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Let = kg of 40% metal needed
= kg of copper in 40% metal
kg of copper in 70% metal





120 kg of the 40% metal are needed
check:





OK

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Is the 1st sentence "Michael is 4 TIMES as old as Jackie." ?
If so,
Let = Michael's age now
Let = Jackie's age now
= Michael's age in 4 years
= Jackie's age in 4 years
-------------------------
given:
(1)
(2)
-----------------------
These are 2 equations with 2 unknowns
so it's solvable
(2)
Substitute (1) into (2)
(2)
(2)
(2)
and, since
(1)
(1)
(1)
--------------
Michael is 8 and Jackie is 2
check:
(2)
(2)
(2)
(2)
OK
(1)
(1)
(1)
OK

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The slow walker has a head start of
1 mile
Let = the distance in miles
that the fast walker covers
--------------
Now time both of them after the head start
Let = the time for both
slow walker:
(1)
Fast walker:
(2)
-------------
Substitute (2) into (1)
(1)
(1)
(1)
------------
and, since
(2)
(2)
(2)
Each walked the entire distance of 8 miles
check:
Slow walker's time during head start
was

So his time for the whole 8 miles was hrs



OK

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Let the numbers be and
given:

complete the square







The number are 109 and 111
check:


OK

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It helps to express it like this:

You can see that the 1st fraction is , so
reduced to lowest terms this is

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Let = distance to friend's house in miles
Let = time in hrs to drive to friend's house
given:
To friend's house:
(1)
Return trip:
(2)
-------------------
Substitute (1) into (20
(2)
(2)
(2)
(2)
(2)
Plug this result back into (1) or (2)
(1)
(1)
(1)
and
Mike traveled 169.412 mi
check:
(2)
(2)
(2)
(2)
(2)
OK

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is called the constant of proportionality
given:
pounds
mi



---------------



The astronaut weighs about 200 pounds

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Whenever you have parentheses, work from
the inside out.
So start with
----------------------
The general rule for this is:
, so
----------------------

--------------------
Now you have:

Note that


answer
---------------------------
You can check this by filling in any
numbers for and







-------------
Plug in same numbers for answer:





This should be the same as



This doesn't PROVE the answer's right
but there is a better than even chance

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Please fill in data

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The 1st car has a head start of
mi
Start a stopwatch when you see the 2nd car
Stop the stopwatch when the 2nd car catches the 1st
Call that time
Let = the distance that the 2nd car travels
-----------------
Equation for 1st car:
(1)
Equation for 2nd car:
(2)
--------------
Substitute (2) into (1)
(1)
(1)
(1) hrs
The 2nd car catches the 1st car in 2 hrs
check answer:
(1)
(1)
(1)
(1) mi
and
(2)
(2)
(2) mi
OK

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This means the line goes through
(6,0) and (0,4)
The slope is ( change in y ) / ( change in x )

----------
The general form is

= slope
= y-intercept
------------------
answer
Here's the plot:

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You only need 2 rules:

and

And, what amounts to the same thing,

---------------------
Simplify a little







Hope I got it

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I've been there. But if you don't say " I'm lost"
when you're lost, you'll just stay lost.
----------
Minimize embarrassment if you can, but math
is something where the basics have to be
very clear or you can't move on.
Good luck

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There must be an odd number of quarters, since
the sum ends in a 5. If there was an even number,
it would end in 0.
------------------
I will try 29 quarters. That comes to
or $.70. That is 7 dimes
coins
-------------------
There are 29 quarters and 7 dimes

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Let = his age now

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Let = the distance in km between house and office
Let = The travel time in hrs which is on time for him
given:
From office to house at 5 km/hr
(1)
From office to house at 6 km/hr
(2)
-----------------------
(1)
(2)
----------------
Substitute (1) into (2)
(2)
(2)
(2)
Plug this result back into (1) or (2)
(1)
(1)
(1)
the distance between his house and office is 10 km
check:
(2)
(2)
(2)
(2)
OK
(1)
(1)
(1)
(1)
(1)
(1)
OK

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Add years to both their ages
given:





( note that 1/3 year = 4 months )
In 1 year and 4 months Jennifer will
be 2x as old as her sister
check:
J will be
Sis will be


OK

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Let = time in hrs for Jim's hose alone to fill pool
= time in hrs for Bob's hose alone to fill pool
------
Add their rates of filling to get rate filling together

Multiply both sides by




40.5 = time in hrs for Jim's hose alone to fill pool
32.4 = time in hrs for Bob's hose alone to fill pool
check:




OK

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Let the speed of the 1st train =
= the speed of the 2nd train
Let = the head start that the 1st train got
4 PM - 2PM is hrs, so

Let = the distance that the 2nd train has to go
to catch the 1st train
----------------
7PM - 4PM = hrs
Use this time for both trains. Figure that
the 1st train already has the head start
-----------------
1st train's equation:

(1)
-----------------
2nd train's equation:
(2)
----------------------
(1)
and
(2)
------------------
Substitute (1) into (2)
(2)
(2)
(2)
also


---------------
The 1st train's speed is 36 mi/hr
The 2nd train's speed is 60 mi/hr
check:



So, the 1st train had a head start of 72 mi
which took 2 hrs. Now it has 3 hrs to go
the rest of the way
2nd train's equation:
(2)
(2)
(2) mi
mi
The 1st train must go 108 mi in 3 hrs at 36 mi/hr


OK

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The 6 painters must have painted rooms

Each painter painted 5 rooms
( That's if they all paint at the same rate , which is assumed )

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There are 4 unknowns but only 2 possible
equations, so this can't be solved directly,
but it still can be figured out.
Let = number of pennies
Let = number of nickels
Let = number of dimes
Let = number of quarters
The equations are:
(1)
(2) ( in cents )
------------------------------------------
The logic steps are:
(a)
There doesn't have to be any pennies,
although there might be, but they will be in
multiples of 5 or 10
(b)
I know there must be less than 9 quarters, since
9 x 25 = 225, which is too much
(c)
I know there must be more than 2 quarters since
2 x 25 = 50, that leaves 13 coins to make $1.70
and even if I have 13 dimes, that is only $1.30
(d)
Likewise, there must be more that 3 quarters
3 x 25 = 75 and 220 - 75 = 145, and
12 x10 = 120 which doesn't make it
(e)
Also, there must be more than 4 quarters
4 x 25 = 100 and 220 - 100 = 120
There are 11 coins left to make $1.20
and 11 x 10 = 110, too small
(f)
It is possible with 5 quarters since
5 x 25 = 125 and 220 - 125 = 95
and 10 x 10 = 100, so it's possible
--------------------------------
5q = 125 which is 5 coins leaving 10
220 - 125 = 95
9d = 90 leaving 1 coin
1 x 5 = 5
So, the answer is
5q
9d
1n
But if I have to use pennies it doesn't work
--------------------------
I'll try
6q
5d
4n
This works, but still no pennies
---------------------------
I'll try
7q
1d
This leaves 7 coins to make 35 cents
and no room for pennies
---------------------
8q
1d
Leaves 6 coins to make 10 cents
that's
1n
5p
That's it
------------
8 quarters, 1 dime, 1 nickel, and 5 pennies
are 15 coins that make $2.20

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First find the head start of the prop plane in miles

mi
Start a stopwatch when the jet leaves
Now they will both fly for the same
amount of time until jet catches prop
Let = distance jet travels
Let = time on stopwatch
--------------------------------
Jet's equation:
(1)
prop's equation:
(2)
---------------------
Substitute (1) into (2)
(2)
(2)
(2) hrs
---------------
(1)
(1)
The planes are 628.57 mi from the start
( unless I miscalculated )

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60% of 35% of the class are the females with blue eyes

21% of the class are females with blue eyes
---------------
If 35% of the class are females, 65% are males
and
38% of 65% have blue eyes

24.7% of the class are males with blue eyes
---------------

45.7% of the class has blue eyes

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Let = number of cookies he started with
After giving some to Mom, he has left

After giving some to Aunt, he has left

After giving some to a friend he has left

He ends up with 6 cookies, so



He started with 55 cookies
check answer:
He gave Mom
,
so he has left
He gave Aunt
,
so, he has left
He gave friend

He has left
OK

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Let = number of boys in original group
Let = number of girls in original group
--------------
When 15 boys leave, you have
boys and girls, and
you are given that
(1)
-------------------
When 45 girls leave, you have
boys
girls
You are given that
(2)
--------------------------
(1)
(1)
and
(2)
(2)
--------------------
Multiply both sides of (1) by
and add the equations
(1)
(2)


There were 40 boys in the original group
check:
(1)
(1)
(1)
(1)
and
(2)
(2)
(2)
(2)
(2)
OK

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Slope = ( change in y ) / ( change in x )
slope = ( 0 - 1 ) / ( 3 - 0 )
slope = -1 / 3

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Let = bird's speed with no wind
Let = the speed of the wind
= half of the wind speed
= bird's speed flying against the wind
= bird's speed flying with the wind
------------
given:
(1)
(2)
-------------------
Multiply both sides of (2) by
(2)
(2)
and
(1)
-----------------
add the equations


and, since
(1)
(1)
(1)
The bird's speed with no wind is 50 km/hr
The speed of the wind is 40 km/hr
check:
(2)
(2)
(2)
(2)
and
(1)
(1)
(1)
OK

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You can write that as

I words, this is:
( what you started with ) - ( what you lost ) = what you have left

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It isn't stated exactly, but there must be a fixed
number of oranges that are being shared.
Let = the number of children
With each one getting 9 oranges, there are
oranges in all.
and If each child gets 6 oranges, there are
oranges in all.
-------------------------
In both cases, the number of oranges hasn't
changed, so



There are 6 children
------------------
check:
If each gets 9 oranges, that is
------------------
If each gets 6 with 18 left over, that is


OK

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There are 4 history courses any 1 of which can be taken
with any 1 of 9 psych courses. That is

Any 1 of these 36 choices can be taken with any 1 of
13 math courses. That is
so far
Any 1 of these 468 choices can be taken with any 1 of
5 business courses. That is
answer
or, you could write it in 1 line: