Question

What is the method to determine whether a 3% or 0.03% rate of change occurred given an initial quantity and a change amount?

Answer 1

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:

1. **Identify the Initial Quantity and Change Amount**:
- Let \( P_0 \) be the initial quantity.
- Let \( \Delta P \) be the change amount.

2. **Calculate the Rate of Change**:
The rate of change can be calculated using the formula:

\[
\text{Rate of Change} = \frac{\Delta P}{P_0}
\]

3. **Convert the Rate of Change to a Percentage**:
To express the rate of change as a percentage, multiply the result by 100:

\[
\text{Percentage Rate of Change} = \left( \frac{\Delta P}{P_0} \right) \times 100
\]

4. **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.

1. **Calculate the Rate of Change**:
\[
\text{Rate of Change} = \frac{30}{1000} = 0.03
\]

2. **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.