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Pipe A can fill a tank in 4 hr. Pipe B can fill the tank in 9 hr less than the time it takes pipe C, a drain pipe, to empty the tank. When all three pipes are open, it takes 2 hr to fill the tank. How much time is required for pipe C to empty the tank if pipes A and B are closed?
:
Pipe C's draining time = x (when Pipes A & B are closed)
:
Pipes A's filling time = 4 hr (given)
:
Pipe B's filling time = (x-9) (Given as 9 hrs less than C)
:
Let the full tank = 1
:
Filling is +, draining is -
:
The equation when all three are open for 2 hrs:
+ - = 1
:
Multiply the equation by 4x(x-9) to get rid of the denominators:
4x(x-9)* + 4x(x-9)* - 4x(x-9)* = 4x(x-9)(1)
Cancel out the denominators and you have:
x(x-9)*2 + 4x(2) - 4(x-9)*2 = 4x(x-9)
:
2x^2 - 18x + 8x - 8x + 72 = 4x^2 - 36x
:
2x^2 - 4x^2 - 18x + 36x + 72 = 0
:
-2x^2 - 18x + 72 = 0
:
x^2 - 9x - 36 = 0; Divide by -2; simplifies and changes the signs
Factors to
(x + 3) (x - 12) = 0
Positive solution
x = 12 hrs for the the drain to empty the tank with A & B closed
:
;
Check solution (B's fill time will be 12-9 = 3 hrs)
+ - =
+ - = = 1

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A company distributes its product by train and by truck. The cost of distributing
by train can be modeled as -0.07x^2+34x-135, and the cost of distributing by
trucks can be modeled as -0.02x^2+24x-180, where x is the number of tons of
product distributed. Write a polynomial that represents the difference between
the cost of distributing by train and the cost of distributing by trucks
:
Let f(x) = difference in cost between trains & trucks
:
f(x) = (-0.07x^2+34x-135) - (-0.02x^2+24x-180)
:
Removing the brackets changes the signs
f(x) = -0.07x^2 + 34x - 135 + 0.02x^2 - 24x + 180
:
f(x) = -.07x^2 + .02x^2 + 35x - 24x - 135 + 180
:
f(x) = -.05x^2 + 11x + 45

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Years after the famous foot race, the hare challenged the tortoise to a rematch
on bicycles. the hare was able to pedal at a rate of 15 MPH while the tortoise
could only pedal at 12 mph. knowing he was slower, the tortoise made up a lame
excuse to be given a 27 mile head start. in the 165 mile race.
:
1. Who won the race? by how much time? explain
:
Can the tortoise travel (165-27) in less time than the rabbit travels 165 mi?
:
Hare's time; (Time = )
= 11 hrs
:
Tortoise's time
= 11.5 hrs
:
Obviously the hare wins by half an hour
:
:
2. How far into the race did the hare overtake the tortoise?
:
When this occurs they will have traveled for the same length of time
Time = distance/speed
:
Hare's time = Tortoise's time
Let d = distance from the starting line of the hare
=
Cross multiply
15(d-27) = 12d
15d - 405 = 12d
15d - 12d = +405
3d = 405
d =
d = 135 miles into the race, the hare overtakes the tortoise
:
:
3. what is the largest head start that tortoise could have been given without being able to beat the hare?
:
Let x = the largest head start which will cause a tie,
Their travel times are equal. Write a time equation:
=
Cross multiply:
15(165-x) = 12 * 165
2475 - 15x = 1980
2475 - 1980 = 15x
495 = 15x
x =
x = 33 mile head start would be a tie
:
Check for equal time:
165/15 = 11 hr
132/12 = 11 hr
;
:
4. suppose a chipmunk entered the race with an 84 mile head start traveling 9
mph. develope an equation for the chipmunk and calculate when the hare would
finally catch up to the chipmunk assuming the race does not end at 165 miles.
:
Another situation where the travel time is equal for both:
Let a = distance from start that the hare catches up with the chipmunk
=
Cross multiply:
15(a-84) = 9a
15a - 1260 = 9a
15a - 9a =+1260
a =
a = 210 miles from start for the hare to catch up with the chipmunk
:
:
5. a confused pig also followed the race course. the pig started at the finish line at a rate of 15v mph traveling backwards. where and when did he meet the hare on the road?
:
I'm not sure if it is 15v or 15 mph, if you mean the pig travels at 15 mph as the hare. They would meet at the half way mark (Assuming they started at the same time)

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prices were $8, $10, and $12 and the total income from the ticket sales was $3700. how many tickets of each type were sold if the combined number of $8 and $10 tickets sold was 7 times the number of $12 tickets sold?
:
It says 400 tickets sold:
x + y + z = 400
:
It says:
"prices were $8, $10, and $12 and the total income from the ticket sales was 3700.
8x + 10y + 12z = 3700
:
It says:
"the combined number of $8 and $10 tickets sold was 7 times the number of $12 tickets sold?"
x + y = 7z
x = (7z-y)
:
Using the 1st equation, substitute (7z-y) for x
(7z-y) + y + z = 400
7z + z - y + y = 400
8z = 400
z =
z = 50 ea $12 ticket sold
;
Substitute 50 for z in the 1st equation:
x + y + 50 = 400
x + y = 400 - 50
x + y = 350
x = (350-y)
:
Subsitute 50 for z in the 2nd equation:
8x + 10y + 12(50) = 3700
8x + 10y + 600 = 3700
8x + 10y = 3700 - 600
8x + 10y = 3100
:
Substitute (350-y) for x in the above equation
8(350-y) + 10y = 3100
2800 - 8y + 10y = 3100
2y = 3100 - 2800
2y = 300
y =
y = 150 ea $10 tickets
:
I'll let you find the number of $8 tickets (x), there are several ways to do it.

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It has been estimated that 1000 curies of a radioactive substance introduced at a point on the surface of the open sea would spread over an area of 36000 km^2 in 40 days. Assuming that the area covered by the radioactive substance is a linear function of (t) time and is always circular in shape, express the radius (r) of the contamination as a function of (t).
:
Take the area of a circle equation; A = pi*r^2
:
we want to solve for r
:
pi*r^2 = A
:
r^2 =
:
r =
:
Find the radius (km) after 40 days using the given area of 36000
r = }
r =
r = 107 km
:
Find the slope using the coordinates of 0,0 (0 days, 0 radius) and 40, 107
m =
:
r = t; is the equation. (t (days) is on the x axis and the radius(km) is on the y axis):
if you graphed it would look like this:

Using your calibrated eyeball, you can see on the graph that after 40 days the radius is about 100 km
:
:
Did this make sense? Did it help you?

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Let x = one number
Let y = a second number
:
Write an equation for each statement;
(where you see the word "is", you can usually put an = sign):
:
"The sum of two numbers is 6 less than twice the first number.
x + y = 2x - 6
Simplify:
y = 2x - x - 6
y = x - 6
:
"Their difference is 10 less than four times the second number.
x - y = 4y - 10
Simplify
x = 4y + y - 10
x = 5y - 10
:
Find each of the numbers
:
Substitute (x-6) for y (from the 1st equation); find x:
x = 5(x-6) - 10
x = 5x - 30 - 10
x = 5x - 40
+40 = 5x - x
40 = 4x
x = +10
:
Find y using the 1st equation
y = 10 - 6
y = 4
:
:
Check solutions using the statement:
"Their difference is 10 less than four times the second number."
10 - 4 = 4(4) - 10
6 = 16 - 10; confirms our answer
:
How about this? Did it seem logical, and do you think you can handle these kind of problems now?

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I have a 10-ounce glass of kool-aid that is 30% sugar. I want to make it less sweet. How much water can I add to the 30% solution to make it only 25% sugar.
:
Let x = amt of water required to make it to reduce the sugar to 25%
:
A simple equation:
.30(10) = .25(x+10)
:
3 = .25x + 2.5
:
3 - 2.5 = .25x
:
.5 = .25x
x =
x = 2 oz of water required
:
:
Check solution:
.3(10) = .25(2+10)
3 = .25(12); confirms our solution

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A painter works on a job for 10 days and is then joined by her helper. Together they finish the job in 6 more days. Her helper could have done the job alone in 30 days. How long would it have taken the painter to do the job alone?
:
Let x = time required by the painter to do the job alone
:
Let the completed job = 1
:
Think of of it this way. The painter has 10/x of the job completed when he's joined by the helper
+ + = 1
same as:
+ = 1
:
Multiply equation by 30x to get rid of the denominator
30x* + 30x* = 30x(1)
:
30(16) + 6x = 30x
480 + 6x = 30x
480 = 30x - 6x
480x = 24x
x =
x = 20 days, painter by himself
:
:
We can check that using the original equation
+ + =
+ + = 1

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7^(2x+3) = 5^(x-4)
:
Use natural logs to solve this:

:
Using the log equivalent of exponents, we can write it:

:
Find nat log (ln) of 7 and 5:
1.9459(2x+3) = 1.6094(x-4)
:
Multiply what's inside the brackets
3.8918x + 5.8377 = 1.6094x - 6.4376
:
Some basic algebra
3.8918x - 1.6094x = -6.4376 - 5.8377
:
2.2824x = -12.2753
x =
x = -5.378
:
;
Check solution on a good calc:
enter 7^(2(-5.378)+3) = 2.7888
enter 5^(-5.378-4) = 2.7865, close enough to check our solution
;
:
Did this make sense? You can use common logs (base 10) if you prefer, the method is the same. (the numbers will be different, but the answer will be the same)

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The publisher of a magazine that has a circulation of 80,000 and sells for 1.60$ a copy decides to raise the price of the magazine because of costs. By surveying the readers, he finds that he will lose 10,000 readers for each .40 $ increase. What price per copy maximizes growth?
;
By growth do they mean maximize revenue, if that is the case we can do it this way;
:
Let x = no. of 40 cent increases
and
Let x = no. of 10,000 reader decreases
:
Revenue = price * no. of readers
:
R = (1.60 + .40x) * (80000 - 10000x)
FOIL
R = 128000 - 16000x + 32000x - 4000x^2
As a quadratic
-4000x^2 + 16000x + 128000 = 0
:
Simplify, divide equation by 4000
-x^2 + 4x + 32 = 0
:
Max will occur at the axis of symmetry: x = -b/(2a); Here: a=-1; b=4
x =
x = +2
:
That would be 2 ea .40 increases for a cost of 2.40 per copy
and
2 ea 10000 readers lost for a total 60000 readers
:
Revenue = 60000 * 2.40 = $144,000 (maximum)
:
If you graphed it:

Multiply the y value by 4000 to get the actual revenue

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If a number is added to the numerator of 3/4 and twice as much is added to the denominator, the result is 4/7. Find the number.
:
Let the number to be added = x
:
=
Cross multiply, and solve for x
4(2x+7) = 7(x+3)
8x + 28 = 7x + 21
8x - 7x = 21 - 28
x = -7
:
Check solution in original equation, substitute -7 for x
=
=
= ; minus into a minus is a plus
:
:
2. The sum of a number an its reciprocal is 13/6. Find the numbers.
x + =
Multiply equation by 6x to get rid of the denominators
6x(x) + 6x* = 6x*
6x^2 + 6 = 13x
:
6x^2 - 13x + 6 = 0; a quadratic equation
:
After a little slogging, we find we can factor this to:
(2x - 3)(3x - 2) = 0
:
2x = 3
x =
and
3x = 2
x =
:
Check solution using x = 3/2:
+ =
+ = ; confirms our solution
Note that the 2nd solution is the reciprocal of the 1st
:
:
3. An inlet pipe can fill an empty swimming pool in 5 hours, and another inlet pipe can fill the pool in 4 hours . How long will it take both pipes to fill the pool?
:
A typical shared work problem:
Let t = time required when both pipes are open
Let the full pool = 1
:
+ = 1
Multiply equation by a common multiple, 20 will work
20* + 20* = 20(1)
Cancel out the denominators and you have:
4t + 5t = 20
:
9t = 20
t =
t = 2.22 hrs or 2 hrs & 13.3 min
:
Check solution using 2.22 for t:
+ = 1
.444 + .555 = .999 ~ 1; confirms our solution
:
:
Could you follow the procedures in all these problems? Let me know if you have any questions

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The Outdoor Furniture Corporation manufactures two products, benches and picnic tables, for use in yards and parks. The firm has two main resources: its carpenters (labor force) and a supply of redwood for use in the furniture. During the next production cycle, 1,200 hours of labor are available under a union agreement. The firm also has a stock of 3,500 feet of good quality redwood. Each bench that Outdoor Furniture produces requires 4 labor hours and 10 feet of redwood: each picnic table takes 6 labor hours and 35 feet of redwood. Completed benches will yield a profit of $9 each, and tables will result in a profit of $20 each. How many benches and tables should Outdoors Furniture produce to obtain the largest possible profit? Use graphical Linear programming approach.
:
Let x = no. of picnic tables
Let y = no. of benches
:
The labor constraint:
6x + 4y <= 1200
4y <= 1200 - 6x; divide equation by 4
y <= 300 - 1.5x
:
Plot above equation for x = 0 and x = 60
x | y
-------
0 | 300
60 | 210
;
:
The material constraint
35x + 10y <= 3500
10y <= 3500 - 35x
y <= 350 - 3.5x; divide equation by 10
:
Plot the above equation for x = 0 and x = 60
x | y
-------
0 | 350
60 | 140
:
Graph will look like this

The area of feasibility is at or below the two graphs, which ever is lowest
Area is bounded by coordinates; 0,0; 0,300; 100,0; and an integer values of 25, 262
Find the profit using each:
Tables + Benches
20(0) + 9(300) = $2700
20(100) + 9(0) = $2000
20(25) + 9(262) = $2858;
:
25 tables and 262 benches will yield max profit;
utilizes:
6(25) + 4(262) = 1190 hrs of labor
35(25) + 10(262) = 3495 ft of red-wood
:
This is the general idea. Check my math here, a lot of chance for mistakes

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Mary wants her $9000 GIC to double in value in nine years. What annual rate of interest is required for her to reach her goal. Assume interest is compounded annually.
:
Let r = decimal value of the interest required
:
Using the annual interest equation: A = Ao*(1+r)^t
:
9000 * (1 + r)^9 = 18000
:
(1 + r)^9 = 18000/9000; divided both sides by 9000
:
(1 + r)^9 = 2
:
9 * ln(1+r) = .693
ln(1 + r) =
ln(1 + r) = .077
:
Find the antilog (e^x) of both sides
1 + r = 1.08
;
r = 1.08 - 1
:
r = .08 which is 8%
;
:
Check solution using a calc
1.08^9 * 9000 = 17991 ~ 18000

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The first cost $99.50 and appreciated by 10% yearly. The second cost $49.50 and appreciated 16% yearly. Algebraically determine how many years will it take for the two models to be of equal value.
:
Use the annual interest equation, t = time in years
:
Cheap car value = expensive car value
:
49.5(1.16)^t = 99.5(1.10)^t
:
Using natural logs:
ln(49.5) + ln(1.16^t) = ln(99.50) + ln(1.10^t)
:
log equiv of exponents:
ln(49.5) + t*ln(1.16) = ln(99.50) + t*ln(1.10)
:
3.902 + .1484t = 4.6 + .0953t
:
.1484t - .0953t = 4.6 - 3.902
:
.0531t = .698
t =
t = 13.14 years
;
:
Find the value of each after 13.14 yrs, using a calc
1.16^13.14 * 49.5 = $348
1.1^13.14 * 99.5 = $348 confirms our solution

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+ = 6
We can get rid of the denominators by multiplying the equation by 6:
6* + 6* = 6(6)
Cancel out the denominators and you have:
3(q+9) + 2(q-3) = 36
:
3q + 27 + 2q - 6 = 36
:
5q + 21 = 36
:
5q = 36 - 21
:
5q = 15
q =
q = 3
:
:
Check in original equation, substitute 3 for q:
+ = 6
+ 0 = 6; confirms our solution

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Assume it's
:

:
; Added to both sides
:
They have the same denominators so we can just write it:
= ; combine like terms, we have quadratic now
That will factor to:

Cancel out the (a+2)'s and we have a simple equation
a - 1 = 3
:
a = 3 + 1
:
a = 4
:
:
Check solution in the original equation, substitute 4 for a



3 is equal to 18/6:
; confirms our solution of a = 4
:
Could you follow what we did here? Any questions?

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sally has f bags of lego in each are 22 blocks .sam has 9 lego blocks spare so gives them to sally.expression for numer of blocks sally has.sally has 119 blocks.form an equation solve the equaation find the number of lego bags sally has
:
They have come up with a wonderful thing called "punctuation", and "capital letters". Hopefully, it will be available in your area soon. Then this problem can be much more understandable.
:
Rewriting, using this wonderful thing:
:
Sally has f bags of lego. In each are 22 blocks. Sam has 9 lego blocks to spare,
so he gives them to Sally. Write an expression for number of blocks Sally has.
Sally has 119 blocks. Form an equation. Solve the equation to find the number of lego bags Sally has.
:
An expression for the number of blocks Sally has after receiving 9 from Sam
22f + 9
:
"Form an equation for the number of blocks Sally has:"
22f + 9 = 119
:
"Solve the equation to find the number of lego bags, Sally has." (Find f)
22f + 9 = 119
:
Subtract 9 from both sides:
22f = 119 - 9
:
22f = 110
f = ; divide both sides by 22
f = 5 bags
:
Check this in the equation we formed:
22(5) + 9 =
110 + 9 = 119 confirms our solution of f = 5
;
:
Did this help you? Any questions?

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Adult tickets sold for five dollars and student tickets cost 3.50. The total proceeds from Thursday's ticket sales were 155. if 37 tickets were sold on Thursday, how many were adult tickets and how many were adult tickets and how many were student tickets?
:
Let a = no. of adult tickets
Let s = no. of student tickets
:
The ticket value$ equation:
5a + 3.5s = 155
;
The total tickets sold equation:
a + s = 37
we can also write it as:
s = (37 - a)
:
Substitute (37-a) for s in the 1st equation:
5a + 3.5(37-a) = 155
:
5a + 129.5 - 3.5a = 155; multiplied what's inside the brackets.
:
5a - 3.5a = 155 - 129.5
:
1.5a = 25.5
a =
a = 17 adult tickets sold
then
37 - 17 = 20 student tickets sold
;
:
Check solution in the $$ equation:
5(17) + 3.5(20) =
85 + 70 = 155; confirms our solutions
:
Did this make sense to you? Any questions?

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Just write an expression or equation for what it says:
"Find three consecutive even integers"
x, (x+2), (x+4)
:
" such that three times the middle integer is six more than the sum of the first and third."
3(x+2) = 6 + x + (x+4)
:
3x + 6 = 2x + 10
:
3x - 2x = 10 - 6
:
x = 4 is the 1st integer, then 6, 8 are the next two
:
:
Check solution in the statement:
"three times the middle integer is six more than the sum of the first and third."
3(6) = 6 + 4 + 8; confirms our solution
:
Not that hard, right?

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Assume the problem is:

:
The common denominator will be:
So we have:
= = ; about all you can do with it

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The complement of an angle is 18 degrees more than 1/3 its supplement. find the measure of the angle.
:
The expression representing the complement of an angle (x):
(90 - x)
:
The expression representing the supplement of an angle (x)
(180 - x)
:
"the complement of an angle is 18 degrees more than 1/3 its supplement."
(90 - x) = (180-x) + 18
:
90 - x = 60 - x + 18; found one third of 180
:
90 - x = 78 - x
:
90 - 78 = -x + x
:
12 = +x
Multiply equation by 3 to get rid of the denominator
3(12) = 3*x
36 = 2x
x =
x = 18 degrees
:
:
Check solution using the statement:
"the complement of an angle is 18 degrees more than 1/3 its supplement."
(90 - 18) = (180-18) + 18
72 = *162 + 18
72 = 54 + 18

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A chemistry experiment calls for a 40% scotch whiskey solution. If your pantry has only a 30% and a 100% solution of a 12 year old scotch, how much of each should be mixed to obtain 14 liters of a 40% for your favorite drink.
:
Let x = 100% solution
The resulting amt is given as 14 liters, therefore:
(14-x) = 30% solution
:
a typical mixture equation:
;
1.0x + .30(14-x) = .40(14)
:
1x + 4.2 - .3x = 5.6
:
1x - .3x = 5.6 - 4.2
:
.7x = 1.4
x =
x = 2 liters of 100% solution
and
14 - 2 = 12 liters of 30% solution
:
:
Check solutions in original equation:
1(2) + .3(12) = .4(14)
2 + 3.6 = 5.6

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Player A can run the 40 yard dash in 4 seconds. Player B can run the 40 yard dash in 5 seconds. If player A catches the football at the 20 yard line running at full speed, and player B at the 15 yard line (also at full speed) starts after player A. How long will it take player B to catch player A. On what yard line will player A be caught on?
:
Player B is slower than Player A; he will never catch him.

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Miranda is making a bow for a Christmas wreath. The bow is made from six yards of ribbon which must be cut into three pieces where two pieces are the same length and the third piece is eight times as long. Find the length of each piece.
:
Come on, it's not that hard
:
Let x = the length one of the small equal pieces
Then
8x = the long piece
:
Long piece + small piece + small piece = 6 yards
8x + x + x = 6
:
10x = 6
x =
x = .6 yds is the length of one of the equal pieces
then
8(.6) = 4.8 yds is the length of the long piece
:
:
Check solution: 4.8 + .6 + .6 = 6 yds

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The perimeter of a rectangle is 64cm. The length of the rectangle is 2cm more than 1 and 1/2 times the width. Find the dimensions of the rectangle.
Underneath this, the teacher has written:
Find and write the two equations.
Solve the system by graphing.
:
Since we will be graphing
Let x = the width
Let y = the length
:
The 1st equation is is easy, the perimeter equation:
2x + 2y = 64
Simplify, divide equation by 2
x + y = 32
Put it in the general (y=) form for graphing
y = 32 - x
Plot it for x = 0 and x = 4
x | y
-------
0 |32
4 |28
:
Get the 2nd equation from the statement:
"The length of the rectangle is 2cm more than 1 and 1/2 times the width."
y = 1.5x + 2
Plot this for x = 0 and x = 4
x | y
--------
0 | 2
4 | 8
:
your graph should look like this:

:
With a calibrated eyeball you can see they intersect at x=12, y=20 and this
is solution to the system
;
You can confirm our solution by substitution in both equations
2(12) + 2(20) = 64
and
20 = 1.5(12) + 2
:
Did you follow what went on here? Any questions?

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At 1 pm your friend leaves for college traveling on the interstate at a constant speed of 65 mph. At 1:24 pm you realize he has forgotten some necessary things for living away at college. You drive after your friend maintaining a speed of 70mph. If neither of you alter your speed, how many miles will you travel before you overtake him?
:
Let d = distance traveled when you catch up with your friend
:
Change 24 min to hrs. 24/60 = .4 hrs
:
When this happens, they will have traveled the same distance (d):
;
Write a time equation: Time =
Friends time - .4 hrs = your time
- .4 =
Multiply equation by 65*70 = 4550
4550* - 4550(.4) = 4550*
Cancel out the denominators and you have:
70d - 1820 = 65d
70d - 65d = 1820
5d = 1820
d =
d = 364 miles
:
That seems like a long way. Check solution by finding the travel times
364/65 = 5.6 hr
364/70 = 5.2 hr
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differ = 0.4 hrs; confirms our solution
:
:
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An employee who produces x units per hour earns an hourly wage of y = 0.35x + 11 (in dollars). Find the hourly wage for an employee who produces 2 units per hour.
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Just substitute 2 for x in the equation and find y
:
y = .35(2) + 11
y = .70 + 11
y = $11.70

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The sum of the digits of a two digit number is 11. The number obtained by reversing the order of the digits is 27 less than the original number. Find the original number.
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Let x = the original number 10's digit
Let y = the original number units digit
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Write an equation for each statement:
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"The sum of the digits of a two digit number is 11."
x + y = 11
or
y = (11 - x); will use this for substitution
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"number obtained by reversing the order of the digits is 27 less than the original number."
10y + x = 10x + y - 27
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10y - y = 10x - x - 27; some basic algebra operations
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9y = 9x - 27
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y = x - 3; simplified, divided equation by 9
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Find the original number.
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Substitute (11-x) for y in the above equation
11 - x = x - 3
:
11 + 3 = x + x
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14 = 2x
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x = 7 is the 10's digit
then
11 - 7 = 4 is the units digit
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74 is the original number
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Check solution using the statement:
"number obtained by reversing the order of the digits is 27 less than the original number."
47 = 74 - 27; confirms our solution
:
:
Could you follow what we did here? This is a good method to use on all these type of problems.

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A man has twenty coins consisting of dimes and quarters. If the dimes were quarters and the quarters were dimes, he would have ninety cents more than he has now. How many dimes and quarters does he have?
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Let x = original number of dimes
Let y = original number of quarters
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Let t = total value of the coins:
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Original equation .10x + .25y = t
Switched equation .25x + .10y = t+.90
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From this write an equation (subtracted t from both sides):
.10x + .25y = .25x + .10y - .90
:
.25y - 10y = .25x - .10x - .90
:
.15y = .15x - .90
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y = x - 6; divide equation .15
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or in the standard form:
x - y = 6
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Add to the total coin equation:
x + y = 20
x - y = 6
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2x = 26
x =
x = 13 dimes, the we have 7 quarters (originally)
:
To check our solution:
Take the switched value equation, subtract the original value equation:
.25(13) + .10(7) = 3.95
.10(13) + .25(7) = 3.05
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The difference is= .90 as given

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There is some strange symbols in your expression I assume that you mean:
:

:
:
Using long division using "......" to keep thing lined up:
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...........................2x - 3/2; (note that you have to multiply 2 by 3/2 to get 3)
.............------------------
2x + 1 | 4x^2 - x + 5
................4x^2 -2x
....................-------------
............................-3x + 5
............................-3x - 3/2
...............................--------
......................................+5 & 3/2 remainder which is 13/2
:
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Janet can shovel snow from her driveway in 65 minutes. Bill can do the same job in 50 minutes. How long would it take if they worked together?
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Minutes, hours, days, years, it does not matter as long as you're consistent.
Don't use minutes for one person and hours for another, for example
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Let t = time to complete the job working together
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Let competed job = 1
:
+ = 1
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Multiply equation by a multiple of 65 and 50; 650 come to mind
650* + 650* = 650(1)
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Cancel out the denominators and you have:
10t + 13t = 650
:
23t = 650
t =
t = 28.26 minutes
:
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Check solution using a calc
+ = .9999 ~ 1