Download Lecture 12: Oct. 21

PERCENTAGES AND DECIMALS
As we saw last day, it is often convenient
(for addition and comparison) to put fractions
over a common denominator.
This leads to the idea of a “standard
denominator”. One very early example of this
is the use of 60ths.
-survives in minutes and seconds of time
(60ths and 3600ths of an hour)
-and in minutes and seconds of arc (60ths and
3600ths of a degree)
Another example is the use of percentages.
1% = 1/100
2% = 2/100 (etc)
Advantages: easy to add and compare
Disadvantages: harder to multiply
not all fractions can be exactly represented.
1/3 is not 33% [It is 331/3%]
1/7 is not 14%. [What is it?]
To convert fractions to percentages:
multiply the numerator by 100 and divide by
the denominator.
If there is as remainder, either round up or
down and state that the percentage is
approximate, or divide the remainder by the
denominator as a fraction or decimal and add
it on.
EXAMPLE: 2/3 = (200/3)%
This is approximately 67%.
It is exactly 66 2/3 %.
EXAMPLE: 1/4 = (100/4)% = 25%
NOTE: Percentages are rarely used for
negative numbers; it’s more common to say “a
loss of 15%” than to say “a profit of –15%”.
To convert percentages to fractions:
divide the “percent number” by 100 and put in
lowest terms. This can always be done exactly
EXAMPLE: 35% = 35/100 = 7/20
EXAMPLE: 27% = 27/100
EXAMPLE: 150% = 150/100 = 3/2
Adding, subtracting, and comparing
percentages: just add, subtract, or
compare the “percent numbers”.
EXAMPLE: 37% + 8% = 45%
EXAMPLE: 35% is less than 43%
Multiplying percentages: EITHER interpret
one directly as a fraction, OR convert both to
fractions.
35% x 50% = half of 35% = 17.5%
35% x 15% = 35/100 x 15/100
= 7/20 x 3/20
= 21/400
= 5.25%
35% x 300% = 3 x 35% = 105%
Dividing percentages: divide the “percent
numbers” and optionally convert back to a
percentage:
35%
25%
130%
26%
= 35
25
=
= 7/5
130/26
= 140%
= 5 (NOT 5%)
Percentage increase and decrease
An increase of p% means that a number is
multiplied by (100+p)% or 1 + (p/100).
A decrease of p% means that a number is
multiplied by (100-p)% or 1 - (p/100).
EXAMPLE: If an item selling at $8 is sold at a
discount of 25%, (think “25% of $8”) the new
price is (100-25)% of $8, or $6
EXAMPLE: If an item is bought for $100 and
sold at a 10% profit (think, “10% of $100”)
the resale price is $110.
EXAMPLE: “70% off” means 30% of the
original price. This is much less than “70% of
original price”
WARNING: You will need to read word
problems carefully! Be familiar with words
such as “discount”, “rebate”, “profit”,
“markup”. These usually refer to the
INITIAL price, no matter which price you
know.
Example: A car sells for $30,000. This
represents a 20% profit for the dealer; what
did the dealer pay for the car?
Answer: The dealer’s profit is 20% of
the initial price (not of $30,000). Thus
$30,000 = 120% x initial price
= 6/5 x initial price
Initial price = 5/6 x $30,000 = $25,000.
WARNING: If two or more percentage
changes are made sequentially, each is based
on the value just before it is applied.
EXAMPLE: “Your marked price is 20% above
your catalog price” said the customer.
“That’s OK, I’ll ring it in as a 20% discount
on the marked price.”
How should the customer react?
Answer: Marked price = 120% x catalog price
= 6/5 x catalog price
Actual price = (100%-20%) x marked price
= 80% of marked price
= 4/5 of marked price
So actual price = 4 x 6 x catalog price
5 5
= 24/25 of catalog price (a 4% discount)
EXAMPLE: “Well, your RRSP value dropped
40% this year, but it rose 50% last year.” Is
this good?
Value last year: (100%+50%) x (2 yrs ago)
= 150% x (2 yrs ago)
= 3/2 x (2 yrs ago)
Value this year = (100%-40%) x (last year)
= 60% of (last year)
= 3/5 x (last year)
Value this year = 3 x 3 x (2 yrs ago)
2 5
= 9/10 x (2 yrs ago)
(a 10% loss)
NOTE: This is NOT because of the order of
the loss and gain. A 40% drop followed by a
50% gain is just as bad as a 50% gain followed
by a 40% drop.
COMPARE: (100-1) (1000+1)  100 x 1000
A 40% drop in value multiplies the value by
3/5. To offset this, the new value must be
multiplied by 5/3 , or nearly 167%. That is an
increase of 67%.
DECIMALS (TERMINATING)
Decimals generalize the “percent” idea, and
are also a natural extension of the place value
concept.
625
12.625
= 12 + 1000
+
+
+
+
1 x 101
2 x 100
6 x 10-1
2 X 10-2
5 X 10-3
101 = 10
100 = 1
10-1 = 1/10
10-2 = 1/100
10-3 = 1/1000
More fractions can be represented exactly as
decimals, but still not all.
EXAMPLES:
1/2 = 0.5
1/5 = 0.2
1/4 = 0.25
3/4 =0.75
1/10 = 0.1
1/8 = 0.125
3/8 = 0.375
1/25 = 0.04
2/5 = 0.4
A decimal always represents a fraction with
denominator a power of 10, though this may
not be in lowest terms.
If there are n digits after the decimal point,
the numerator is the integer obtained by
removing the decimal point and the
denominator is 10n.
So 10.132 = 10132/1000 = 2533/250
-5.31
= -531/100
(Putting the fraction into lowest terms,
we need only check for common factors of 2
and 5, which are easy to test!)
p/q (in lowest terms) can be represented as a
decimal if and only if q = 2a 5b.
Hint: any such number is a power of 2, a
power of 5, or one of these followed by zeros.
EXAMPLES: 37/6250 has an exact decimal
form, 0.0592
12643/640 has an exact decimal
form, 19.7546875
2/17 does not have an exact decimal
form.
COMPUTING WITH DECIMALS
We have two approaches to this. We can
convert to fractions, or we can extend our
“place value” algorithms.
4.2
21.23
3.22
1.1 4.62
+5.65
x1.2
4.4
4
26.88
0.644
.22
3.22
.22
.2
3.864
Keeping track of place value is, as always,
essential!
NON-TERMINATING DECIMALS
Remember that the integer number line is
discrete, with gaps of width 1:
The rational number line is “dense” with
infinitely many rational numbers between
each pair:
The “terminating decimal number line”
is also dense:
But it has gaps in it!
EXAMPLE: between 0.267 and 0.268 lie
infinitely many other points, such as 0.2675,
0.2673, 0.26758376487484, and so on.
No terminating decimal has a “next door
neighbour”.
But the rational number 1/3 lies in a “gap”
in that line! That gap isn’t bounded by two
terminating decimals, but it’s bounded by two
sets of terminating decimals:
{d: 3d < 1} < 1/3 < {d: 3d>1}
Note that 3 x m can never be an integer,
10n
(try for yourself!) so every terminating
decimal is in one of these two sets.
For any given number of decimal places, there
is always a largest terminating decimal with
that number of places that’s less than 1/3.
0.3, 0.33, 0.333, 0.3333 are all < 1/3
0.4, 0.34, 0334, 0.3334 are all > 1/3.
There is also always a smallest terminating
decimal with that number of places that’s
greater than 1/3.
*These give us as good an approximation
to 1/3 as we like.
* None of them actually equal 1/3. We
should NEVER write 1/3 = 0.33 or even
1/3 = 0.33333333.
0.3, 0.33, 0.333, 0.3333 are all < 1/3
0.4, 0.34, 0334, 0.3334 are all > 1/3.
The first sequence (best lower bounds with
1,2,3... decimal places) are extensions of each
other- we just add digits as we go. So any
one of them records all the ones before it.
In the same way, an infinite string of 3’s
contains all the approximations. We write
1/3 = 0.3333333333333333333333333333...
We call this a nonterminating decimal
expansion.
NOTATION PROBLEMS WITH
NONTERMINATING DECIMALS
The decimal representation of 1/3 has
infinitely many digits, all “3”.
*We can’t write them all out
*If we just write out a few, there are
always many ways to continue a pattern.
3.14... could be  or 31/7.
0.33... could be 1/3 or 331/999
To make it really clear, we can use a bar
over a set of digits to indicate that they are
repeated infinitely often:
1/3 = 0.3
= 0.3333....
1/7 = 0.142856
= 0.142856142856...
1/6 = 0.16
= 1.6666...
1 = 0.9
as well as 1.0
This representation is EXACT; it gives a
full description of the number.
WARNING:
When we write “x = 1.234...” what we mean is
“The number x has a decimal expansion that
begins with 1.234”
This is NOT the standard use of the equals
sign. In particular, it’s not transitive:
 = 3.14...
31/7 = 3.14...
but it is not true that  = 31/7.
Alternatively, don’t think of 3.14... as a
number. Think of it as meaning something like
“something between 3.14 and 3.15”
Were we just lucky that 1/3 could be
expressed exactly in finitely many symbols
using a “repeat bar”?
NO! Every rational number has a
terminating or repeating decimal expansion.
Proof: We can find any number of places of
p/q by long division. If the remainder at any
step is a power of 10 times one we’ve seen
before, the expansion starts to cycle.
EXAMPLE: 3
1/3 = 0.3
0.33
1
0.9
0.3
0.1
0.09
0.03
0.01
...
EXAMPLE: 2/11 = 0.18
11
0.1818...
2
1.1
0.9
0.88
0.02
0.011
0.009
0.0088
.
0.1
0.08
0.001
0.0008
.
.
Conversely, any repeating decimal is rational.
This follows from the observation that
1/9 = 0.111...,
1/99 = 0.010101...,
1/999 = 0.001001001...
and so on.
EXAMPLE:
0.777... = 7/9
0.232323... = 23/99
0.457457457... = 457/999.