At a price of $217 per crate, the demand for plumbuses is 5315 crates. At a price of $206 per crate, the demand for plumbuses is 6378 crates.

Break Down the Problem

1. Identify the two price-demand pairs:

   • Price 1: $P_1 = 217$, Demand 1: $Q_1 = 5315$
   • Price 2: $P_2 = 206$, Demand 2: $Q_2 = 6378$

2. Determine the change in price and demand:

   • Change in price: $\Delta P = P_2 - P_1$
   • Change in demand: $\Delta Q = Q_2 - Q_1$

Relevant Concepts

1. Calculate the price elasticity of demand (PED) using the formula: $PED = \frac{\Delta Q / Q_{avg}}{\Delta P / P_{avg}}$ where:
   • $Q_{avg} = \frac{Q_1 + Q_2}{2}$
   • $P_{avg} = \frac{P_1 + P_2}{2}$

Analysis and Detail

1. Calculate $\Delta P$ and $\Delta Q$:

   • $\Delta P = 206 - 217 = -11$
   • $\Delta Q = 6378 - 5315 = 1063$

2. Calculate $Q_{avg}$ and $P_{avg}$:

   • $Q_{avg} = \frac{5315 + 6378}{2} = \frac{11693}{2} = 5846.5$
   • $P_{avg} = \frac{217 + 206}{2} = \frac{423}{2} = 211.5$

3. Calculate PED: $PED = \frac{1063 / 5846.5}{-11 / 211.5}$

Verify and Summarize

1. Calculate the components:

   • $\frac{1063}{5846.5} \approx 0.1817$
   • $\frac{-11}{211.5} \approx -0.0519$

2. So, $PED \approx \frac{0.1817}{-0.0519} \approx -3.50$

Final Answer

The price elasticity of demand (PED) for plumbuses is approximately $-3.50$, indicating that the demand is elastic; a change in price leads to a proportionally larger change in demand.