Beginning Algebra
Tutorial 16:
Percent and Problem
Solving
 Learning Objectives
|
After completing this tutorial, you should be able to:
-
Convert percents into decimal numbers.
-
To convert decimal numbers into percents.
-
Use Polya's four step process to solve word problems involving percents.
-
Work problems involving pie charts and percents.
-
Work problems involving tables and percents.
|
Introduction
|
| In this tutorial we will be solving problems involving
percentages.
Since we are still problem solving, I will use Polya’s four steps to
Problem
Solving as introduced in Tutorial
15: Introduction to Problem Solving to step us through the
percent
problems in this tutorial. It is a good idea to be comfortable working
with percents, you never know when you will be confronted with
them.
Let's see how we can help you out with percents. |
Tutorial
|
| Percent means per hundred.
% is the symbol that we use to notate percent.
Some examples of percentages are:
15% = 15/100 = .15
25% = 25/100 = .25
100% = 100/100 = 1.00
|
Writing a Percent as
a Decimal Number
|
| When you are going from percent to decimal,
drop the percent
sign and then move your decimal two places to the left. |
 Example
1: Write 57% as a decimal. |
| Dropping the percent sign and then moving the
decimal two places
to the left we get:
57% = . 57
|
| Example
2: Write 145% as a decimal. |
| Dropping the percent sign and then moving the
decimal two places
to the left we get:
145% = 1.45
|
| Example
3: Write .34% as a decimal. |
| Dropping the percent sign and then moving the
decimal two places
to the left we get:
.34% = .0034
|
Writing a Decimal Number
as a Percent
|
| When you are going from decimal to percent, move
your decimal place
two to the right and then put a % sign at the end of the number. |
| Example
4: Write .78 as a percent. |
| Moving the decimal place two to the right and then
putting a % sign
at the end of the number we get:
.78 = 78%
|
| Example
5: Write 8 as a percent. |
| Moving the decimal place two to the right and then
putting a % sign
at the end of the number we get:
8 = 800%
|
| Example
6: Write .0325 as a percent. |
| Moving the decimal place two to the right and then
putting a % sign
at the end of the number we get:
.0325 = 3.25%
|
Polya’s Four-step
Process
for Problem Solving
(revisited)
|
| Example
7: Find 45% of 125. |
| Make sure that you read the question carefully several
times.
We are looking for a number that is 45% of 125, we
will let
x = the value we are
looking for
|
 |
*Multiply |
56.25 is 45% of 125.
FINAL ANSWER:
The number is 56.25.
|
| Example
8: The number 5.25 is what percent of 35? |
| Make sure that you read the question carefully several
times.
We are looking for the percent we would have to take of
35 to get 5.25.
x = the percentage we
are looking for
|
 |
*Inverse of mult by 35 is div. by
35
|
5.25 is 15% of 35.
FINAL ANSWER:
The answer is 15%.
|
| Example
9: 32 is 40% of what number? |
| Make sure that you read the question carefully several
times.
We are looking for the number that you when you take 40%
of it you would
get 32.
x = the number we are
looking for
|
 |
*Inverse of mult by .4 is div. by
.4
|
| 32 is 40% of 80.
FINAL ANSWER:
The number is 80.
|
| Example
10: A math class has 30 students. Approximately
70% passed their last math test. How many students passed the
last
math test? |
| Make sure that you read the question carefully several
times.
We are looking for how many students passed the last
math test,
we will let
x = number of
students
|
 |
*Multiply |
21 is 70% of 30.
FINAL ANSWER:
21 students passed the last math test.
|
| A pie chart or circle graph is another way to
give a visual representation
of the relationship of data that has been collected.
It is made up of a circle cut up in sectors. Each
sector represents
the percentage that a category of data is of the whole pie.
Keep in mind that a circle is 360 degrees.
The graph below is a pie chart:
Each sector of the circle represents the percentage of
profits that
the given ice cream flavor made.
The top sector shows that chocolate made 41% of the
profits in 2002.
The bottom right sector shows that vanilla made 29% of
the profits in
2002.
The bottom left sector shows that strawberry made 30% of
the profits
in 2002.
With all of this talk about
pies and ice cream,
is anyone else hungary????
|
| Example
11: The pie chart or circle graph below shows the
total
enrollment of students at State College during the Fall 2002 semester,
broken down into seniors, juniors, sophomores, and freshmen. Use
the graph to answer questions 11a - 11c.
11a. In the Fall 2002 semester, what was the ratio
of freshmen
to seniors at the college?
11b. If the number of sophomores in the Fall 2002
semester was
20% higher than the number of sophomores in the Fall 2001 semester, how
many sophomores were enrolled in Fall 2001?
11c. If the areas of sectors in the circle graphs
are drawn in
proportion to the percentages shown, what is the measure, in degrees,
of
the central angel sector representing the percentage of juniors?
|
11a. In the Fall 2002 semester, what was the
ratio of freshmen
to seniors at the college?
(return to pie chart) |
| When setting up a ratio you need to write the number
that corresponds
to the first part first and then compare it to the number that
corresponds
to the second part of the ratio.
What do you think the first part of the ratio,
freshmen or seniors?
Since freshmen are listed first, that is what our first number of our
ratio
has to correspond to.
What is the percentage attached to freshmen?
Looking on
the pie chart, I believe it is 40%.
That leaves the number associated with seniors to be our
second part
of the ratio. Looks like that will be 12%.
So the ratio of freshman to seniors would be 40 to 12.
You can think of ratios as fractions, and simplify them in the same
manner.
Since 40 and 12 have a greatest common factor of 4, we can reduce this
to be 10 to 3.
Note that if you had started with 12 to 40, this would
be incorrect.
12 to 40 would be the ratio of seniors to freshman. You write a
ratio,
just like you read it, left to right.
The simplified ratio of freshmen to seniors would be
10 to 3.
|
11b. If the number of sophomores in the Fall
2002 semester
was 20% higher than the number of sophomores in the Fall 2001 semester,
how many sophomores were enrolled in Fall 2001?
(return to pie chart) |
| Wow, where do we start? Since we know the total
number and percent
of sophomores from Fall 2002, we can start by finding the number of
sophomores
there were in the Fall 2002 semester.
What percentage were sophomores in the Fall 2002
semester? If
you said 30% you are correct!!!
So what would be the number of sophomores for
the Fall 2002
semester? When we take a percentage of a number, we write
the percentage in decimal form and then multiply it times the
number
we are taking the percentage of.
Taking 30% of the total of 6542 we get:
(.3)(6542) = 1962.6 which rounds up to 1963.
1963 is the number of sophomores in the Fall 2002
semester.
Using this found information we need to find out how
many sophomores
were enrolled in the Fall 2001 semester.
The problem says that the Fall 2002 semester has 20%
more sophomores
than the Fall 2001 semester.
We are going to let x
be the number
of sophomores in Fall 2001.
We are needing an equation that represents the English
phrase "the Fall
2002 semester has 20% more sophomores than the Fall 2001 semester".
Going
left to right, the Fall 2002 semester would be 1963, has would be our =
sign, 20% more than the Fall 2001 semester, would be starting with the
Fall 2001 semester, which is x and
adding on
20% of that, which is .2x.From all of
this
we get the following equation:
Solving this equation for x
we get:
|
 |
*Add like terms
*Divide both sides by 1.2
|
| The number of sophomores in the Fall 2001 semester
would round up
to be 1636. |
11c. If the areas of sectors in the circle
graphs are drawn
in proportion to the percentages shown, what is the measure, in
degrees,
of the central angle sector representing the percentage of juniors?
(return to pie chart) |
| On this problem, the key is to know that a circle
measures 360 degrees.
So if we know the percentage of the circle that a sector represents,
then
we can take that percentage of 360 degrees and find the measure of just
that sector.
What percentage of the students were juniors in the
Fall 2002 semester?
If
you said 18% you are correct!!!
So what would be the measure of the central angle for
juniors for
the Fall 2002 semester?
Since a full circle is 360 degrees, we are basically wanting
to know what 18% of 360 degrees is.
As shown above, when we take a percentage of a number,
we write
the percent in decimal form and then multiply it times the
number
we are taking the percentage of.
Taking 18% of the total of 360 degrees we get:
(.18)(360degrees) = 64.8 degrees
The central angle sector for the juniors measures
64.8 degrees.
|
| A table is another way to give a visual representation
of the relationship
of data that has been collected.
A table can have one, two, three or more columns of
data.
The graph below is a table:
Yummy Ice Cream Profits
|
Flavor
|
2001
(%)
|
2002
(%)
|
|
Vanilla
|
35.3
|
29
|
|
Chocolate
|
40
|
41
|
|
Strawberry
|
24.7
|
30
|
| |
100.0%
|
100.0%
|
|
Total Profits:
|
$98 million
|
$105 million
|
The first column identifies the flavors of ice cream
that made a profit.
The second column represents the percentage of profits
that each flavor
made in 2001 as well as the total profits in dollars.
The third column represents the percentage of profits
that each flavor
made in 2002 as well as the total profits in dollars.
Vanilla made 35.3% of the profits in 2001 and 29% of the
profits in
2002.
Chocolate made 40% of the profits in 2001 and 41% of the
profits in
2002.
Strawberry made 24.7% of the profits in 2001 and
30% of the profits
in 2002.
|
| Example
12: The table below shows the results of a survey on
beverage
preference taken with customers of the Good Eats Café in 2001
and
2002. Each customer voted for only one beverage. Use the
table
to answer questions 12a - 12c.
12a. Approximately how many customers preferred
Sprite in 2002?
12b. By approximately what percent did the
preference of root
beer decrease from 2001 to 2002?
12c. What was the difference between the number of
votes for Coca
Cola in 2001 versus 2002?
Survey of Customer’s Beverage Preference at the Good
Eats Café.
Each customer voted for only one beverage.
|
Beverage
|
2001
(%)
|
2002
(%)
|
|
Coca Cola
|
35
|
30
|
|
Diet Coke
|
22.3
|
23
|
|
Sprite
|
15.9
|
14.4
|
|
tea
|
12
|
15
|
|
raspberry tea
|
11.5
|
12
|
|
root beer
|
2.7
|
1.1
|
|
Diet Sprite
|
.6
|
4.5
|
| |
100.0%
|
100.0%
|
|
Total number of customers:
|
8950
|
9432
|
|
| Remember that the numbers in the table are
percentages.
What percent of customers in 2002 voted for
Sprite? Looking
at the third column (2002), it looks like it is 14.4%.
How many votes were taken in 2002? Looking
at the bottom
of the third column (2002), it says that the total number of votes in
2002
is 9432.
When we take a percentage of a number, we write
the percentage in decimal form and then multiply it times the
number
we are taking the percentage of.
Taking 14.4% of the total of 9432 we get:
(.144)(9432) = 1358.208 which rounds down to 1358.
Approximately 1358 customers voted for Sprite in 2002.
|
12b. By approximately what percent did the
preference of root
beer decrease from 2001 to 2002?
(return to table) |
| Basically we are looking for the difference in
percent. That
means we will not have to take a percentage of any numbers. We
just
need to find the difference between those two percents.
What was the percent of customers that voted for root
beer in 2001?
If
you said 2.7, you are correct. You find that by going to the
second
column (2001) and going down to root beer.
What was the percent of customers that voted for root
beer in 2002?
If
you said 1.1, you are correct. You find that by going to the
third
column (2002) and going down to root beer.
So what is their difference? 2.7 - 1.1 =
1.6
There was a 1.6% decrease of votes for root beer from
2001 to 2002.
|
12c. What was the difference between the
number of votes for
Coca Cola in 2001 versus 2002?
(return to table) |
| Now we are looking for a difference in the number of
votes, so we will
have to do a little bit more work here then in 12b above. We will
have to take the appropriate percentage of the corresponding totals for
each year and find the number of votes for each year. Then we
will
have the numbers that we need to take the difference of.
What percent of customers in 2001 voted for Coca
Cola? Looking
at the second column (2001), it looks like it is 35%.
How many votes were taken in 2001? Looking
at the bottom
of the second column (2001), it says that the total number of votes in
2001 is 8950.
When we take a percentage of a number, we write
the percentage in decimal form and then multiply it times the
number
we are taking the percentage of.
Taking 35% of the total of 8950 we get:
(.35)(8950) = 3132.5 which rounds up to 3133.
Approximately 3133 customers voted for Coca Cola in
2001.
What percent of customers in 2002 voted for Coca
Cola? Looking
at the third column (2002), it looks like it is 30%.
How many votes were taken in 2002? Looking
at the bottom
of the third column (2002), it says that the total number of votes in
2002
is 9432.
When we take a percentage of a number, we write the
percentage in decimal
form and then multiply it times the number we are taking the percentage
of.
Taking 30% of the total of 9432 we get:
(.3)(9432) = 2829.6 which rounds up to 2830.
Approximately 2830 customers voted for Coca Cola in
2002.
Finding the difference between the two values that we
found we get:
3133 - 2830 = 303
There was a 303 difference between the number of
customers that voted
for Coca Cola in 2001 versus 2002.
|
Practice Problems
|
| These are practice problems to help bring you to the
next level.
It will allow you to check and see if you have an understanding of
these
types of problems. Math works just like
anything
else, if you want to get good at it, then you need to practice
it.
Even the best athletes and musicians had help along the way and lots of
practice, practice, practice, to get good at their sport or instrument.
In fact there is no such thing as too much practice.
To get the most out of these, you should work the
problem out on
your own and then check your answer by clicking on the link for the
answer/discussion
for that problem. At the link you will find the answer
as well as any steps that went into finding that answer.
|
 Practice
Problems 1a - 1b:
Write each percent as a decimal.
|
|
Practice
Problems 2a - 2b:
Write each decimal as a percent.
|
|
Practice
Problems 3a - 3c:
Solve the percent problem.
|
3c. A local furniture store is having a terrific
sale.
They are marking down every price 45%. If the couch you have our
eye on was $800 before the markdown, find the decrease and the sale
price.
(answer/discussion
to 3c) |
|
Practice
Problems 4a - 4c:
The pie chart or circle graph below
shows the profit
breakdown of the paper products sold by ABC Paper Company in 2001.
Use the graph to answer questions 4a
- 4c.
|
4b. If the profit for napkins in 2001 was 35%
lower than its
profit in 2000, how much profit was made from napkins in 2000?
(answer/discussion
to 4b) |
4c. If the areas of sectors in the circle graphs
are drawn in
proportion to the percentages shown, what is the measure, in degrees,
of
the central angle sector representing the percentage of profit of
tissues?
(answer/discussion
to 4c) |
Need Extra Help on These Topics?
|
All contents copyright (C) 2001 - 2008, WTAMU and Kim Seward. All rights reserved.
Last revised on July 4, 2003 by Kim Seward. |
(Back
to the THEA Math Help Page)
