Problem 55 - Applied Calculus

Question

School organizations raise money by selling candy door to door. When the price is \(\$ 1\) a school organization sells 2765 candies and when the price goes up to \(\$ 1.25\) the quantity of sold candy drops down to 2440 (a) Find the relative change in the price of candy. (b) Find the relative change in the quantity of candy sold. (c) Find and interpret the ratio \(\frac{\text { Relative change in quantity }}{\text { Relative change in price }}\)

Step-by-Step Solution

Answer

Relative change in price is 25%, relative change in quantity sold is approximately -11.74%, and the ratio is approximately -0.47.

Step 1: Calculate the Relative Change in Price

To find the relative change in price, we use the formula:\[\text{Relative Change in Price} = \frac{\text{New Price} - \text{Old Price}}{\text{Old Price}}\]Substituting the given values, the old price is \(1\) and the new price is \(1.25\):\[\text{Relative Change in Price} = \frac{1.25 - 1}{1} = 0.25\]This indicates a \(25\%\) increase in price.

Step 2: Calculate the Relative Change in Quantity Sold

To find the relative change in quantity sold, we use the similar formula:\[\text{Relative Change in Quantity} = \frac{\text{New Quantity} - \text{Old Quantity}}{\text{Old Quantity}}\]Substituting the given values, the old quantity is \(2765\) and the new quantity is \(2440\):\[\text{Relative Change in Quantity} = \frac{2440 - 2765}{2765} \approx -0.1174\]This represents approximately an \(-11.74\%\) decrease in quantity sold.

Step 3: Calculate and Interpret the Ratio of Relative Changes

Now we calculate the ratio of the relative change in quantity to the relative change in price:\[\frac{\text{Relative Change in Quantity}}{\text{Relative Change in Price}} = \frac{-0.1174}{0.25} \approx -0.4696\]This ratio \(-0.4696\) suggests that, for each \(1\%\) increase in price, the quantity sold decreases by approximately \(0.47\%\).

Key Concepts

Price Elasticity

Price elasticity is a key economic concept that measures how the quantity demanded of a good responds to changes in the price of that good. It's a vital tool for businesses and economists to understand consumer behavior. Price elasticity is calculated as the ratio of the percentage change in quantity demanded to the percentage change in price. In simpler terms, it shows how sensitive customers are to price changes.

When the price of candy increased from \(1.00 to \)1.25, the quantity sold decreased significantly. This indicates that the demand for candy is somewhat elastic, meaning that consumers' buying decisions are affected by price changes.

• If the absolute value of the elasticity is greater than one, demand is considered elastic; a price change will significantly affect the quantity demanded.
• If the elasticity is less than one, the demand is inelastic; quantity demanded is not as sensitive to price changes.
• An elasticity equal to one indicates unit elastic demand; a proportional change in quantity for a change in price.

Based on the exercise, the ratio \(-0.4696\) suggests that the demand for candy is relatively inelastic. Consumers do not drastically change their purchasing habit in response to a price increase.

Percentage Change

Understanding percentage change is crucial for evaluating how different factors influence each other. Calculating percentage change can help measure growth, decline, or any other transformations in numerical data over time.

In the context of the exercise, percentage change helps reveal the impact of a price increase on the sale of candies. For instance, the price of candy increased by \(25\%\), moving from \\(1.00\ to \\)1.25\. The formula for percentage change is:

\[ \text{Percentage Change} = \frac{\text{New Value} - \text{Old Value}}{\text{Old Value}} \times 100 \% \]

• For the price, the new value is \\(1.25\ and the old is \\)1.00\, which leads to \(25\%\) increase using the formula.
• For the quantity, this method shows an \(-11.74\%\) decrease as the quantity decreased from \2765\ (old) to \2440\ (new).

Understanding these calculations can easily help deduce how any variable change can significantly impact other variables linked through economic activities.

Quantity Sold Analysis

Analyzing the quantity sold involves assessing how different factors, like price changes, influence the number of goods sold. In the exercise, when analyzing candy sales, the quantity sold dropped from \2765\ to \2440\ when the price increased from \\(1.00\ to \\)1.25\. This change reflects a \(-11.74\%\) decline, calculated using the relative change formula for quantities.

Changes in quantity sold are often influenced by customers' price sensitivity. This understanding helps businesses make informed decisions regarding pricing strategies, inventory management, and overall market strategies.

• Understanding customer behavior towards price changes can help in forecasting demand.
• Analyzing past data on quantity sold can reveal patterns that are beneficial for setting prices competitively.
• Keen analysis on quantity sold data provides insights that inform promotional or discount strategies to maximize sales.

Insights from analyzing the quantity sold guide businesses to anticipate market responses and optimize for better financial outcomes.