Problem 14 - Algebra Form and Function

Question

Catherine has the following phone bills over a twelvemonth period: \(\$ 32, \$ 27, \$ 20, \$ 40, \$ 33, \$ 20, \$ 32, \$ 30,\) \(\$ 36, \$ 31, \$ 37, \$ 22\) (a) What is the average phone bill? (b) Suppose Catherine spends \(\$ 5\) more on phone bills each month. What happens to her average phone bill? What if she spends \(\$ 10\) more each month? (c) Suppose she spends \(\$ 60\) more on the highest phone bill, but the same amount on the other 11 bills. What happens to her average phone bill? What if she spends \(\$ 120\) more on the highest bill?

Step-by-Step Solution

Answer

Answer: (a) The average phone bill is $30. (b) If Catherine spends $5 more each month, her average phone bill becomes $35. If she spends $10 more each month, her average phone bill becomes $40. (c) If Catherine spends $60 more on her highest phone bill, her average phone bill becomes $35. If she spends $120 more on her highest phone bill, her average phone bill becomes $40.

Step 1: Calculate the average phone bill for 12 months

To find the average of the given set of phone bills, we need to sum up the bills and divide by the number of bills (12). Mean (average) \(=\frac{32+27+20+40+33+20+32+30+36+31+37+22}{12}\)

Step 2: Calculate the sum and the mean value

Now we calculate the total sum of the phone bills and divide by 12 to get the average. Total sum \(= 32+27+20+40+33+20+32+30+36+31+37+22 = 360\) Mean (average) \(= \frac{360}{12} = 30\) The average phone bill over 12 months is \(30\). (a) Answer: The average phone bill is \(\$ 30\).

Step 3: Calculate the new average when Catherine spends \(\$ 5\) and \( \$ 10\) more on her phone bills each month.

Catherine spends extra \(\$ 5\) or \(\$ 10\) on phone bills each month, so we need to add that amount to each bill and find the new average for each scenario.

Step 4: Scenario 1: (\(\$ 5\) more each month)

New Total sum \(= 360 + (5 * 12) = 420\) Mean (average) \(= \frac{420}{12} = 35\)

Step 5: Scenario 2: (\(\$ 10\) more each month)

New Total sum \(= 360 + (10 * 12) = 480\) Mean (average) \(= \frac{480}{12} = 40\) (b) Answer: If Catherine spends \(\$ 5\) more each month, her average phone bill becomes \( \$ 35\). If she spends \(\$ 10\) more each month, her average phone bill becomes \( \$ 40\).

Step 4: Calculate the new average when Catherine spends \(\$ 60\) and \(\$ 120\) more on her highest phone bill

In this step, we need to find the new average of the phone bills after adding \(\$ 60\) and \(\$ 120\) to her highest phone bill (which is \(\$ 40\)).

Step 7: Scenario 1: (\(\$ 60\) more on the highest phone bill)

Highest phone bill becomes (\(40 + 60 = 100\)) New Total sum \(= 360 - 40 + 100 = 420\) Mean (average) \(= \frac{420}{12} = 35\)

Step 8: Scenario 2: (\(\$ 120\) more on the highest phone bill)

Highest phone bill becomes (\(40 + 120 = 160\)) New Total sum \(= 360 - 40 + 160 = 480\) Mean (average) \(= \frac{480}{12} = 40\) (c) Answer: If Catherine spends \(\$ 60\) more on her highest phone bill, her average phone bill becomes \( \$ 35\). If she spends \(\$ 120\) more on her highest phone bill, her average phone bill becomes \( \$ 40\).

Key Concepts

Mean Calculation

The mean, or average, is a common statistical measure to find a central tendency in a set of numbers. To calculate the mean of Catherine's phone bills over 12 months, add up all the monthly bill amounts and then divide by the total number of months.
For Catherine, her bills were \(32, 27, 20, 40, 33, 20, 32, 30, 36, 31, 37, 22\). First, sum these numbers:

• \(32 + 27 + 20 + 40 + 33 + 20 + 32 + 30 + 36 + 31 + 37 + 22 = 360\)

Then divide this total sum by the 12 months:

• Mean (average) \(= \frac{360}{12} = 30\)

Thus, the average monthly phone bill for Catherine is \(\$30\). Calculating the mean helps understand the typical monthly expenditure Catherine incurs on her phone bills.

Impact of Changes on Average

Changes in individual bill amounts affect the average. These can be consistent changes across all items (e.g., monthly bills) or significant changes in a single item.Consider if Catherine spends \\(5 more each month. Add \\)5 to each of the 12 monthly bills. This increases the sum by \(5 \times 12 = 60\), resulting in a new total sum of \(420\). The new mean becomes:

• \(\frac{420}{12} = 35\)

If the increase is \\(10 each month, the sum increases by \(10 \times 12 = 120\), giving:

• \(\frac{480}{12} = 40\)

These examples show how regular adjustments across a set affect the average.
Now, consider a one-time increase, like adding \\)60 to the highest bill (\\(40 becomes \\)100):

• New sum: \(360 - 40 + 100 = 420\)
• Resulting mean: \(\frac{420}{12} = 35\)

These examples reveal how both constant and single item variations can adjust averages compared to their initial calculations.

Monthly Spending Analysis

Analyzing monthly spending provides insights into budgeting and financial adjustments. Knowing the average phone bill allows Catherine to assess her spending trends and prepare for potential increases.
When changes occur, like spending \\(5 or \\)10 more per month, the average reflects these adjustments, offering foresight into how consistent incremental changes might accumulate over time. Such an analysis helps in managing monthly budgets and preparing for future financial demands.When Catherine considers making a one-time higher payment, her spending analysis focuses on one bill. For example, adding \\(60 or \\)120 more on the highest bill causes less predictable budget changes.Taking such scenarios into account informs Catherine whether her uptrend in bills is temporary or part of a larger pattern. Comparing the average stats makes Catherine's budgeting resilient by foreseeing larger expenditure patterns and adapting to them with balance.