Percents
Percent means “per one hundred”
So 32 percent or 32% means “32 per 100” or “32 out of 100”or
0.32
Paul Koester ()
MA 111, Percents
32
100
September 14, 2011
or
1 / 15
Percents
Percent means “per one hundred”
So 32 percent or 32% means “32 per 100” or “32 out of 100”or
0.32
32
100
or
What does 25% mean?
Paul Koester ()
MA 111, Percents
September 14, 2011
1 / 15
Percents
Percent means “per one hundred”
So 32 percent or 32% means “32 per 100” or “32 out of 100”or
0.32
32
100
or
What does 25% mean?
25
100
(which reduces to 14 ) OR 0.25
Paul Koester ()
MA 111, Percents
September 14, 2011
1 / 15
Percents
Percent means “per one hundred”
So 32 percent or 32% means “32 per 100” or “32 out of 100”or
0.32
32
100
or
What does 25% mean?
25
100
(which reduces to 14 ) OR 0.25
What does 2845% means?
Paul Koester ()
MA 111, Percents
September 14, 2011
1 / 15
Percents
Percent means “per one hundred”
So 32 percent or 32% means “32 per 100” or “32 out of 100”or
0.32
32
100
or
What does 25% mean?
25
100
(which reduces to 14 ) OR 0.25
What does 2845% means?
2845
100
OR 28.45
Paul Koester ()
MA 111, Percents
September 14, 2011
1 / 15
Percents
Percent means “per one hundred”
So 32 percent or 32% means “32 per 100” or “32 out of 100”or
0.32
32
100
or
What does 25% mean?
25
100
(which reduces to 14 ) OR 0.25
What does 2845% means?
2845
100
OR 28.45
What does 0.047% means?
Paul Koester ()
MA 111, Percents
September 14, 2011
1 / 15
Percents
Percent means “per one hundred”
So 32 percent or 32% means “32 per 100” or “32 out of 100”or
0.32
32
100
or
What does 25% mean?
25
100
(which reduces to 14 ) OR 0.25
What does 2845% means?
2845
100
OR 28.45
What does 0.047% means?
0.047
100
OR 0.00047
Paul Koester ()
MA 111, Percents
September 14, 2011
1 / 15
Percents
32
32 percent of 25 means “ 100
· 25” OR “0.32 · 25”, which is 8.
Paul Koester ()
MA 111, Percents
September 14, 2011
2 / 15
Percents
32
32 percent of 25 means “ 100
· 25” OR “0.32 · 25”, which is 8.
What does 230 percent of 800 mean?
Paul Koester ()
MA 111, Percents
September 14, 2011
2 / 15
Percents
32
32 percent of 25 means “ 100
· 25” OR “0.32 · 25”, which is 8.
What does 230 percent of 800 mean?
230
100
· 800 = 1840
Paul Koester ()
MA 111, Percents
September 14, 2011
2 / 15
Percents
32
32 percent of 25 means “ 100
· 25” OR “0.32 · 25”, which is 8.
What does 230 percent of 800 mean?
230
100
· 800 = 1840
What does 0.03 percent of 1, 550, 000 mean?
Paul Koester ()
MA 111, Percents
September 14, 2011
2 / 15
Percents
32
32 percent of 25 means “ 100
· 25” OR “0.32 · 25”, which is 8.
What does 230 percent of 800 mean?
230
100
· 800 = 1840
What does 0.03 percent of 1, 550, 000 mean?
0.03
100
· 1, 550, 000 = 45
Paul Koester ()
MA 111, Percents
September 14, 2011
2 / 15
Percents
72
78 percent of 200 equals 156 means “ 100
· 200 = 156”
This can be thought of as
78
156
=
100
200
In general, P % of N equals A means
P
·N =A
100
OR
P
A
=
100
N
Thus far, P and N were given and we found A. In applications, any
two of P, A, N could be given. The above formula can be used to find
the remaining variable.
Paul Koester ()
MA 111, Percents
September 14, 2011
3 / 15
Percents
At a certain high school, 72% of entering freshman earn a diploma.
Suppose 306 students finish with a diploma. How large was the
entering class?
Paul Koester ()
MA 111, Percents
September 14, 2011
4 / 15
Percents
At a certain high school, 72% of entering freshman earn a diploma.
Suppose 306 students finish with a diploma. How large was the
entering class?
72
306
=
100
A
so
72 · A = 306 · 100
so
A=
Paul Koester ()
306 · 100
= 425
72
MA 111, Percents
September 14, 2011
4 / 15
Percents
Tuition for resident students at a certain university was $3, 735 in 2001
and is currently $9, 128.
Express the current tuition as a percentage of the 2001 tuition.
Paul Koester ()
MA 111, Percents
September 14, 2011
5 / 15
Percents
Tuition for resident students at a certain university was $3, 735 in 2001
and is currently $9, 128.
Express the current tuition as a percentage of the 2001 tuition.
9128
P
=
100
3735
so
9128
· 100 = 244.39%
3735
Thus, the current tuition is 244.39% of the 2001 tuition.
P =
Paul Koester ()
MA 111, Percents
September 14, 2011
5 / 15
Percents
Tuition for resident students at a certain university was $3, 735 in 2001
and is currently $9, 128.
Express the 2001 tuition as a percentage of the current tuition.
Paul Koester ()
MA 111, Percents
September 14, 2011
6 / 15
Percents
Tuition for resident students at a certain university was $3, 735 in 2001
and is currently $9, 128.
Express the 2001 tuition as a percentage of the current tuition.
3735
P
=
100
9128
so
3735
· 100 = 40.92 . . . %
9128
Thus, the 2001 tuition is 40.92% of the current tuition.
P =
Paul Koester ()
MA 111, Percents
September 14, 2011
6 / 15
Setting up the correct percentage
The previous two examples used the same data, but a subtle change in
wording switched the role of N and A.
How can we check if we set up the fractions correctly?
The current tuition is more than the 2001 tuition, thus, if the current
tuition is expressed as a percentage of the 2001, this percentage had
better be greater than 100%
Likewise, the 2001 tuitiion is less than the current tuition, thus, if the
2001 tuition is expresses as a percentage of the current tuition, this
percentage had better be less than 100%
Paul Koester ()
MA 111, Percents
September 14, 2011
7 / 15
Percent Increase / Percent Decrease
In applications, it is very common to express only the percent change
between two quantities, not the absolute percentages.
The current tuition is 244.39% of the 2001 tuition, which could be
expressed by saying
“Tuition increased by 144.39%”
Paul Koester ()
MA 111, Percents
September 14, 2011
8 / 15
Percent Increase / Percent Decrease
In applications, it is very common to express only the percent change
between two quantities, not the absolute percentages.
The current tuition is 244.39% of the 2001 tuition, which could be
expressed by saying
“Tuition increased by 144.39%”
If the population of a town was 1000 people and the population
increased by 25%, what is the new populaton?
Paul Koester ()
MA 111, Percents
September 14, 2011
8 / 15
Percent Increase / Percent Decrease
In applications, it is very common to express only the percent change
between two quantities, not the absolute percentages.
The current tuition is 244.39% of the 2001 tuition, which could be
expressed by saying
“Tuition increased by 144.39%”
If the population of a town was 1000 people and the population
increased by 25%, what is the new populaton?
25
· 1000 = 250
100
The population increased by 250 people, therefore the total population
is
1000 + 250 = 1250
(Original Population + Increase = New Population)
Paul Koester ()
MA 111, Percents
September 14, 2011
8 / 15
Percent Increase / Percent Decrease
You buy a new computer. The list price is $1200. You have to pay 6%
sales tax. How much do you actually pay?
Paul Koester ()
MA 111, Percents
September 14, 2011
9 / 15
Percent Increase / Percent Decrease
You buy a new computer. The list price is $1200. You have to pay 6%
sales tax. How much do you actually pay?
6
· 1200 = 72. Thus, you pay
METHOD 1: The amount of tax is 100
1200 + 72 = $1272.
Paul Koester ()
MA 111, Percents
September 14, 2011
9 / 15
Percent Increase / Percent Decrease
You buy a new computer. The list price is $1200. You have to pay 6%
sales tax. How much do you actually pay?
6
· 1200 = 72. Thus, you pay
METHOD 1: The amount of tax is 100
1200 + 72 = $1272.
METHOD 2: You pay 100% for the computer plus 6% for the tax, thus
the actual price is (100 + 6) = 106% of the list price. Thus, you pay
106
· 1200 = $1272
100
Notice
106
100
6
6
=
+
=1+
100
100 100
100
thus, the final price could also be expressed as
6
1200 · 1 +
= $1272
100
Paul Koester ()
MA 111, Percents
September 14, 2011
9 / 15
Percent Increase / Percent Decrease
A new pair of shoes sells for $75. The store advertises a “ 30% off
everything” sale. What is the sale price of the new shoes?
(A) $45.00
(B) $22.50
(C) $72.00
(D) $72.75
(E) $52.50
(F) $97.50
Paul Koester ()
MA 111, Percents
September 14, 2011
10 / 15
Percent Increase / Percent Decrease
Your textbook gives these two formulas:
If you start with a quantity Q and increase it by x% then you end up
with the quantity I,
x I = 1+
·Q
100
If you start with a quantity Q and decrease it by x% then you end up
with the quantity D,
x D = 1−
·Q
100
You may find it easier to only remember the first quantity and to think
of decreases as “negative increases”.
Paul Koester ()
MA 111, Percents
September 14, 2011
11 / 15
Percent Increase / Percent Decrease
On Monday morning, the price of gold was $1845.00 per ounce. On
Tuesday morning the price of gold was $1841.00 per ounce. Find the
percent increase / percent decrease in the price of gold.
Paul Koester ()
MA 111, Percents
September 14, 2011
12 / 15
Percent Increase / Percent Decrease
On Monday morning, the price of gold was $1845.00 per ounce. On
Tuesday morning the price of gold was $1841.00 per ounce. Find the
percent increase / percent decrease in the price of gold.
Using the percent increase formula with Q = 1845 and I = 1841 we find
x 1841 = 1 +
· 1845
100
x 1841
=
= 0.997831 . . .
1+
100
1845
x
= 0.997831 − 1 = −0.0021680 . . .
100
x = −0.0021680 . . . · 100 = −0.21680 · · ·
Thus, the price of gold decreased by 0.21680%.
Paul Koester ()
MA 111, Percents
September 14, 2011
12 / 15
Back to where we started?
An item typically sells for $100.00 but is on sale, at 7% off. The sales
tax is 7%. After the sale and the tax, do you pay $100.00 for this item?
Why or why not?
Paul Koester ()
MA 111, Percents
September 14, 2011
13 / 15
Back to where we started?
An item typically sells for $100.00 but is on sale, at 7% off. The sales
tax is 7%. After the sale and the tax, do you pay $100.00 for this item?
Why or why not?
First, deduct 7% from the original price. The sale price is therefore
7
= $93.00
100 · 1 −
100
The price after tax is obtained by adding 7% of the sale price, so the
final price is
7
= $99.51
93 · 1 +
100
Paul Koester ()
MA 111, Percents
September 14, 2011
13 / 15
How did this happen?
We subtracted 7% then added 7%. Why isn’t this 0%?
Shouldn’t 7% − 7% = 0?
Paul Koester ()
MA 111, Percents
September 14, 2011
14 / 15
How did this happen?
We subtracted 7% then added 7%. Why isn’t this 0%?
Shouldn’t 7% − 7% = 0?
We need to pay special attention to the bases of the percentages. In
particular,
7% of 100 − 7% of 100IS equal to 0
BUT our problem is concerned with
7% of 93 − 7% of 100 = −0.49
Thus, the combined effect of the two percent changes was to lower the
price by $0.49
Paul Koester ()
MA 111, Percents
September 14, 2011
14 / 15
A more complex example
The Dow Jones Industrial average started the week at 11260 points.
On Monday it increased 1.9%. On Tuesday it decreased 0.7%. On
Wednesday it decreased 3.5%. On Thursday it decreased 2.7% On
Friday it increased 1.4%. What was the total percent change over the
week? Is this a percent increase or a percent decrease?
Paul Koester ()
MA 111, Percents
September 14, 2011
15 / 15
A more complex example
The Dow Jones Industrial average started the week at 11260 points.
On Monday it increased 1.9%. On Tuesday it decreased 0.7%. On
Wednesday it decreased 3.5%. On Thursday it decreased 2.7% On
Friday it increased 1.4%. What was the total percent change over the
week? Is this a percent increase or a percent decrease?
The percent change factors are
1.9
1+
= 1.019
add since increase
100
0.7
1−
= 0.993
subtract since decrease
100
3.5
1−
= 0.965
subtract since decrease
100
2.7
1−
= 0.973
subtract since decrease
100
1.4
1+
= 1.014
add since increase
100
Paul Koester ()
MA 111, Percents
September 14, 2011
15 / 15
A more complex example
The Dow Jones Industrial average started the week at 11260 points.
On Monday it increased 1.9%. On Tuesday it decreased 0.7%. On
Wednesday it decreased 3.5%. On Thursday it decreased 2.7% On
Friday it increased 1.4%. What was the total percent change over the
week? Is this a percent increase or a percent decrease?
For the whole week, the percent change factor is the product
1.019 · 0.993 · 0.965 · 0.973 · 1.014 = 0.9634
Now, if the percent change over the entire week is x% then
1+
x
= 0.9634
100
x
= 0.9634 − 1 = −0.0366
100
x = −0.0366 · 100 = −3.66%
The percent change is 3.66% and this is a decrease since x was negative.
Paul Koester ()
MA 111, Percents
September 14, 2011
16 / 15