Problem 117 - Elementary and Intermediate Algebra

Question

To determine the average afternoon wait time in security lines at an airport, officials monitored four passengers, each at a different gate. The time that each passenger entered a security line and the time the same passenger cleared the checkpoint was recorded, as shown below. Find the average (mean) wait time for these passengers. $$ \begin{array}{|l|c|c|} \hline & {\text { Time }} & {\text { Time }} \\ \hline & {\text { entered }} & {\text { cleared }} \\ \hline \text { Passenger at Gate A } & {3: 05 \text { pm }} & {3: 21 \text { pm }} \\ \hline \text { Passenger at Gate B } & {3: 03 \text { pm }} & {3: 13 \text { pm }} \\ \hline \text { Passenger at Gate C } & {3: 01 \text { pm }} & {3: 09 \text { pm }} \\ \hline \text { Passenger at Gate D } & {3: 02 \text { pm }} & {3: 16 \text { pm }} \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The average wait time is 12 minutes.

Step 1: Calculate Wait Times

First, determine the wait time for each passenger by subtracting the time they entered the line from the time they cleared the checkpoint. - Gate A: 3:21 pm - 3:05 pm = 16 minutes - Gate B: 3:13 pm - 3:03 pm = 10 minutes - Gate C: 3:09 pm - 3:01 pm = 8 minutes - Gate D: 3:16 pm - 3:02 pm = 14 minutes

Step 2: Add Wait Times

Next, add up all the individual wait times for the passengers. - Total wait time = 16 + 10 + 8 + 14 = 48 minutes.

Step 3: Calculate the Average

Divide the total wait time by the number of passengers to get the average wait time.- Average wait time = \( \frac{48}{4} \) = 12 minutes.

Key Concepts

Subtracting Time

When dealing with time, subtracting one time from another is often necessary to find the duration between events. For example, if you know when a passenger entered a security line and when they cleared the checkpoint, you can calculate their wait time by subtracting the earlier time from the later time.

To do this, make sure both times are in the same format, such as 24-hour or AM/PM, and align them up properly. Then, work through the time starting from minutes and then to hours, similar to how you would handle carrying over in subtraction problems.

• Subtract the minutes. If the result is negative, borrow 1 hour from the hours portion and add 60 minutes to the current total.
• Subtract the hours.

Step through each of these carefully to avoid mistakes, especially when crossing between hours and minutes.

Division

Division is a mathematical operation where you take two numbers and determine how many times one number, the divisor, goes into another number, the dividend. When you divide, you essentially distribute the total into equal parts. This operation is essential in finding averages.

In our case, after finding the total waiting time for four passengers, you divide this sum by the number of passengers to get the average wait time.

• Identify the total: Here, the total wait time is 48 minutes.
• Identify the divisor: The number of contributing elements—in this case, 4 passengers.
• Divide: Use the formula \[ \text{Average} = \frac{\text{Total Wait Time}}{\text{Number of Passengers}} \]

Simple division problems such as this often involve whole numbers and yield neat, rounded results. However, when dealing with real-world data, division could lead to non-exact results, requiring further rounding.

Arithmetic Mean

The arithmetic mean, commonly referred to as the average, is a central value of a data set. To find the arithmetic mean, you sum up all data points and then divide by the number of data points.

The concept of the arithmetic mean is used frequently when you need to summarize a set of values with a single average value.

• Sum all the values together: First, collect the wait times, adding them to get a total wait time.
• Count the number of values: Determine how many data points, or in our case, passengers, contributed to the total wait time.
• Divide: Use the sum and the count to calculate the mean by division.

The arithmetic mean is an effective measure to understand the central tendency of time or any other data set, providing a simple but powerful summary of overall trends.