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How much water must be evaporated from 500 grams of a 10% salt solution
to obtain a 15% salt solution?
;
Let w = amt of water to be evaporated
:
Write an amt of salt equation (the amt remains the same, only the % changes)
.10(500) = .15(500-w)
50 = 75 - .15w
.15w = 75 - 50
w =
w = 166 grams to be evaporated
:
:
Check: That leaves: 500 - 166.67 = 333.33 gram solution after evaporation
.10(500) = .15(333.33)
50 = 50; confirms our answer

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the difference of nine times a number and the quotient of six and the same number...I need to know what that would be if you put it into a expression :
:
9x -

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Sonya walked at a rate of 1/t+a mph for bt+ab hours.
How far did she walk?
Dist = rate * time, therefore
d = * bt + ab
Factor
d = * b(t + a)
Cancel (t+a)
d = b

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I owe a court settlement of $46,000 @ 6% interest and I pay $500.00 a month. How many months is this going to take.
:
To pay off a $46,000 loan at 6% will take about 124 month or 10 yr and 4 months.
:
If I were to offer a lump sum now, would would the present value be, and/or a structured Settlement be?
:
During this time you would pay $16,000 in interest (above the principal of $46,000)
:
I assume that you are charged interest on the unpaid balance, therefore paying it off right away would save you as much as $16,000
:
As to a structured settlement, that is negotiated, like perhaps they would except $40,000 if you paid it right now for example. Be sure get it all in writing if you do that.

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How far can airplane fly from its base, and yet return within 3 hours,
if its outgoing speed is 600 kilometers per hour and its returning speed is 400 kilometers per hour?
:
let d = the one-way distance
:
Write a time equation, Time = dist/speed
out time + return time = 3 hrs
+ = 3
multiply by 1200 to get rid of the denominators, results
2d + 3d = 3(1200)
5d = 3600
d =
d = 720 mi the plane can fly from it's base and return
:
:
Check by finding the times
720/600 = 1.2 hrs
720/400 = 1.8 hrs
-------------------
total time: 3 hrs; confirms our solution of 720 mi

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let x = "a number"
:
Just write an equation for what it says:
"Two less than three times a number is equal to 14 more than five times the number."
3x - 2 = 5x + 14
Obviously this will be a negative number
-2 - 14 = 5x - 3x
-16 = 2x
x =
x = -8 is the number
:
:
Check solution in the original statement: x=-8
3(-8) - 2 = 5(-8) + 14
-24 - 2 = -40 + 14
-26 = -26

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When u reverse the digits of age of father u will get the age of son.
10x + y = father's age
10y + x = son's age
:
one year ago the age of father was twice that of son's age.
10x + y - 1 = 2(10y + x - 1)
10x + y - 1 = 20y + 2x - 2
10x - 2x = 20y - y - 2 + 1
8x = 19y - 1
x = y -
Only one set of single digit integers will satisfy this equation
y = 3; x = 7
:
what are the current ages of father and son?
Dad is 73, son is 37
:
Check one year ago
Dad is 72, son is 36, half Dad's age

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A mechanical engineer drilled a hole through a metal cube.
If the bases of the right cylinder representing the hole are inscribed in the
bases of the cube and the volume of the cube is 256 m, find the volume of the
remaining solid figure after the hole has been drilled.
:
Find the length of a side of the cube, find the cube root of 256
s =
s = 6.3496 m
this is also the diameter of the drilled hole and the height of the hole
r = 6.3496/2
r = 3.1748 is the radius
:
Find volume of the hole (a cylinder)
v =
v = 201 cu/m
:
256 - 201 = 55 cu/m
:
But would it still be a solid figure if you did this; wouldn't it fall apart
into 4 pieces?

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British sterling silver is a copper-silver alloy that is 7.5% copper by weight.
How many grams of pure copper and how many grams of British sterling silver
should be used to prepare 200 grams of a copper-silver alloy that is 10% copper by weight?
:
Let x = amt of pure copper required
then
(200-x) = amt of 7.5% copper alloy required
:
Write a percent copper equation in decimal form
x + .075(200-x) = .10(200)
x + 15 - .075x = 20
x - .075x = 20 - 15
.925x = 5
x =
x = 5.4 grams of pure copper required
then
200 - 5.4 = 194.6 grams of 7.5% alloy required
:
:
Check solution
5.4 + .075(194.6) = .10(200)
5.4 + 14.595 = 20
19.995 ~ 20

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A hiker walks with an average speed of 1.2 m/s
What distance in kilometers does the hiker travel in a time of one hour?
:
3600 sec in 1 hr
:
d =
d = 4.32 km/hr

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e walked 5mph then jogged 8mph , total time 2hours and 15 min ,
how many miles did e go?
:
convert 2 hr 15 min to 2.25 hrs
:
With the limited amt of information, find the average speed
:
Let s = the average speed
:
s =
s = 6.5 mph
:
Distance = Speed * time
d = 6.5 * 2.25
d = 14.625 miles

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the measure of the supplument of an angle is 60 less than 3x the measure of the
complement of the angle. find the measure?
:
Let x = the angle
we know that
(180-x) = the supplement
and
(90-x) = the complement
:
With is in mind just write an equation for the statement:
"the measure of the supplement of an angle is 60 less than 3x the measure of the complement of the angle. "
(180-x) = 3(90-x) - 60
180 - x = 270 - 3x - 60
-x + 3x = 270 - 180 - 60
2x = 30
x =
x = 15 degrees is the angle
:
:
confirm this in the statement above
180 - 15 = 3(90-15) - 60
165 = 3(75) - 60
165 = 225 - 60
165 = 165;

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At 10 AM, Francisco is asked by his uncle to weed the garden.
From past experience, Francisco knows that this will take him 4 hours, working alone.
His older cousin, Rodrigo, when it is his turn to do this job, requires 6 hours.
Since Rodrigo wants to go see the game with Francisco and have already bought
the tickets for the game a 1 PM, he agrees to help Francisco.
Assuming no gain or loss of efficiency, when will they finish if they work together?
Can they make it to the game after doing the assignment?
:
This is shared work problem
:
Let t = time required to do the job when working together
Let the completed job = 1 (a weed-free garden)
:
Each will do a fraction of the job, the two fractions add up to 1
+ = 1
Multiplying by 12 gets rid of the denominators, results:
3t + 2t = 12
5t = 12
t =
t = 2.4 hrs, since it is 3 hrs until game time, they may be able to make it.
:
(Forget the shower and drive fast!!)

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How do you add rational expressions when one of he denominators is a trinomial and one is a binomial.
ex:
Both denominators will factor

The common denominator; (y-2)(y-1)(y+1), so we have:

:
=

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Voyager 2 traveled approximately 4.4 billion miles in 2.0 years.
What was Voyager 2's average speed, expressed in km/s?"
:
On my calc, I entered:
= 112.2463 km/sec, I think you're right

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question and this is how she gave them give the equation, through (1,-6) slope=3
Use the point/slope form: y - y1 = m(x - x1)
In this problem: x1=1; y1=-6; m=3
y -(-6) = 3(x - 1)
y + 6 = 3x - 3
y = 3x - 3 - 6
y = 3x - 9; (now it is the slope intercept form)
:
You can prove this: replace x with 1, find y
y = 3(1) - 9
y = 3 - 9
y = -6, which is the value given for y
:
:
3. find equation- vertical line thru ( 0, -3)
this vertical line goes thru 0 and all values for y. therefore
x = 0 is the equation
:
4. find equation- thru (-3,6)and (1,-2)
Find the slope using the slope equation: m =
assign the given values as follows
x1=-3; y1=6
x2=1; y2=-2
m = = = = -2 is the slope
Use the point/slope form again:
y - 6 = -2(x - (-3))
y - 6 = -2(x + 3)
y - 6 = -2x - 6
y = -2x - 6 + 6
y = -2x , is the equation
Here again, you can confirm this, replace x with -3, find the y value
:
:
5. find equation- horizontal line thru (0,2)
this horizontal line goes thru 2, and all values of x, therefore
y = 2
:
;
7. find equation - slope= -3 and y intercept 3
Here you can just substitute in the slope intercept form; y = mx + b
y = -3x + 3
;
:
10. find equation- thru ( -2, -3) and perpendicular to 3x-5y=1
First we have to find the slope of the equation,
put it in the slope intercept form
3x - 5y = 1
-5y = -3x + 1
y has to be positive, multiply by -1 to change the signs
5y = 3x - 1
Divide by 5
y = x -
then slope: m1 =
:
The relationship between the slopes of perpendicular lines is: m1*m2 = -1
*m2 = -1
m2 = is the slope of the perpendicular line
Find the equation for the line where: m=; x1=-2; y1=-3
y -(-3) = (x - (-2))
y + 3 = (x + 2)
y + 3 = x -
y = x - - 3
y = x - -
y = x -
In standard form, multiply by 3
3y = -5x - 19
5x + 3y = -19 is perpendicular to 3x - 5y = 1, goes thru x/y; -2,-3
:

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Going into the final exam, which counts as two grades, Kendra has test scores of
84, 78, 64, and 88.
What score does Kendra need on the final exam in order to have an average of 80?
:
Let x = score she need to get an 80
:
It counts as two grades, call it 2x and set up the equation for 6 test scores
= 80
:
= 80
Mult both sides by sides by 6
2x + 314 = 6(80)
:
2x + 314 = 480
:
2x = 480 - 314
:
2x = 166
x =
x = 83 is the grade to get an 80 average

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A bus traveled at an average rate of 50 mph and then reduced its rate to 40 mph for the rest of the trip.
If the 220-mile trip took 5 hours, determine how long the bus traveled at each rate.
;
Let t = time at 50 mph rate
then
(5-t) = time at 40 mph rate
:
Write a distance equation: dist = speed * time
:
50 mph dist + 40 mph dist = 220 mi
50t + 40(5-t) = 220
50t + 200 - 40t = 220
50t - 40t = 220 - 200
10t = 20
t = 2 hrs at 50 mph
then
5 - 2 = 3 hrs at 40 mph
;
:
Check solution by finding the distances
50(2) = 100
40(3) = 120
--------------
total: 220

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The area of a circular ring is 90
. If the radius of the outer circle forming the ring is 9,
what is the radius of the inner circle of the ring?
(Assume the circles forming the ring have the same center.)
:
Find the total area of the circle
A =
A = 254.469
:
The the area of the small circle: 254.469 - 90 = 164.469
:
Find the radius (r) of the small circle
= 164.469
r =
r = 7.235 is the radius of the small circle
:
:
Check this:
overall area - small circle area
254.469 - = 90.0

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8X SQUARED PLUS 8XY PLUS 8XZ PLUS 3X PLUS 3Y PLUS 3Z
:
8x^2 + 8xy + 8xz + 3x + 3y + 3z
:
We have two groups of common values 8x and 3
(8x^2 + 8xy + 8xz) + (3x + 3y + 3z)
:
Factor out 8x and 3
8x(x + y + z) + 3(x + y + z)
;
Now we can factor out (x+y+z) from 8x and 3 and we have
(x + y + z)(8x + 3)

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A man age is now his son’s age in reverse.
His son was born when he was 45. How old is his son?
:
Let x = the 10's digit
Let y = the units digit
:
10x + y = 10y + x + 45
10x - x = 10y - y + 45
9x = 9y + 45
Simplify divide by 9
x = y + 5
Several options here, we know y < 4, and x > 5
Man is 72, son is 27
or
Man is 61, son is 16
or
Man is 83, son is 38
or
Man is 94, son is 49

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a small company finds that it costs $3,200 to produce 100 units per day and
$9,600 to produce 500 units per day
:
a) assuming the relationship between cost and number of units produced per day is linear.
find a linear equation that expresses the cost as a function of the number of
units produced per day.
:
Find the slope using the slope formula; m =
Assign the given values as follow
x1=100, y1=3200
x2-500, y2=9600
m = = = 16 is the slope
:
Use the point/slope formula to find the equation: y - y1 = m(x - x1)
y - 3200 = 16(x - 100)
y - 3200 = 16x - 1600
y = 16x - 1600 + 3200
y = 16x + 1600
Cost equation
f(x) = 16x + 1600, where x = no. of unit produced per day
:
b) how many units per day can the company produce for $10,000?
16x + 1600 = 10000
16x = 10000 - 1600
16x = 8400
x = 8400/16
x = 525 units cost $10000

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My explanation:
Complementary angles are angles that add up to 90.
For example, 55 degrees and 35 degrees are complementary, they add up to 90
If you were asked,"What is the complement of 25 degrees?" you would
say 90 - 25 = 65 degrees and is the complement of 25
;
The sum of the measures of two complementary angles is 90 degrees. (we just said that)
Call the two angles, x & y, they are complementary therefore:
x + y = 90
:
Find the measures of two complementary angles whose measures are in the ratio of 1:4.
Therefore:
=
Cross multiply
y = 4x
:
return to the equation x + y 90
Replace y with 4x,
x + 4x = 90
5x = 90
x =
x = 18 degees
We know y = 4x, therefore
y = (4*18)
y = 72 degrees
:
These two angle add up to 90 and have the ratio of 1:4
;
Did this shed some light on complementary angles??

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From a rectangle , corner is cut in triangular shape to give a pentagon .
The length of the sides are 13,19,20,25 and 31 in cm . lengths are not in order
. find the area of the pentagon.
:
I assigned the dimensions as follow:
starting with the hypotenuse of the triangle = 13
the remaining part of that side = 19
bottom = 25
vert side = 31
top = 20
:
Find the area of the rectangle without the triangle removed
25 * 31 = 775
:
The triangle dimensions: 5 by 12 by 13,
prove it is a right triangle: = 13
A = .5 * 5 * 12 = 30
:
Pentagon area: 775 - 30 = 745

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A rectangular swimming pool is surrounded by a walk.
The area of the pool is 323 square feet and the outside dimensions of the walk
are 20 feet by 22 feet.
How wide is the walk?
:
let x = the width of the walkway
then the dimensions of the pool will be: (20-2x) by(22-2x)
Area of the pool given as 323, therefore:
:
(20-2x) * (22-2x) = 323
FOIL
440 - 40x - 44x + 4x^2 = 323
Arrange as a quadratic equation
4x^2 - 84x + 440 - 323 = 0
4x^2 - 84x + 117 = 0
this will factor to:
(2x - 3)(2x - 39) = 0
Only one solution makes sense
2x = 3
x = 1.5 ft is the width of the walkway
:
:
Check solution by finding the area
A = (20-3)(22-3)
A = 19 * 17
A = 323 sq/ft confirms our solutions

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I am looking for the process of converting m/s to mi/h, as in the speed of sound.
:
One way is convert meters (m) to feet, multiply meters by 3.28
3600 sec in one hr, 5280 ft = 1 mi
therefore:
Mph =

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S.A. = 6(6^2) = 216 sq/km

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Trail is approximately 1200 miles between Old Franklin, Missouri, and Santa Fe, New Mexico.
Suppose that a group of hikers start from each town and walk the trail toward each other.
They meet after a total hiking time of 240 hours.
If one group travels at 1/2 mile per hour slower than the other group,
find the rate of each group.
:
When they meet the distance of each group totals 1200 mi
:
Let s = speed of one group
then
(s+.5) = speed of the other group
:
Write a distance equation, dist = speed * time
240s + 240(s+.5) = 1200
240s + 240s + 120 = 1200
480s = 1200 - 120
480s = 1080
s =
s = 2.25 mph is one group
and
2.25 + .5 = 2.75 mph is the other group
:
:
See if that is true; find the distance traveled by each group
240(2.25) = 540 mi
240(2.75) = 660 mi
------------------
total dist: 1200 mi

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On a recent trip, Sarahs car traveled 20 mph faster on the first 200 miles that
it did on the remaining 80 miles.
The total time for the trip was 4 hours. Find the speed of Sarahs car on the first part of the trip.
:
Let s = speed on the 1st part of the trip (200 mi)
then
(s-20) = speed on the last part (80 mi)
:
Write a time equation: time = dist/speed
:
Fast speed time + slow speed time = 4 hrs
+ = 4
multiply equation by s(s-20),results:
200(s-20) + 80s = 4s(s-20)
:
200s - 4000 + 80s = 4s^2 - 80s
:
280s - 4000 = 4s^2 - 80s
A quadratic equation
0 = 4s^2 - 80s - 280s + 4000
:
4s^2 - 360s + 4000 = 0
simplify divide by 4
s^2 - 90s + 1000 = 0
Solving this with quadratic formula got two solutions: s=13 and s=77
Obviously s = 77 mph is the one
:
:
Check it find the time at each speed
200/77 = 2.6 hr
80/57 = 1.4 hrs
-----------------
total time 4 hrs

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Start by getting rid of the fraction, multiply both sides by 3
x = 3(36x+7)
:
x = 108x + 21
:
-21 = 108x - x
:
-21 = 107x
x =
x = -.19626 (approx) in decimal form
:
:
Check solution in original equation
=
-.06542 = -7.06536 + 7
-.06542 ~ -.06536; close enough to confirm solution x =

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Two pipes fill a storage tank in 9 hours.
If the larger pipe fills the tank three times as fast as the smaller one,
how long would it take the larger pipe to fill the tank alone?
:
Let t = time required by the larger pipe alone
then
3t = time required by the smaller pipe alone
:
Let the completed job = 1; (a full tank)
:
Each pipe will fill a fraction of the tank, the two fractions add up to 1.
:
+ = 1
Multiply by 3t, results:
3(9) + 9 = 3t
27 + 9 = 3t
36 = 3t
t =
t = 12 hrs required by the large pipe alone
:
:
See if that's true (36 hrs for small pipe)
+ = 1
+ = 1