
x = 2 and 3 are the numbers
:
Check in statement
"the product of the sum and difference of the numbers plus 8 is the sum of their squares.
(2+3) * (3-2) + 8 = 2^2 + 3^2
5 + 8 = 4 + 9

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Write an equation in slope-intercept form of line that passes thru -5,7
And is perpendicular to the graph of y = x - 1
:
Find the slope of the perpendicular line (m2):
m1 * m2 = -1
:
* m2 = -1
m2 = -1 *
m2 =
:
use the point/intercept formula: y - y1 = m(x - x1)
where
m =
x1 = -5
y1 = 7
:
y - 7 = (x - (-5))
y - 7 = (x + 5)
y - 7 = x + (*5)
y - 7 = x +
y = x + + 7
y = x + +
y = x +
which is
y = 2x + 20

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What is the area of the trapezoid with lower base of 12 cm, upper base of 8 cm if a leg 4 cm long makes an angle of 60 degrees with the lower base?
:
Area of trapezoid: A =
where
a = 12
b = 8
we have to find h
We can use the sine of 60 to find h
sin(60) =
.866 =
h = .866 * 4
h = 3.46 cm
:
A =
A =
A =
A = 34.64 sq/cm
:
Hope this made your Christmas Merry!

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A median of a triangle measures 1 m. What is the measure of the corresponding median of a similar triangle with an area 3 times the area of the original triangle?
:
When you want 3 times the area, the sides are multiplied by
This would also be true for the median. 1 * = 1.732 meters

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Two numbers are such that their difference, their sum, and their product are to
each other as 1:7:24. Their product must equal what number?
:
Two numbers a & b
Let x = the multiplier
:
a - b = 1x
a + b = 7x
a * b = 24x
:
Add the 1st two equations
a - b = x
a + b = 7x
2a = 8x
a = 4x
or
x = .25a
:
a * b = 24x
Replace 24x; a = 4x therefore:
a * b = 6a
b = 6
;
Using the 1st equation
a - b = 1x
Replace b with 6 and x with .25a
a - 6 = .25a
a - .25a = 6
.75a = 6
a =
a = 8
:
Find the multiplier
a - b = x
8 - 6 = 2
:
Check this
a - b = 2 (1*2)
a + b = 14; (7*2)
a * b = 48: (24*2)
:
The numbers are 8 and 6; their products = 48

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A 10‐foot piece of wire is cut into two pieces.
One piece is bent to form a square.
The other is bent to form a circle inscribed in the square.
Approximately how long are the two pieces of wire?
:
let x = length of 1 piece, use this for the square
and
(10-x) = length of the other piece, for the circle
:
.25x = one side of the square
:
The side of the square will be the diameter of the circle
:
Find the circumference of the circle
c =
:
Circle + square = 10
() + x = 10
.7854x + x = 10
1.7854x = 10
x =
x = 5.6" is the one piece
and
10 - 5.6 = 4.4" is the other piece
:
I guess that would be (d), Merry Christmas!!

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A dairy farmer had 54 liters of milk in 3 churns.
He wanted to make sure each churn contained 18 liters.
First, he poured 1/4 of the firs churn into the second churn.
Then he poured 1/2 of the second churn into the third churn.
Finally, he poured 1/3 of the third churn into the first churn.
How many liters did each churn contain before he started pouring?
:
Let x, y, z = the amts in each churn originally
:
"A dairy farmer had 54 liters of milk in 3 churns."
x + y + z = 54
:
"he poured 1/4 of the first churn into the second churn.
then
1st churn = .75x
2nd churn = (y+.25x)
:
"Then he poured 1/2 of the second churn into the third churn."
then
2nd churn = .5(y+.25x) = (.5y+.125x)
3rd churn = z+.5y+.125x
:
" he poured 1/3 of the third churn into the first churn."
then
3rd churn = (z+.5y+ .125x)
1st churn = .75x + (z+.5y+.125x)
:
At this point we assume all churns = 18 gal
:
1st churn
.75x + (z+.5y+.125x) = 18
Multiply by 3:
2.25x + z + .5y + .125x = 54
2.375x + .5y + z = 54
:
2nd churn
.5y + .125x = 18
.125x + .5y = 18
.5y = 18 - .125x
multiply by 2, can use this for substitution
y = (36 - .25x)
:
3rd churn
(z+.5y+ .125x) = 18
multiply by 3
2(z+.5y + .125x) = 54
2z + y + .25x = 54
.25x + y + 2z = 54
Replace y with (36-.25x)
.25x + (36-.25x) + 2z = 54
.25x - .25x + 2z = 54 - 36; (x is conveniently eliminated)
2z = 18
z = 9 gal originally
:
Using the total equation
x + y + 9 = 54
x + y = 54 - 9
x + y = 45
we also know, (use these two equation for elimination)
.125x + .5y = 18
Multiply by 2, subtract x + y = 45
.25x + y = 36
x + y = 45
-----------------subtraction eliminates y
-.75x = -9
x =
x = 12 gal originally
then
12 + y + 9 = 54
y = 54 - 21
y = 33 gal originally
;
:
See if that works
"he poured 1/4 of the first churn into the second churn.
then
1st churn has 9 gal
2nd churn has 36 gal
:
"poured 1/2 of the second churn into the third churn."
then
2nd churn has 18 gal
3rd churn has 27 gal
:
" he poured 1/3 of the third churn into the first churn."
then
3rd churn = 18
1st churn = 18
:
How many liters did each churn contain before he started pouring?
x = 12; y = 33, z = 9
:
Merry Christmas, hope this made sense!!

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Suppose a population of initial size 100 grows at the rate of 8% per year forever.
What is the size of the population at the end of year 1?
What is the size of the population at the end of year 2?
What is the size of the population at the end of year 3?
What is the size of the population at the end of year n (for any integer n)?
What algebraic equation would you need to solve to find the number of years x that it would take for our population to reach 200? Use a calculator to solve to x.
I get year one would be 108 and year two would be 116.64 and year three would be 125.9712 but I cannot get year n
:
You can use a equation similar to compound interest formula
:
P = 100*(1.08)^n
:
where n = the year

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two airplanes start out 10,000 miles apart.
one plane heads east and the other heads west traveling at a rate 100 mph faster than the other plane. after how many hours will they pass each other?
:
There are multiple solutions to this problem
Let s = speed of the slower plane
then
(s+100) = speed of the faster speed
:
Both planes travel the same length of time (t)
:
Write a distance equation Dist = speed * time:
:
Slow plane dist = fast plane dist = 10,000 miles
ts + t(s+100) = 10000
ts + ts + 100t = 10000
2ts + 100t = 10000
t(2s + 100) = 10000
t = = =
:
t =
:
Enter the speed (s) of the slower plane to get the time (t)
For example
s = 200 mph; t = 20 hrs
s = 1200 mph; t = 4 hrs

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A diver jumps from a 48 foot cliff into the ocean below with an initial velocity of 8 ft/sec. The height of the diver from the ocean after t sec is given by h = -16t2 + 8t + 48. When does the diver hit the water?
:
h = the height at t seconds
:
When it hits the water h = 0, therefore:
-16t^2 + 8t + 48 = 0
Simplify and change the signs, divide by -8
2t^2 - t - 6 = 0
Factor
(2t + 3)(t - 2) =
the positive solution
t = 2 sec
:
Check solution in original equation
h = -16(2^2) + 8(2) + 48
h = -16(4) + 16 + 48
h = -64 + 64
h = 0 when it hits the water
:
:
Swimmer will hit the water in 2 seconds from a height of 48 ft

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$1000 to the first born and 1/10 of what then remains,
then $2000 to the second born and 1/10 of what then remains
then $3000 to the third born and 1/10 of what then remains,
and so on. When this was done each child had the same amount
How many children were there?
:
let a = original amt to be divided among his children:
:
1sh. 1000 + .1(a-1000) = (1000 + .1a - 100)
(.1a + 900) = 1st child amt
:
Subtract that from original amt:
a - (900+.1a) = (.9a - 900)
:
2sh. 2000 + .1[(.9a-900)-2000] = 2000 + .1(.9a-2900) = (2000 + .09a - 290)
(.09a + 1710) = 2nd child amt
;
It says the shares are equal, therefore we can find a:
1st child sh = 2nd child sh
.1a + 900 = .09a + 1710
.1a - .09a = 1710 - 900
.01a = 810
a =
a = $81,000; original amt to be divided
:
Find the share given to the 1st child using the equation
1sh = .1a + 900
1sh = .1(81000) + 900
1sh = 8100 + 900
1sh = $9000 amt to 1st child
:
Check to see if the 2nd child equation yields the same amt
2sh = .09a + 1710
2sh = .09(81000) + 1710
2sh = 7290 + 1710
2sh = $9000 amt to 2nd child, so we are on the right track
:
It asks how many children are there
= 9 children each getting 9000

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A and B, traveling at respective rates of a and b kilometers per hour, head directly towards each other across a distance of 120 kilometers. If both start at 10:30 a.m., they will meet at noon. If A starts at 10:00 a.m. and B starts at 11:00 a.m., they will also meet at noon. Find the ordered pair (a,b).
:
When a meets b, their total travel dist = 120 km;
:
Travel time of both starting at 1:30: 1.5 hrs
When a starts at 10: 2 hrs
When b starts at 11: 1 hr
:
Write two distance equations; Dist = speed * time
1.5a + 1.5b = 120
and
2a + 1b = 120
or
b = 120-2a
:
Substitute above for b in the 1st equation, find a
1.5a + 1.5(120-2a) = 120
1.5a + 180 - 3a = 120
1.5a - 3a = 120 - 180
-1.5a = -60
a =
a = 40
and
b = 120 - 2(40)
b = 120 - 80
b = 40
:
Ordered pair, 40,40
:
:
Check solution in 2nd equation
2(40) + 1(40) = 120

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A car travels at 1/2 miles per hour for 8 minutes, then 1 mph for 4 minutes,
then 2 mph for 2 minutes, then 4 miles per hour for 1 minute, and so forth:
stopping after traveling 128 miles per hour for 1/32 minutes.
How many miles does the car travel?
:
Interesting that at each stage, the car travels the same distance; 1/15 of a mile
Converting minutes to hours:
.5(8/60) = (4/60) = (1/15)mi
1(4/60) = (1/15)mi
The last one
128(1/32)*(1/60) = (128/1920) = (1/15) mi
:
Total of all 9 stages
9(1/15) = .6 mi

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by increasing the cost of entry ticket to a fair in the ratio 10:13,
the number of visitors to the fair has decreased in the ratio 6:5
in what ratio has the total collection increased or decreased;
:
Let original Tickets cost $10 and and number of visitors is 600
Original revenue: 10 * 600 = $6000
then
Tickets cost $13 and number of visitors is 500
New Revenue: 13 * 500 = $6500
:
Increase of total collection ratio: reduces to: 13:12

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If water freezes at 32 degrees fahrenheit, what does it freeze at converting it to the celsius scale?
:
The formula: C = (F - 32)
C = (32 - 32)
C = 0 degrees

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" A rancher wants to yield the max amount of edible beef off of his square mile of land. At first he finds that as he adds more cattle, his yield goes up.
However, if he overgrazes, the yield goes down. An agricultural agent tell him that his yield will vary quadratically with the # of animals that he grazes.
For 5 head of cattle, his production is 8750 pounds of beef, and for 10 head of cattle, he reaps 15,000 pounds of beef.
He had no production when he was grazing no cattle."
Find the particular equation of this function expressing pounds of beef produced in terms of the number of cattle.
:
Using the form; ax^2 + bx + c = y, c=0, find a and b
x=5 (no. of cattle); y=8750 (lb of beef)
a(5^2) + 5b = 8750
25a + 5b = 8750
and
x=10; y=15000
100a + 10b = 15000
:
Multiply the 1st equation by 2, subtract from the above equation
100a + 10b = 15000
50a + 10b = 17500
---------------------subtraction eliminates b, find a
50a = -2500
a =
a = -50
;
;
Find a using the 1st equation, replace a with -50
25a + 5b = 8750
25(-50) + 5b = 8750
-1250 = 5b = 8750
5b = 8750 + 1250
5b = 10000
b =
b = 2000
:
The quadratic function (lb of beef for No. of cattle): f(x) = -50x^2 + 2000x
;
:
Check solution using x=10
y = -50(10^2) + 2000(10)
y = -50(100) + 20000
y = -5000 + 20000
y = 15000 as given in the problem

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i am a ski racer and i found out that im going 9 ft per second how many mph would that be?
:
= 6.136 mph

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Here's another way to do this.
:
The product of two two-digit numbers is 1484.
The product of their units digits is 24 and the product of their tens digits is 10.
The two numbers can be?
:
1st two digit number 10a + b
2nd two digit number 10c + d
:
"The product of two two-digit numbers is 1484."
(10a + b)&(10c + d) = 1484
FOIL
100ac + 10ad + 10bc + bd
:
"The product of their units digits is 24"
b * d = 24
:
"the product of their tens digits is 10."
a * c = 10
We know that a is 5 and c is 2 or vice versa
:
In the 1st equation, we find we can replace bd with 24 and ac with 10
100(10) + 10ad + 10bc + 24 = 1484
1000 + 10ad + 10bc + 24 = 1484
10ad + 10bc + 1024 = 1484
10ad + 10bc = 1484 - 1024
10ad + 10bc = 460
Simplify, divide by 2
5ad + 5bc = 230
:
Assume a=2, c=5
2(5d) + 5(5b) = 230
10d + 25b = 230
10d = 230 - 25b
d = 23 - 2.5b
Only two single digit integer solutions to this equation:
b=8; d=3 (the only pair that satisfy eq: b * d = 24)
and assuming; a=2; c=5
therefore:
28, 53 can be the numbers
:
check: 28 * 53 = 1484