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In these kind of problems, you add the
RATES that each works to get their rates
working together.
(portion of job completed by #1)/(time to complete that portion) +
(portion of job completed by #2)/(time to complete that portion) +
(portion of job completed by #3)/(time to complete that portion) =
(portion of job completed by all working together)/(time to complete that portion)
Let r[a] = Anne's rate of working in (portion of job)/(days to do that portion)
Let r[l] = Lucy's rate of working in (portion of job)/(days to do that portion)
Let r[p] = Phoebe's rate of working in (portion of job)/(days to do that portion)
given:
(1) (note that 1 means the whole job)
I can also say:
equals Anne's rate x days worked = portion of job done by her
equals Lucy's rate y days worked = portion of job done by her
equals Phoebe's rate z days worked = portion of job done by her
given:
(2) (1 means whole job)
(3)
-------------------------------
The problem wants to find rate for each = (1 job)/(time to do 1 job)
There are 3 equations and 3 unknowns, so it's solvable
First, I divide both sides of (2) by
(2)
(2)
Now, I subtract this from (1)
(1)
(2)
----------------------------------------------




Now I put this value back into (1)
(1)
(1)
(1)
Now I subtract this from (3)
(3)
(1)
---------------------------------




Now I put and back into (2)
(2)
(2)
(2)

Anne can do the job in 6 days working alone
Lucy can do the job in 8 days working alone
Phoebe can do the job in 12 days working alone
check answer:
(3)
(3)

OK

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Note that


answer

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Let = number of quarters she has
Let = number of nickels she has
given:
(1)
------------------
= value of quarters in cents
= value of nickels in cents
From (1):



The total value in cents of the nickels and
quarters is 30 times the number of nickels
plus 150 cents

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Let = ml of heavy cream that needs to b added
given:
Butterfat in heavy cream = ml
Amount of 20% light cream = ml
Butterfat in ml of light cream= ml
--------------------
In words:
(total ml of butterfat I end up with)/(total ml of mixture) = .36






4000 ml or 4 liters of heavy cream must be added
check answer:




OK

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(1)
(2)
first, I will get (1) into the same form as (2)
(1)
Then I multiply both sides of (2) by
(2)
Now I multiply both sides of (1) by
(1)
Now I multiply both sides of (2) by
(2)
Now I add (1) and (2)


Then I substitute this result back into (1)
(1)




The answer is (-3,3)

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Let = wind speed
Let = speed of plane without wind
= speed of plane flying into wind
= speed of plane flying with the wind
given:


This is 2 equations with 2 unknowns so it's solvable
-------------------
Divide both sides of each by

add the equations
-------------------


mi/hr
and, since


mi/hr
The speed of the plane without wind is 223.445 mi/hr
check:



close enough
and




OK

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Call the first page
The 2nd page will be
given:


Complete the square:



Take the square root of both sides





The pages are 48 and 49
check:

OK

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Locate the x and y intercepts for both equations
x-intercept:


solve for


-----------
y-intercept:

solve for


The intercepts are at (-4,0) and (0,3)
drw a straight line through these points
-------------
Now repeat the process for
and
draw a straight line through the points

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You need 2 equations:


and


You are given that , so

Multiply both sides by




and, since



The speed of train A is 46 mi/hr
The speed of train B is 54 mi/hr
check answers:



hrs
and



hrs

OK

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answer
Since square roots can be either (+) or (-), there
are other valid solutions also:

2nd solution
and

3rd solution
and

4th soution

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The quadratic formula for finding roots is

You need to know what a, b, and c refer to
This formula assumes that the equation is in the form

Your equation is already in this form:




plugging in the numbers:



I'm guessing that should be and not . That would make the
solution




and


These are the 2 roots.
If for real, then stick with

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The slope would be -(1/slope of line)







OK

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the formula is:





I'll find out when the sum is greater than 500000



Multiply both sides by

I can take the log of both sides

If , the left side is close to , so





Putting this back into formula,





The next whole number for is
I think that would put over
seems to be too small

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Let = amount invested in each bank





She invested $50,000 in each

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percentage means "per one hundred"
Let
In this equation is the percent that is of
Solve for

Use calculator

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given:
Katy:


Chad:


By substitution:



mi/hr
and

mi/hr
How long did they travel?


hrs
and


hrs
Katy traveled for 8 hrs
Chad traveled for 6 hrs

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You need 2 equations, one going with the wind
and one going against the wind
Let = wind speed
Let = speed of plane in still air
given:
with the wind:
hrs
mi
against the wind:

mi
--------------


This is 2 equations with 2 unknowns, so it's solvable
(1)
(2)
Multiply both sides of (1) by
and add (1) and (2)
I can't find my calculator, so you can do it

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You can look up on calculator

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Let = speed of southbound plane in mi/hr
= speed of northbound plane in mi/hr
The are separating at the rate of

After 5 hrs






The speeds of the planes are:
southbound: 150 mi/hr
northbound: 250 mi/hr

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Let = the original price/dozen of apples
given:
= purchaser's savings after reduction
--------------------
(savings)*(dozens bought) = amount spent



The reduced price/dozen is


The reduced price/dozen is $30

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Wow, this looks so confusing, I'm not sure what it says.
The is where to start, since it looks simple.
It says "I am an exponent to the base that gives
I know that , so

Now I have

This is saying "An exponent which is , with a base that is
gives me
I'll rewrite it as:

Now take the square root of both sides
answer
and
not answer
Only the positive result would work here, because what would the
answer to this be?

Can't solve it

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You have to be able to read this. First you must
know that **logs are exponents**.
If you have , then is a log to the base
says the log to the base which gives
me is .
You know is the exponent because
log(something) =
If is the exponent, and is the base, I can write

Now you have to know what to do with a negative exponent
The rule is:
, so
, and


So far, I have
, so

Remember, also, that there is no way you could have
something like

This is saying
There is no possible that can give you a minus result
Good luck - just pound away at this stuff- It's like
cement- It will crumble after a while

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The formula is:

where( x[1],y[1]) = (2,-1)
plugging in numbers:

Multiply both sides by




Here's the plot:

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c)6 is the answer

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The average salary for the 149 adults is
(their total income)/149 =
given:




This is the total salary for 149 people (not including millionaire)

the average salary for the remaining 149 adults in that town is $47000

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When the wrench hits the ground,
since the height is zero.

The minus in front of the squared term tells
me the parabola solution has a maximum peak
Divide both sides by


Complete the square



sec
The wrench will hit the ground in 2 seconds

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First, write

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Let = liters of 20% acid solution needed
Let = liters of 80% acid solution needed
given:
liters
liters
--------------------------
acid in final solution:


subtract

liters
and, since

liters
30 liters of 20% solution and
70 liters of 80% solution are needed
check:




OK

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Multiply both sides by

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Let = cost of garden table
Let = cost of bench
given:


By substitution:




and, since



The bench costs $450

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You need a linear equation, which has the form

= cost of a textbook during a particular year
is a particular year, calling



etc.
All you have to do is find and in the equation
is the slope, or rate of increase
When increases from to (2005- 2010),
increases from to
The slope is (change in C)/(change in y)



So far the equation is

Now to find ,
Use , and (a point on the line)




So, the equation is

The plot of this equation is: