To determine whether a 3% or 0.03% rate of change occurred given an initial quantity and a change amount, you can use the following method:
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Identify the Initial Quantity and Change Amount:
- Let \( P_0 \) be the initial quantity.
- Let \( \Delta P \) be the change amount.
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Calculate the Rate of Change: The rate of change can be calculated using the formula:
\[ \text{Rate of Change} = \frac{\Delta P}{P_0} \]
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Convert the Rate of Change to a Percentage: To express the rate of change as a percentage, multiply the result by 100:
\[ \text{Percentage Rate of Change} = \left( \frac{\Delta P}{P_0} \right) \times 100 \]
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Compare the Result:
- If the result equals 3%, then a 3% rate of change occurred.
- If the result equals 0.03%, then a 0.03% rate of change occurred.
Example
Suppose the initial quantity \( P_0 \) is 1000 and the change amount \( \Delta P \) is 30.
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Calculate the Rate of Change: \[ \text{Rate of Change} = \frac{30}{1000} = 0.03 \]
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Convert to Percentage: \[ \text{Percentage Rate of Change} = 0.03 \times 100 = 3% \]
In this case, the rate of change is 3%, not 0.03%.
Conclusion
To determine whether a change represents a 3% or a 0.03% rate of change, calculate the change as a proportion of the initial quantity and convert it to a percentage. Then, compare against the two percentage values.