Question

During her eight-hour shift on Wednesday, the salesperson sold items that had an average price of \(\$ 8.10\). To the nearest tenth of an item, what is the number of items she sold per hour on Wednesday?

Short answer

The salesperson sold \(1\) item per hour during her eight-hour shift on Wednesday.

Step-by-step solution

Determine the total sales during her shift.

Since we know that the average price per item was \(\$8.10\) and she sold items over an eight-hour shift, we can determine her total sales by multiplying the average price by the number of hours she worked. We will call the total number of sold items x: Total sales= \(\$ 8.10 * 8\) hours

Calculate the total number of items sold.

We'll use the formula for calculating an average, which is total cost/average price: x = Total sales / Average price x = \((8.10 * 8) / 8.10\) x = \(8 * 1\) x = \(8\)

Find the number of items she sold per hour.

Now that we have found the total number of items sold during her 8-hour shift, we can find out the number of items she sold per hour by dividing this number by 8: Number of items sold per hour = x / 8 Number of items sold per hour = \(8 / 8\) Number of items sold per hour = \(1\) So, the salesperson sold\(\bold{1}\) item per hour during her eight-hour shift on Wednesday.

Key concepts

Average Price Calculation

Understanding how to calculate the average price is pivotal in solving various practical problems, especially those related to sales and finance. The average price is simply computed by dividing the total cost of items by the number of items. In the case of our salesperson's shift problem, the calculation involved the average price per item, which is given as \( \(8.10 \). When you come across such a problem, it's important to note that the average price provides an central value for the price of items, which implies that not all items need to be exactly priced at \( \)8.10 \).

To contextualize, suppose you visit a store where twenty pens are sold at \( \(1 \) each, and ten notebooks at \( \)2 \) each, making for thirty items in total. The total cost would be \( \(40 \) (\( \)20 \) from pens and \( \(20 \) from notebooks). The average price per item would be the total cost divided by the number of items, which would yield approximately \( \)1.33 \) per item. This exercise of calculating average prices is a fundamental aspect of handling arithmetic word problems that involve sales and transactions.

Salesperson Shift Problem

The salesperson shift problem usually involves determining how many items were sold in a certain period or deducing the revenue generated within a given time frame. In this scenario, we focus on pinpointing the number of items a salesperson sells per hour. These problems help illustrate how arithmetic word problems apply in real-world scenarios. To solve such problems, the relationship between the total number of items sold and the hours worked should be clear.

For instance, if a salesperson sells \(40\) items in an \(8\)-hour shift, to find the average, you would divide \(40\) by \(8\), which results in \(5\) items per hour. This problem required the understanding that if the total number of items sold and the average price are known, as well as the duration of the shift, the calculation becomes straightforward: simply divide the number of items by the hours worked to determine the average rate per hour.

Arithmetic Word Problems

Arithmetic word problems are an important part of mathematical education as they bridge the gap between abstract mathematical concepts and real-world applications. They require the student to translate the text into mathematical language and use mathematical operations to find a solution. To tackle such problems effectively, identifying keywords and the relevant numerical information is crucial.

When faced with arithmetic word problems like the one presented here, it's advisable to break them down step by step. Look for clues such as 'average price', 'shift duration', or 'total sales', which help to form the equations needed for the solution. Then use basic arithmetic operations like addition, subtraction, multiplication, and division to reach the answer. Remember to consider units of measure, as these can often provide hints about the operations needed. In our textbook problem, 'per hour' suggested a division of the total items by the hours worked, which ultimately led to the correct solution.