Learn with Amar
Teaching video
A teaching recording for this chapter has not been published yet. Start with the worked example below and explore the related animations where available.
Browse grades and teaching videos01 · Read and understand
What you will learn
- Define a variable and write a cost model.
- Solve a budget inequality while preserving its meaning.
- Check that the answer fits the context and the units.
Before you start
Multiplication, subtraction, and solving one-step equations.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
A fictional community workshop charges a $36 registration fee plus $8 per visit. With a $100 total budget, what is the greatest number of visits you can afford?
Why this math matters
Many everyday costs have a fixed part and a part that grows with use. A linear model makes the pattern visible and helps compare choices without guessing repeatedly. A budget is a limit, so an inequality often expresses the real question better than an equation.
| Visits | Total cost | Within $100? |
|---|---|---|
| 0 | $36 | Yes |
| 4 | $68 | Yes |
| 8 | $100 | Yes, exactly |
| 9 | $108 | No |
Set up the model
A useful answer starts with clear assumptions:
- Registration is paid once, and each visit costs exactly $8.
- The fictional prices include all charges; there are no taxes or additional fees in this example.
- Visits must be whole nonnegative numbers.
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.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
Build a budget with a linear equation
PausedQuestion: Start with the question. Paused.
Question
Start with the question
A fictional community workshop charges a $36 registration fee plus $8 per visit. With a $100 total budget, what is the greatest number of visits you can afford?
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Define the input and output
v = number of visits; C(v) = 36 + 8v dollars
The fixed $36 is the cost when v = 0. Each additional visit adds $8, so the rate of change is $8 per visit.
Represent the budget
36 + 8v ≤ 100
The total can equal $100 or be less than $100. The ≤ sign captures both possibilities.
Isolate the variable
8v ≤ 64; v ≤ 8
Subtract $36 from both sides, then divide by the positive price of $8 per visit. There is $64 left for visits.
Check the boundary
C(8) = 36 + 64 = $100; C(9) = 36 + 72 = $108
Eight visits fit exactly. Nine exceed the budget. Feasible counts are the whole numbers 0 through 8.
The result
You can afford at most 8 visits within the $100 budget.
If the budget were $99, the algebra would give v ≤ 7.875. Since a fraction of a visit is not sold in this model, the greatest affordable count would be 7. For an affordability limit, round a noninteger count down.
Common mistakes to catch
- Dividing the entire $100 by $8 ignores the fixed registration fee.
- Rounding 7.875 up to 8 would spend more than a $99 budget.
- The algebraic line accepts any real input, but the context permits only whole, nonnegative visit counts.
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.
Practice 1
A different workshop has a $24 fixed fee and a $6 per-visit price. How many visits fit within $78?
Show a hint
Write 24 + 6v ≤ 78 and solve for v.
Reveal answer and explanation
At most 9 visits
Subtracting 24 gives 6v ≤ 54, so v ≤ 9. Checking: 24 + 6(9) = 78 dollars.
Practice 2
For the original $36 plus $8-per-visit plan, how many visits fit within $95?
Show a hint
After subtracting $36, divide the remaining budget by $8. Apply the whole-visit restriction.
Reveal answer and explanation
At most 7 visits
(95 − 36)/8 = 7.375. Seven visits cost $92, leaving $3; eight cost $100 and exceed the budget.
Take the idea with you
Create a fictional cost model for printing, transport, or club membership. Identify the fixed fee, the cost per unit, and any restrictions before comparing totals.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: Work out a discount, then a tax
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