Question: Textbook prices have a seasonal structure on Ebay. At the end of a term, the supply of used books outstrips demand, and the price is lower. Near the start of a term, many students are looking for the textbook and the price is higher. Suppose we can classify sales for a particular chemistry textbook into these two time periods and an "other" time period in the proportions shown below. We have also listed the average price the textbook sells for in each of these three time periods
Sales proportion | Start-of-term | End-of-term | Other
Sales proportion | 0.45 | 0.31 | 0.24
Average price | $82.52 | $49.12 | $65.23
For example, about 45% of Ebay auctions for this chemistry textbook occur at the start of the term and the books sell for an average of $82.52 during this time. Using the information above, compute the average price of the textbook over all seasons. (Do not put a dollar sign in your answer.)
1. **State the problem:** We want to find the overall average price of the chemistry textbook over all seasons given the sales proportions and average prices for each season.
2. **Formula used:** The weighted average price is calculated by multiplying each season's average price by its sales proportion and then summing these products:
$$\text{Average price} = p_1 \times x_1 + p_2 \times x_2 + p_3 \times x_3$$
where $p_i$ are the sales proportions and $x_i$ are the average prices.
3. **Given data:**
- Start-of-term proportion $p_1 = 0.45$, average price $x_1 = 82.52$
- End-of-term proportion $p_2 = 0.31$, average price $x_2 = 49.12$
- Other proportion $p_3 = 0.24$, average price $x_3 = 65.23$
4. **Calculate each product:**
$$0.45 \times 82.52 = 37.134$$
$$0.31 \times 49.12 = 15.2272$$
$$0.24 \times 65.23 = 15.6552$$
5. **Sum the products to find the weighted average:**
$$37.134 + 15.2272 + 15.6552 = 68.0164$$
6. **Final answer:** The average price of the textbook over all seasons is approximately **68.02** (rounded to two decimal places).