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Let = average monthly summer gas bill
Let = average monthly winter gas bill
Let = average monthly summer electric bill
Let = average monthly winter electric bill
given:
(1)
(2)
(3)
(4)
-----------------
There are 4 equations and 4 unknowns, so it's solvable
Substitute (1) in (3)
(3)

Substitute (2) in (4)
(4)

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


For July and August, he should set aside $298

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Let = amount invested @ 3% interest
Let = amount invested @ 4% interest
given:
(1)
(2)
-------------------
(2)
(2)
Multiply both sides of (1) by
and subtract (1) from (2)
(2)
(1)

and, since
(1)
(1)

$4,000 = amount invested @ 3% interest
$8,000 = amount invested @ 4% interest
check answer:
(2)
(2)


OK

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given:


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







During the 15th day 35% have heard of the crime
check answer:






OK

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They intersect where


There can be no intersection, because there is no way
the left side can equal the right
can only be positive
-------------------
Here's the graph:

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Let = theregular price
given:




The regular price was $82.34

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(1)
None of the digits can be , , or , since
the sum of any arrangement of is too small
What is left is all possible arrangements of:
994
985
976
886
877
----
The arrangements are:
994
949
499 - 3 possible
---
985
958
895
859
598
589 - 6 possible
---
976 - 6 possible
886 - 3 possible
877 - 3 possible
--------------
There are 21 3-digit numbers whose digits add up to 22

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Let = highway miles he has to drive
In words:
(total miles driven)/(gallons used) = efficiency in mi/gal
-------------
given:
Gallons of gas used in city =
Gallons of gas used on highway =


Multiply both sides by

Multiply both sides by



He has to drive 100 miles on the highway
check answer:






OK

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The slope is
Here's the plot:

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Subtract from both sides

Divide both sides by

Now you can complete the square.
Take 1/2 of the coefficient of ,
square it, and add it to both sides


Both sides are now a perfect square

Take the square root of both sides
( this is the (+) square root)

and
(this is the (-) square root)

The solutions are and

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Let = the price of a box of 16 peaches
Let = the price of a bag of 12 apples
Let = the price of 2 boxes of peaches and 5 bags of apples
given:
(1) (in cents)
(2) (in cents)
(in cents)

( in cents)
-----------------------
(1)
(2)
Multiply both sides of (1) by and add (1) and (2)
(1)
(2)


and, since
(1)



Each box of 16 peaches costs $7.65
Each bag of 12 apples costs $5.64
check:
(2)
(2)


OK

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Let width =
Let length =
given:


By substitution:








the length is 16 m

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There are 26 red cards and 26 black cards in a deck
Hearts and diamonds are red, so if the 1st card is a
heart, it is red.
There are then 25 red cards left, and there are 25 + 26 = 51 cards
left. The probability of drawing a 2nd card that is red
is

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Let the 10s digit of Father's age =
Let the units digit of Father's age =
given:
Father's age =
Age of Son =



This is 1 equation and 2 unknowns, so it's
not solvable directly.
I see that . I also see that the left
side will always be even, so must be
odd for that to be.
So, can be 3,5,7
( can't be , since
can be even or odd
---------------------


does not divide for any
------------------------


does not divide for any
------------------------




---------
The father is 73 and the son is 37
check:




OK

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(4,4)

The formula is:

Multiply both sides by




check:
Does the line go through (4,4) ?



OK

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1 child would take 10 days to do a job
1 adult would take 6 days to do a job
---------------------
Building the tower, let = the rate that
children work.
Let = the rate that an adult works
44 adults in 5 days to build a tower
is adults/day to build a tower





The rate for 40 adults is
The rate for 80 children is
---------------------
Add the rates to get their rate working together
(1 tower in d days)

Multiply both sides by




It will take 50 adults and 80 children 2.5 days to build a tower

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Let = number of necklaces
Let = number of bracelets
given:
(1)
(2)
----------------
Multiply both sides of (1) by
and subtract (1) from (2)
(2)
(1)


and
(1)


Don bought 70 necklaces and 50 bracelets
check:
(2)
(2)


OK

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Let





---------


and


--------------
check answers:



and




OK

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Let = the grade she needs




Looks like she can't do it unless she gets a 104

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Get logs on both sides.
First, express as a log to the base

Rewrite the equation:






--------------
check answers:



and

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The letter used for slope is .
If the slope of a given line is , then
any line perpendicular to this line will have
a slope equal to
given:

slope of perpendicular line =

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Let = voltage of 2nd battery
= voltage the 1st battery
= voltage of 3rd battery





The batteries are
6v, 12v, and 36v

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This says that is greater than
or equal to minus 5 1/3. The negative
odd integers that satisfy this are
-5, -3, and -1
--------------
check:




OK

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The 1st day:


The 2nd day:


------------
Since the distance are the same:



and, since



The distance was 280 mi
check answer:




OK

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I think it may be simpler than it looks.
The man could choose from the categories in
any order he wishes except for the 1st and last night.
------------------
The problem is hard to interpret. I think they are talking about
5 successive nights that he reads a single short story.
I don't think I can go beyond 5 successive nights because
I don't know how many short stories are in each book.
-------------------
I'll use 1 letter for each category:
M - 5
N - 4
T - 6
R - 3
H - 5
------
total - 23
---------------
The 1st night, he chooses between
M - 5
T - 6
-------
total - 11 ways to choose
2nd night: there are 23 - 1 = 22 possible ways he can choose
3rd night: there are 23 - 2 = 21 possible ways he can choose
4th night: there are 23 - 3 = 20 possible ways he can choose
5th night: there are 11 - 1 = 10 possible ways he can choose
--------
The choices must be multiplied to get all the possible orders

There are 1,016,400 possible orderings of short stories he can
read on 5 successive nights
-------------------------
You should get a 2nd opinion on this, too. These kind of
problems can be very tricky, but they are good for the brain.

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Let = Melissa's earnings/hr
Let = Jonathan's earnings/hr
Let = Louise's earnings/hr
(1)
(2)
-----------
(2)

(1)
(1)
(1)
(1)

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Let = time for larger pipe to fill oil tank
given:
= time for smaller pipe to fill tank
---------
Add their rates of filling to get rate working together

Multiply both sides by




The solutions are
(can't use negative time)

and

The smaller pipe takes 12 min
The larger pipe takes 6 min

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Let = miles driven
Let = rental cost
given:
Cheap Truck:

Trucks 4 Less:

Set the costs equal to each other



The costs are the same driving 250 miles
check:




and




OK

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Let the larger number =
Let the smaller number =
given:
(1)
(2)
----------------
Multiply both sides of (2) by
(2)
Substitute this into (1)
(1)
(1)

and
(1)

The numbers are 4 and 13

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When the equation has the form ,
The discriminant is
In your equation,
,





discriminant = 97

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Scientific notation always looks like:
a.aaa * 10^b
There is only 1 digit to the left of the decimal point
You can do it in steps:


answer

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Remember that logs are exponents
The way you read this is:
The exponent that gives me is .931, and
the base of the exponent is