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This question set appears to deal with the weight of a puppy and the concept of calculating weight differences.

Here’s an analysis of each question:

1. **Context Summary**:
   - A puppy weighed **6.5 pounds** when it was first brought home.
   - **\( w \)** represents the current weight of the puppy.

2. **Question Breakdown**:
   - **(a)** If we substitute **\( w = 16.3 \)** into the expression **\( w - 6.5 \)**, what are we calculating?
      - This expression calculates the **increase in weight** since the puppy was brought home. By substituting \( w = 16.3 \), we get:
        \[
        16.3 - 6.5 = 9.8 \text{ pounds}
        \]
        So, the puppy gained **9.8 pounds**.

   - **(b)** If we substitute **\( w = 24.9 \)** into the expression **\( w - 6.5 \)**, what are we calculating?
      - This expression also calculates the weight gain, but with the puppy now at **24.9 pounds**:
        \[
        24.9 - 6.5 = 18.4 \text{ pounds}
        \]
        Thus, the puppy gained **18.4 pounds**.

   - **(c)** In general, what can we say the expression **\( w - 6.5 \)** represents?
      - This expression, **\( w - 6.5 \)**, represents the **total weight gain** of the puppy since it was first brought home, where **\( w \)** is the current weight.

Would you like further details on any of these calculations?

### Related Questions
1. How would the calculations change if the initial weight was different?
2. What if the puppy's weight decreased instead of increased? How would that affect the result?
3. How can this type of expression be used to calculate weight gain or loss over time?
4. What if \( w \) were a function of time, \( w(t) \)? How would \( w(t) - 6.5 \) then represent weight gain over time?
5. How would you express the percentage increase in weight from the initial weight?

**Tip**: In calculations like these, it's often helpful to interpret expressions as representing "changes" or "differences" from an initial state. This approach makes it easier to apply the same logic to a variety of problems.

1. Suppose that a puppy was 6.5 pounds when you brought it home from the breeder. Let w represent the weight of the puppy in pounds.
   a. If we substitute w = 16.3 into the expression w - 6.5, what are we calculating?
   b. If we substitute w = 24.9 into the expression w - 6.5, what are we calculating?
   c. In general, what can we say the expression w - 6.5 represents?

Learn how to calculate the weight gain of a puppy using algebraic expressions. This question guides you through substituting values into an expression to determine the increase in weight over time, suitable for students in Grades 6-8.

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