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EVERYTHING YOU
NEED TO ACE
MATH
IN ONE BIG
FAT NOTEBOOK
Flexibound paperback
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978-0-7611-6096-0 • No. 16096
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August
2016
RATIONAL NUMBERS
AND THE NUMBER LINE
EXAMPLE:
All rational numbers can be
Similarly, because
placed on a
NUMBER LINE.
than
-3 is smaller than -2 and also smaller
-1, it is placed to the left of those numbers.
A number line is a line that orders
and compares numbers. Smaller
numbers appear on the left and
-3
larger numbers on the right.
-3
-2
-1
0
1
2
3
-2
-1
0
1
2
3
EXAMPLE:
Not only can we place integers on a number line, we can
put fractions, decimals, and all other rational numbers on a
EXAMPLE:
Because
number line, too:
2 is larger than 1 and also larger than 0, it is placed
to the right of those numbers.
-2.38
-3
6
-2
-1
0
1
2
3
-3
-2
3
-4
-1
1
2
0
5
1
2
π
3
7
Absolute value bars are also grouping symbols, so you
EXAMPLE:
must complete the operation inside them first, then
(The absolute value of
take the absolute value.
EXAMPLE:
- | -1 6 | = - 1 6
−16 is 16.
Then we apply the negative symbol
on the outside of the absolute value
| 5 -3 | = | 2 | = 2
bars to get the answer
−16.)
Sometimes there are positive or negative symbols outside
A number in front of the absolute value bars means
an absolute value bar. Think: inside, then outside-first
multiplication (like when we use parentheses).
take the absolute value, then apply the outside symbol.
EXAMPLE:
| |
EXAMPLE: - 6 = - 6
(The absolute value of
2• 4 = 8
(The absolute value of
−4 is 4.)
(Once you have the value inside
we apply the negative symbol on the
the absolute value bars, you
outside of the absolute value bars to
can solve normally.)
get the answer
NOW THIS CHANGES
EVER YTHING.
20
6 is 6. Then
2|-4|
−6.)
Multiplication can be shown in a few different ways—not
just with x. All of these symbols mean multiply:
2 x 4 = 8
2 • 4 = 8
(2)(4) = 8
2(4) = 8
If you use VARIABLES, you can put variables next to
each other or put a number next to a variable to indicate
multiplication, like so:
ab = 8
VARIABLE: a letter or symbol
3x = 15
is used in place of a quantity
we don’t know yet
21
150 miles in 3 hours. At this rate,
how far would you travel in 7 hours?
EXAMPLE:
You drive
150 miles
X miles
-= 3 hours
7 hours
150 •7 = 3 •x
1050 = 3x (Divide both sides by 3 so you can get x alone.)
350 = x
You’ll travel
EXAMPLE: A recipe requires 6 cups of water for 2
pitchers of fruit punch. The same recipe requires
15 cups
5 pitchers of fruit punch. How many cups
of water are required to make 1 pitcher of fruit punch?
of water for
We set up a proportion:
6 cups
2 pitcher
=
15 cups
X cups
- or 5 pitcher
1 pitcher
=
X cups
1 pitcher
By solving for x in both cases, we find out that the
350 miles in 7 hours.
answer is always:
3 cups.
We can also see unit rate by using a table. With the data
Sometimes a proportion stays the same, even in different
1
scenarios-for example, Tim runs - a mile, and then
2
he drinks 1 cup of water. If Tim runs 1 mile, he needs 2
cups of water. If Tim runs
1.5 miles, he needs 3 cups of
from the table, we can set up a proportion:
EXAMPLE: Daphne often jogs laps at the track. The
table below describes how much time she jogs, based on
water (and so on). The proportion stays the same, and we
how many laps she finishes. How many minutes does
multiply by the same number in
Daphne jog per lap?
each scenario (in this case, times
2). This is known as the CONSTANT
OF PROPORTIONALITY or the
Whenever you see
“at this rate,”
set up a proportion!
CONSTANT OF VARIATION and is
closely related to UNIT RATE (or UNIT PRICE).
Total minutes jogging
Total number of laps
28 minutes
4 laps
Solving for
100
28 42
=
X minutes
1 laps
4
or
42 minutes
6 laps
6
=
X minutes
1 laps
x, we find out that the answer is: 7 minutes .
101
u h-o h
...
exaMple of coMMiSSion: my sister got a summer
job working at her favorite clothing store at the mall.
12% commission on her total sales.
at the end of her first week, her sales total is $3,500.
Again, the more your bill
is, the more the gratuity
or commission will be—
they have a proportional
relationship.
her boss agrees to pay
how much will she earn in commission?
12% = 0.12
exaMple of gratuity:
$3,500 x 0.12 = $420.00
at the end of a meal,
your server brings the final bill, which is
to leave a
$25. you want
15% gratuity. how much is the tip in dollars
she earned
$420.
and how should you leave in total?
Don’t forget: you can also solve these
15% = 0.15
$25 x 0.15 = $3.7 5
the tip is
$3.7 5.
$25 + 3.7 5 = $28.7 5
the total is
142
$28.7 5.
problems by set ting up proportions, like this:
12
x
- = 100 3500
100x = 42,000
x = $420
143
EXAMPLE:
In order to purchase your first used car,
$3,000. He deposits it in a bank
that offers an annual interest rate of 4%. How long
EXAMPLE:
Joey has
$11,000. Your bank agrees to loan
you the money for 5 years if you pay 3.25% interest
does he need to leave it in the bank in order to
each year. How much interest will you have paid after
earn
you need to borrow
the
5 years?
P = $11,000
R = 3.25% = 0.0325
T = 5 years
$600 in interest?
I = $600
P = $3,000
R = 4% (use .04)
T=x
I=PxRxT
I = ($11,000) (0.0325) (5)
$600 = $3,000(.04)T
I = $1,7 87.50
$600 = $120T
$1,7 87.50 in interest alone!
With this in mind, what will be the total price of the car?
150
$12,7 87.50 in total.
don’t know the length of time.
We use
(Divide both sides by 120
to get
5=T
So, Joey will earn
x to represent
information we know.)
T by itself.)
$600 after 5 years.
$11,000 + $1,7 87.50 = $12,7 87.50
The car will cost
the interest will be, but we
time and fill in all the other
I=PxRxT
You’ll have to pay
(In this case, we know what
HAS IT BEEN
5 YEARS YET?
IT’S BEEN
2 HOURS.
BANK
151
caution!
We can only use tables if rates are
PROPORTIONAL! Otherwise, there is no
ratio or proportion to extrapolate from.
If each runner’s rates are proportional, how would
their coach find out who runs faster? Their coach must
complete the table and find out how much time it would
take Tim to run 1 lap and how much time it would take
Linda to run 1 lap, and then compare them. The coach
exaMple: linda and tim are racing around a track.
can find out the missing times with proportions:
their coach records their times below:
Linda:
Linda
1 2
-=x 8
nuMber of lapS
total MinuteS run
1
2
6
?
8 minutes
24 minutes
So, it takes Linda
4 minutes to run one lap.
Tim:
1
3
- = x 15
Tim
nuMber of lapS
total MinuteS run
1
3
4
?
160
x=4
15 minutes
20 minutes
x=5
So, it takes Tim
5 minutes to run one lap.
w o o-
h o o!
Linda runs faster than Tim!
161
exaMple:
7 (x + 8) =
Think about catapulting
the number outside the
parentheses inside to simplify.
FaCtoring is the reverse of the distributive property.
instead of get ting rid of parentheses, factoring allows
us to include parentheses (because sometimes it’s simpler
to work with an expression that has parentheses).
(x + 8) = 7 (x) + 7 (8) = 7x + 56
exaMple: 15y + 12 = 3(5y + 4)
Step 1: ask yourself, “What is the greatest common factor
the DistributiVe ProPerty oF multiPliCation
of both terms?” in the above case, the greatest common
oVer subtraCtion looks like this
factor of
a(b - c) = ab - ac.
15y and 12 is 3. (15y = 3 • 5 • y) (12 = 3 • 4)
it says that subtracting two numbers inside parentheses,
then multiplying that difference times a number outside
Step 2: Divide all terms by the greatest common factor
the parentheses is equal to first multiplying the number
and put the greatest
outside the parentheses by each of the numbers inside the
common factor on
parentheses and then subtracting the two products.
the outside of the
exaMple:
exaMple:
9(5 - 3) = 9 (5) - 9(3)
parentheses.
You can always check your answer by
using the Distributive propertY.
Your answer should match the
expression you started with!
12a + 8 = 6(2a + 3)
(both expressions equal 18.)
exaMple:
6 (x - 8) =
the greatest common factor of
so, we divide all terms by
12a and 18 is 6.
6 and put it outside of
the parentheses.
(x - 8) = 6 (x) - 6 (8) = 6x - 48
176
177