Detect a fixed fee in a table

Set up the model

A useful answer starts with clear assumptions:

• The stated model is a fixed fee plus a constant unit price.
• All example prices include every charge.

02 · Work through the example

Follow the reasoning, one step at a time.

Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.

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Detect a fixed fee in a table

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Question

Start with the question

A fictional printer charges $7, $9, and $11 for 1, 2, and 3 posters. Assuming a fixed fee plus a constant per-poster price, find the cost rule and decide whether it is proportional.

Before you calculate

Read what is known and what you need to find. Make a prediction before moving to the first calculation.

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1. 1

   Represent the quantities

   unit price = 9 − 7 = 11 − 9 = $2

   Equal input increases of one poster cause equal output increases of two dollars.

2. 2

   Apply the relationship

   fixed fee = 7 − 2(1) = $5; C(n) = 5 + 2n

   Remove the one-poster variable charge to recover the initial cost.

3. 3

   Check and interpret

   C(0) = $5, so the rule is not proportional

   A proportional rule starts at zero, even though this rule has a constant rate of change.

Common mistakes to catch

• Equal increases alone establish neither a zero intercept nor proportionality.
• The first listed cost is not necessarily the starting fee.

03 · Practice independently

Try it before revealing the answer.

Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.