Opening the book…
LEARNING GOALS
By the end of this lesson, you should understand
- the relationship between decimals and percents.
By the end of this lesson, you should be able to
- convert between decimals and percents.
- perform arithmetic operations with percents.
FUNDAMENTALS OF THE LESSON
"Percent" comes from the latin Per Centum. The latin word Centum means 100. For example, a century is 100 years. A penny (a cent) is part of a dollar since there are 100 cents in one dollar. It is a portion of the whole amount.
40 percent, or 40%, means ‘40 out of 100’. 40% of the whole means 40 out of each hundred contained in the whole. Since 40% can be understood as 40 out of one hundred, it can also be represented as a fraction as well as a decimal. For example:
\[40\% = \dfrac{40}{100} = 0.40\nonumber \]
If 40 is divided by 100, the decimal equivalent of 0.40 is obtained. Likewise,
\[15\% = \dfrac{15}{100} = 0.15\;or\;2\% = \dfrac{2}{100} = 0.02\nonumber \]
\[100\% = \dfrac{100}{100} = 1.00\nonumber \]
Notice this is equal to 1 as 100% indicates the whole.
Percentages can be greater than 100%. For example:
\[110\% = \dfrac{110}{100} = 1.10\nonumber \]
110% is more than 1. 100% is the whole, so 110% is actually more than the whole. A percent can be interpreted as a ratio of a portion and a whole. Whenever the portion is greater than the whole, the percentage will be greater than 100%.
This activity will cover three basic operations involving percents: addition, subtraction, and multiplication.
Addition or subtraction happens when two portions of the whole are added or subtracted together. For example, if you are holding two quarters (each quarter is 25 cents, or 25% of a dollar), what percentage of a dollar do you have? To calculate the percentage of a dollar that you are holding, you can add 25% and 25% to get 50% of a dollar, which is 50 cents.
Problems may involve multiplying percentages. This occurs when you are taking a percent of a percent. For example, you know that 50% of the time you lose in the game “rock, paper, scissors”. You also know that 30% of the games that you lose, you are holding out “rock”. You can use this information to determine the percentage of all games that you lose AND are holding out “rock”. This can be calculated by multiplying 50% x 30%. This is 0.50 x 0.30 = 0.15 = 15%. So, 15% of the time that you play the game you lose and are holding out “rock”.
1. Let’s begin by practicing converting percentages to decimals.
A. Write 20% as a decimal.
B. Write 5% as a decimal.
C. Write 250% as a decimal.
D. Will 95% be greater than 1 or less than 1, and why?
NEXT STEPS
Working with percents
2. Saranne, Antonio, and Kera are work-study students in the admissions office of their college. One of their main tasks is filing new student information.
A. On a certain workday, Saranne files 50% of the information while Antonio files 50%. What percentage of the student information has been filed?
B. Antonio files 35% of the documents and his two associates each file 10%. What percent has NOT been filed yet?
C. While filing student information, Kera notices a note that states that students are charged a 2% late fee for tuition if a payment arrives late but within 10 days of the due date. That is, 2% of the amount due is added to the account if the payment is late but arrives within 10 days of the due date. If a payment is more than 10 days late, the late fee percentage is tripled. Suppose a student sends in their tuition payment 15 days late. What percentage of the tuition payment will be charged as a late fee?
D. Kyle submitted his tuition payment 18 days late. His tuition payment was $978. What late fee will Kyle be charged?
3. Harper wants to raise money for breast cancer awareness. Her goal is to raise $10,000 from her community. She works so hard she ends up raising $11,000. Her friends tell her that she raised 110% of her goal. Is her friend right? If so, what does that mean? Hint: What is her fundraising goal? What is 110% of her goal?
4. 70% of students who apply to Upstate New York Community College are originally from New York State. Of these New York State applicants, 30% are from Madison County. What percent of students who apply to Upstate New York Community College are from Madison County? Write the answer as a percentage. Hint: Do we add percents, or find a percent of a percent to solve this problem?
5. Approximately 2% of the applicants who apply to Upstate New York Community College are international students. Of the international applicants, about 75% of them are from Japan or China. What percentage of all of the students are from Japan or China?
6. Antonio makes quarterly payments to his insurance company and is charged a 2% late fee if his payment is received after the due date. If his payment is $558, how much will he be charged if his payment is late?
FURTHER APPLICATIONS
7. Kera is a college-student who lives at home with her parents. Her parents are generous and pay for most of her expenses so that she can save her earnings from her work-study job. The bank closest to both her home and college is Simplicity Bank. Banks calculate the interest earned on a savings account in a fairly complicated way. Interest is the amount of money the bank pays the customer for investing their money with the bank. The annualized percentage yield (APY) describes how much money a savings deposit earns in interest in one year. If the APY applied to Kera’s account at Simplicity Bank is 1.05%, then the amount of interest Kera will earn in a year is 1.05% of the amount of money she deposited. Kera’s account balance after one year can be found by adding the interest she earned to the initial amount she deposited.
A. If Kera only invests $100 in Simplicity Bank over an entire year, how much money should she expect to have in her account after 1 year?
B. Kera would like to earn more interest and decides that she needs to save and invest more money. What would her account balance be after one year if she invested $1000?
C. How much would she have if she invested $5000 for one year?
Questions: Percents–Operations
1. There are several types of colleges and universities in the United States. Colleges can be public or private, and can be for-profit or non-profit. The table below displays the percentages of recent high school graduates from a large city who were enrolled in college in 2012 and 2016, and the type of college the graduates were enrolled in.
|
2012 |
2016 |
|
|
Private 4-year non-profit |
16.9 |
17.8 |
|
Public 4-year |
38 |
35.9 |
|
Public 2-year |
35.5 |
40.1 |
A. In 2012, what type of college were most recent high school graduates enrolled in?
B. Notice the percentages for 2012 do not add to 100%. What percentage of recent graduates in 2012 are not represented in this data?
C. There were 15,492 recent graduates in the city’s 2012 class. How many of them enrolled in a private 4-year non-profit college? Round to the nearest student.
D. Which type of college had the greatest increase in percent of enrollment from 2012 to 2016?
E. What is the increase in the percentage of enrollment in public 2-year colleges from 2012 to 2016? Write the increase in terms of percentage points.
2. The table below contains nutrition facts for Almond Milk and 1% Lowfat Milk. The percent daily values of a nutrient indicates the percentage of a daily 2000-calorie diet which a 1-cup serving of each product covers. For example, 1 cup of Almond Milk covers 10% of the amount of Vitamin A a person should consume in one day.
A. Per 1 cup serving, which product provides a greater percentage of calcium? How much more?
B.If you had 5 servings of Almond Milk in one day, would you be meeting or exceeding your percent daily value of calcium for a 2,000 calorie diet?
C. What percentage of your total calories would you have met for a 2,000 calorie diet if you had 5 servings of Almond Milk?
D. If you had 5 servings of 1% Lowfat Milk in one day, would you be meeting or exceeding your percent daily value of calcium for a 2,000 calorie diet?
E. What percentage of your total calories would you have met for a 2,000 calorie diet if you had 5 servings of 1% Lowfat Milk?
This page titled 5.16: Fundamentals 15 - Percents–Operations was last modified on Wed, 15 May 2024 07:38:56 GMT and is shared under a CC BY-NC 4.0 license and was authored, remixed, and/or curated by Carnegie Math Pathways (WestEd) .
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