4.3: Corequisite- Interest, Doubling Time
- Page ID
- 148602
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By the end of this lesson, you should understand that
- population growth can be measured in terms of doubling time.
- doubling times can be used to compare population growth during different periods.
- quantitative reasoning and math skills can be applied in various contexts.
- creditworthiness affects credit card interest rates and the amount paid by the consumer.
- reading quantitative information requires filtering out unimportant information (introductory level).
By the end of this lesson, you should be able to
- calculate quantities in the billions.
- use data to estimate a doubling time.
- compare and contrast population growth via population doubling times.
- recognize common mathematical concepts used in different contexts.
- identify a complete response to a prompt that asks for connections between mathematical concepts and a context.
- write a formula in a spreadsheet.
- convert percentages to decimals
- decide when simple interest is appropriate and how to calculate it
- use and interpret the terms “millions,” “billions,” and “trillions.”
PROBLEM SITUATION 1: MEASURING POPULATION GROWTH
In this lesson, you will look at doubling times to determine how the human population of the earth has changed over time. The doubling time of a population is the amount of time it takes a population to double in size. Calculating doubling time helps you understand how fast a population is growing. Comparing doubling times helps you understand how growth is changing over time.
The table below gives historical estimates of the human population. The population in 8,000 BCE was estimated to be 5 million people. Two thousand years later, in 6,000 BCE, the population had doubled to 10 million people. Therefore, the population doubling time for 8,000 BCE is about 2,000 years.
Population Estimates Throughout History5
| Year |
World Population |
Year |
World Population |
|
| 10,000 BCE | 1 | 1850 | 1,262 | |
| 9,000 BCE | 3 | 1900 | 1,650 | |
| 8,000 BCE | 5 | 1950 | 2,519 | |
| 7,000 BCE | 7 | 1960 | 2,982 | |
| 6,000 BCE | 10 | 1970 | 3,692 | |
| 5,000 BCE | 15 | 1980 | 4,435 | |
| 4,000 BCE | 20 | 1985 | 4,831 | |
| 3,000 BCE | 25 | 1990 | 5,263 | |
| 2,000 BCE | 35 | 1995 | 5,674 | |
| 1,000 BCE | 50 | 2000 | 6,070 | |
| 500 BCE | 100 | 2005 | 6,454 | |
| 1 CE | 200 | 2010 | 6,922 | |
| 1000 CE | 310 | 2015 | 7,339 | |
| 1800 CE | 978 | 2020 | 7,753 |
(1) Using the information in the table, estimate the doubling times of Earth’s human population. Enter your answers in the table below. Start with the year given below and estimate how long it took for the population in that year to double. The first entry is done for you. Be prepared to explain how you arrived at your answers.
| Year | Doubling Time | Year | Doubling Time |
| 8,000 BCE | 2,000 years | 1800 CE | (d) |
| 6,000 BCE | (a) | 1850 CE | (e) |
| 3,000 BCE | (b) | 1900 CE | (f) |
| 1 CE | (c) | 1970 CE | (g) |
(2) What do you notice about the doubling times? What does this tell you about how the human population has changed over time?
A Writing Tip
One of the skills you will learn in this course is how to write quantitative information. A writing principle that you will use throughout the course is given below and followed by Question 3, which offers examples of how to use this principle.
Writing Principle: Use specific and complete information. Readers should understand what you are trying to say even if they have not read the question or writing prompt. This includes
- information about context, and
- quantitative information.
(3) Which of the following statements best describes the change in doubling times before 1800 CE?
(a) The doubling times decreased.
(b) Before 1800 CE, estimated population doubling times decreased from 2,000 to 1,000.
(c) The doubling times decreased from 2,000 to 1,000.
(4) Write a statement that describes the change in doubling times after 1800 CE.
PROBLEM SITUATION 2: UNDERSTANDING CREDIT CARDS
When you use a credit card, you can pay off the amount you charge each month. If you do not pay the full amount, you are borrowing money from the credit card company. This is called credit card debt. Many people in the United States are concerned about the amount of credit card debt both for individuals and for society in general. In this lesson, you will use skills and ideas from previous lessons to think about some issues related to credit cards.
(5) The statement below came from a website that reported statistics for credit card debt in 2022:
- “American household debt hit a record $14.6 trillion in the spring of 2021, according to the Federal Reserve.”6
Does the statement “US households had $14,600 billion in debt in 2021,” express the same amount of debt? Justify your answer with an explanation.
Understanding the Fine Print
When you apply for a new credit card, you usually receive a disclosure in the mail along with the new credit card. The information below is excerpted from the disclosure that came with a new credit card. Use this information to answer Questions 2 and 3.
| Annual Percentage Rate (APR) for Purchases |
0.00% introductory APR for 6 months from the date of account opening. After that, your APR will be 10.99% to 23.99% based on your creditworthiness. This APR will vary with the market based on the Prime Rate. |
A credit score is a number assigned to a person that indicates to lenders his or her ability to repay a loan. A high credit score indicates good credit. In the following questions, you will explore how your credit score can affect how much you have to pay in interest for the purchases you make with your credit card.
(6) Juanita and Brian both have a credit card with the terms in the above disclosure form. They have both had their credit cards for more than six months.
(a) Juanita has good credit and gets the lowest interest rate possible for this card. She is not able to pay off her balance each month, so she pays interest. Estimate how much interest Juanita would pay in a year if she maintained an average balance of $5,000 each month on her card. Juanita is past the 6 month 0.0% introductory APR rate. Explain your estimation strategy.
(b) Brian has a very low credit score and has to pay the highest interest rate. He is not able to pay off his balance each month, so he pays interest. Calculate how much interest he would pay in a year if he maintained an average balance of $5,000 each month. Show your calculation. Round your answer to two decimal places.
(c) What are some things that might affect your credit score?
(7) The Annual Percentage Rate (APR) of a credit card is the interest rate for a full year. The periodic rate of a credit card is the interest rate for a month. You can find the periodic rate by dividing the APR by the number of months in a year.
(a) What is the periodic rate for Juanita’s card? Round to two decimal places.
(b) Juanita has a balance of $1,082 on her January credit card statement. The statement says the minimum payment required in January is $100. Juanita makes only the minimum payment, so the rest of the balance carries over to February. Which of the following is the best estimate of how much interest she will owe in February?
(i) Less than a dollar
(ii) $5-$10
(iii) $10-$20
(iv) More than $20
Credit Cards and Cash Advances
You will use the following information from the credit card disclosure for Question 8. A cash advance is when you use your credit card to get cash instead of using it to make a purchase.
| Annual Percentage Rate (APR) for Purchases | After that, your APR will be 10.99% to 23.99% based on your creditworthiness. This APR will vary with the market based on the Prime Rate. |
| APR for Cash Advances | 28.99%. This APR will vary with the market based on the Prime Rate. |
(8) Discuss each statement below and decide if it is reasonable.
(a) Jeff pays the highest interest rate for purchases he makes with his credit card. For a cash advance, he would pay $0.05 more for each dollar he receives in cash than he pays for purchases he makes with his credit card.
(b) The APR for cash advances is about two-and-a-half times as much as the lowest APR for purchases.
(9) Brian used a spreadsheet to record his credit card charges for a month.
Brian used the following expression to calculate his interest for that month’s charges.
\[\dfrac{0.2399}{12}(B2 + B3 + B4 + B5) \nonumber\]
Which of the following statements best explains what the expression means in terms of the context?
(i) Brian added his individual charges. Then he divided 0.2399 by 12. Then he multiplied the two numbers.
(ii) Brian found the interest charge for the month by dividing 0.2399 by 12 and multiplying it by the sum of Column B.
(iii) Brian added the individual charges to get the total amount charged to the credit card. He found the periodic rate by dividing the APR by 12 months and multiplied the rate by the total charges. This gave the interest charge for the month.
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5 https://www.census.gov/population/international/data/worldpop/table_history.php