# Beginning Algebra Tutorial 16

Beginning Algebra
Tutorial 16: Percent and Problem Solving

Learning Objectives

After completing this tutorial, you should be able to:
1. Convert percents into decimal numbers.
2. To convert decimal numbers into percents.
3. Use Polya's four step process to solve word problems involving percents.
4. Work problems involving pie charts and percents.
5. 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

Percents

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)

As mentioned above, since we are still problem solving, we will use the exact same four step process we used in Tutorial 15: Introduction to Problem Solving.  To refresh your memory, here they are again:

Step 1:  Understand the problem.

Step 2:  Devise a plan (translate).

Step 3:  Carry out the plan (solve).

Step 4:  Look back (check and interpret).

Percent Problems

Whenever you are working with a percent problem you need to make sure you write your percent in an equivalent decimal form as shown above.

When you are looking for a percent, make sure that you convert your decimal into a percent, as shown above, for the final answer.

Example 7:  Find 45% of 125.

Step 1: Understand the problem.

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

Step 2:  Devise a plan (translate).

Step 3:  Carry out the plan (solve).

*Multiply

Step 4:  Look back (check and interpret).

56.25 is 45% of 125.

The number is 56.25.

Example 8:  The number 5.25 is what percent of 35?

Step 1: Understand the problem.

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

Step 2:  Devise a plan (translate).

Step 3:  Carry out the plan (solve).

*Inverse of mult by 35 is div. by 35

Step 4:  Look back (check and interpret).

5.25 is 15% of 35.

Example 9:  32 is 40% of what number?

Step 1: Understand the problem.

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

Step 2:  Devise a plan (translate).

Step 3:  Carry out the plan (solve).

*Inverse of mult by .4 is div. by .4

Step 4:  Look back (check and interpret).

32 is 40% of 80.

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?

Step 1: Understand the problem.

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

Step 2:  Devise a plan (translate).

Step 3:  Carry out the plan (solve).

*Multiply

Step 4:  Look back (check and interpret).

21 is 70% of 30.

21 students passed the last math test.

Pie Charts and Percents

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 angle sector representing the percentage of juniors?

11a.  In the Fall 2002 semester, what was the ratio of freshmen to seniors at the college?

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?

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:

*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?

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.

Tables and Percents

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

12a.  Approximately how many customers preferred Sprite in 2002?

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?

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?

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.

1a.  82%
1b.   325%

Practice Problems 2a - 2b: Write each decimal as a percent.

2a.  .64
2b.   .0003

Practice Problems 3a - 3c: Solve the percent problem.

3a.    54 is 60% of what number?
3b.  50.4 is what percent of 120?

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.

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.

4a.  In 2001, what was the ratio of profit of toilet paper to profit of paper cups?

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?

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?

Need Extra Help on these Topics?

The following webpage can assist you in the topics that were covered on this page:

http://www.purplemath.com/modules/percents.htm
This webpage goes over percentages.

Go to Get Help Outside the Classroom found in Tutorial 1: How to Succeed in a Math Class for some more suggestions.

Last revised on July 27, 2011 by Kim Seward.