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Sure, let’s go through the step-by-step solution to find the Price Elasticity of Demand.
### Step 1: Calculate the Percentage Change in Quantity
First, we need to determine the initial and final quantities. Here, the initial quantity ([tex]\(Q_i\)[/tex]) is 3000 units and the final quantity ([tex]\(Q_f\)[/tex]) is 2800 units.
[tex]\[ \text{\% change in quantity} = \frac{Q_f - Q_i}{\frac{Q_i + Q_f}{2}} \times 100 \][/tex]
Substitute the given values:
[tex]\[ \text{\% change in quantity} = \frac{2800 - 3000}{\frac{3000 + 2800}{2}} \times 100 \][/tex]
Calculate the average quantity:
[tex]\[ \frac{3000 + 2800}{2} = 2900 \][/tex]
Calculate the change in quantity:
[tex]\[ 2800 - 3000 = -200 \][/tex]
Now, plug these values back into the formula:
[tex]\[ \text{\% change in quantity} = \frac{-200}{2900} \times 100 \][/tex]
Simplify the fraction:
[tex]\[ \frac{-200}{2900} \approx -0.06896551724137931 \][/tex]
Convert to a percentage:
[tex]\[ -0.06896551724137931 \times 100 \approx -6.896551724137931 \][/tex]
### Step 2: Calculate the Percentage Change in Price
Next, determine the initial and final prices. Here, the initial price ([tex]\(P_i\)[/tex]) is [tex]$70 and the final price (\(P_f\)) is $[/tex]60.
[tex]\[ \text{\% change in price} = \frac{P_f - P_i}{\frac{P_i + P_f}{2}} \times 100 \][/tex]
Substitute the given values:
[tex]\[ \text{\% change in price} = \frac{60 - 70}{\frac{70 + 60}{2}} \times 100 \][/tex]
Calculate the average price:
[tex]\[ \frac{70 + 60}{2} = 65 \][/tex]
Calculate the change in price:
[tex]\[ 60 - 70 = -10 \][/tex]
Now, plug these values back into the formula:
[tex]\[ \text{\% change in price} = \frac{-10}{65} \times 100 \][/tex]
Simplify the fraction:
[tex]\[ \frac{-10}{65} \approx -0.15384615384615385 \][/tex]
Convert to a percentage:
[tex]\[ -0.15384615384615385 \times 100 \approx -15.384615384615385 \][/tex]
### Step 3: Calculate the Price Elasticity of Demand
Finally, the Price Elasticity of Demand (PED) is calculated as the percentage change in quantity divided by the percentage change in price:
[tex]\[ \text{Price Elasticity of Demand} = \frac{\text{\% change in quantity}}{\text{\% change in price}} \][/tex]
Substitute the percentages we found:
[tex]\[ \text{Price Elasticity of Demand} = \frac{-6.896551724137931}{-15.384615384615385} \][/tex]
Simplify the division:
[tex]\[ \text{Price Elasticity of Demand} \approx 0.44827586206896547 \][/tex]
### Conclusion
So, the calculations yield the following results:
- Percentage change in quantity: [tex]\(-6.90\%\)[/tex]
- Percentage change in price: [tex]\(-15.38\%\)[/tex]
- Price Elasticity of Demand: [tex]\(0.45\)[/tex]
These values indicate that the demand is inelastic since the absolute value of the Price Elasticity of Demand is less than 1.
### Step 1: Calculate the Percentage Change in Quantity
First, we need to determine the initial and final quantities. Here, the initial quantity ([tex]\(Q_i\)[/tex]) is 3000 units and the final quantity ([tex]\(Q_f\)[/tex]) is 2800 units.
[tex]\[ \text{\% change in quantity} = \frac{Q_f - Q_i}{\frac{Q_i + Q_f}{2}} \times 100 \][/tex]
Substitute the given values:
[tex]\[ \text{\% change in quantity} = \frac{2800 - 3000}{\frac{3000 + 2800}{2}} \times 100 \][/tex]
Calculate the average quantity:
[tex]\[ \frac{3000 + 2800}{2} = 2900 \][/tex]
Calculate the change in quantity:
[tex]\[ 2800 - 3000 = -200 \][/tex]
Now, plug these values back into the formula:
[tex]\[ \text{\% change in quantity} = \frac{-200}{2900} \times 100 \][/tex]
Simplify the fraction:
[tex]\[ \frac{-200}{2900} \approx -0.06896551724137931 \][/tex]
Convert to a percentage:
[tex]\[ -0.06896551724137931 \times 100 \approx -6.896551724137931 \][/tex]
### Step 2: Calculate the Percentage Change in Price
Next, determine the initial and final prices. Here, the initial price ([tex]\(P_i\)[/tex]) is [tex]$70 and the final price (\(P_f\)) is $[/tex]60.
[tex]\[ \text{\% change in price} = \frac{P_f - P_i}{\frac{P_i + P_f}{2}} \times 100 \][/tex]
Substitute the given values:
[tex]\[ \text{\% change in price} = \frac{60 - 70}{\frac{70 + 60}{2}} \times 100 \][/tex]
Calculate the average price:
[tex]\[ \frac{70 + 60}{2} = 65 \][/tex]
Calculate the change in price:
[tex]\[ 60 - 70 = -10 \][/tex]
Now, plug these values back into the formula:
[tex]\[ \text{\% change in price} = \frac{-10}{65} \times 100 \][/tex]
Simplify the fraction:
[tex]\[ \frac{-10}{65} \approx -0.15384615384615385 \][/tex]
Convert to a percentage:
[tex]\[ -0.15384615384615385 \times 100 \approx -15.384615384615385 \][/tex]
### Step 3: Calculate the Price Elasticity of Demand
Finally, the Price Elasticity of Demand (PED) is calculated as the percentage change in quantity divided by the percentage change in price:
[tex]\[ \text{Price Elasticity of Demand} = \frac{\text{\% change in quantity}}{\text{\% change in price}} \][/tex]
Substitute the percentages we found:
[tex]\[ \text{Price Elasticity of Demand} = \frac{-6.896551724137931}{-15.384615384615385} \][/tex]
Simplify the division:
[tex]\[ \text{Price Elasticity of Demand} \approx 0.44827586206896547 \][/tex]
### Conclusion
So, the calculations yield the following results:
- Percentage change in quantity: [tex]\(-6.90\%\)[/tex]
- Percentage change in price: [tex]\(-15.38\%\)[/tex]
- Price Elasticity of Demand: [tex]\(0.45\)[/tex]
These values indicate that the demand is inelastic since the absolute value of the Price Elasticity of Demand is less than 1.
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