4.4.2: Exercise 4.4

MAKING CONNECTIONS TO THE COLLABORATION

(1) Which of the following was one of the main mathematical ideas of the collaboration?

(i) Algebra is based on rules that have no connection to thinking about numbers or context.

(ii) Spreadsheets are the only useful application of algebra.

(iii) An increase of 23% can be calculated by multiplying the reference value by 0.23.

(iv) Having a good understanding of how to work with numbers is helpful in algebra.

DEVELOPING SKILLS AND UNDERSTANDING

(2) The following is an excerpt from an article about tuition increases at Rutgers University.¹⁰

After hearing the concerns of students, faculty and staff about rising tuition and salary freezes, the Board of Governors decided to raise in-state tuition for the upcoming fiscal year by 1.8 percent at its meeting yesterday in Winants Hall on the College Avenue campus …

Aside from tuition, mandatory fees for next term will rise 0.6 percent, and room and board will see a 3-percent rise. The overall working budget, originally set at about $2 billion … will be around $12 million less. Out-of-state tuition will rise 5 percent.

The excerpt contains information about the relative change in four different quantities. For each one, identify the quantity and write an expression that could be used to calculate the new value. Carefully define any variables that you use, including units. One is done for you as an example.

Example:

Quantity: In-state tuition in dollars

Expression: 1.018t

Variable: t = old in-state tuition in dollars

(a) Quantity:

Expression: 1.006f

Variable: f = old mandatory fees (in dollars)

(b) Quantity: Cost of room and board (in dollars)

Expression: 1.03r

Variable: r =

(c) Quantity: Out-of-state tuition (in dollars)

Expression:

Variable: T = out-of-state tuition (in dollars)

(d) In addition to the relative changes you listed above, the excerpt also gives information about the absolute change in the working budget. Which of the following is the best estimate for the relative change in the budget?

(i) Decrease of less than 1 percent.

(ii) Decrease of between 1 and 5 percent.

(iii) Decrease of about 10 percent.

(iv) Decrease of more than 10 percent.

(3) Read the following excerpt from an article about changes in gas rates in Charleston, West Virginia.¹¹

Mountaineer Gas wants to lower rates for customers for the second consecutive year.

Gas rates for residential customers will likely drop by 5.24 percent, starting
Nov. 1.

Mountaineer Gas filed for the rate reduction earlier this month, citing a decline in natural gas prices.

If the state Public Service Commission approves the lower rate, the average residential customer's monthly gas bill would drop by about $7.50 a month.

“Natural gas has been relatively cheaper for the last couple of years,” said Byron Harris, who heads the PSC’s Consumer Advocate Division. “Gas rates are definitely going in the right direction, while electricity rates keep going up.”

Mountaineer’s business customers will see their gas rates decline by about 6 percent, while industrial customer rates will drop 9.7 percent.

Suppose you are creating the spreadsheet below to track the gas rates.

(a) Enter each type of customer in Column A.

(b) Assume the original rates would be in Column B (you do not need to enter anything in
Column B). Write a spreadsheet formula for the new rates in Column C.

(c) Based on the information in the article, what was the average monthly gas bill for a residential customer before the rate change? Round to the nearest cent.

(4) One of your friends missed class, and you are helping her get caught up. Which of the following is a good explanation for the answer for 2.45x in Question 4 of Collaboration 4.4?

(i) If you convert 2.45 to a percent, you get 245%. So multiplying by 2.45 is an increase of 245%.

(ii) 2.45x increases x by 245% because x + 1.45x = 2.45x.

(iii) Think of 2.45x as adding 1x + 1.45x, where 1x is 100% of x and 1.45x is 145% of x. So you are adding 100% and 145%, which is an increase of 145% over the original 100%.

(iv) Think of 2.45x as a percent, so when you multiply x by 2.45%, you are actually decreasing x by 97.55%.

(5) The value of investments in the stock market changes daily. Suppose you buy a stock for $1,000. It increases in value by 4.5% and then falls 4.5%. What is the new value? Round to the nearest cent.

MAKING CONNECTIONS ACROSS THE COURSE

(6) In Preparation 4.4, you wrote formulas for Marissa’s budget spreadsheet. Her spreadsheet was set up to calculate the amount to be put into savings and then used to find the amount left for expenses. Write a new formula for Cell B2 in the spreadsheet shown below that will calculate the amount for expenses in one step. Marissa puts 5% of her take home pay into savings.

(7) Historically, women in the United States have earned less than men.¹² The table and graph below show national data on women’s earnings as a percentage of men’s earnings from 1992 to 2021.

Use the table and graph to create a linear equation that will model the data. Use the following
variables:

E = Women’s earnings as a percentage of men’s

t = Time in years after 1992

Use the following two points:

Point 1: Year 1992 , Percentage 75.8

Point 2: Year 2021 , Percentage 83.1

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¹⁰ https://dailytargum.com/article/2011/07/bog-agrees-on-1-8-percent-tuition-increase

¹¹ http://www.highbeam.com/doc/1P2-29398083.html

¹² https://www.bls.gov/opub/ted/2022/median-earnings-for-women-in-2021-were-83-1-percent-of-the-median-for-men.htm