A reliable method
Translate at most as an upper limit and at least as a lower limit. Solve the real-valued inequality first, then decide which integer values the context permits.
The common trap
Rounding a maximum purchase count to the nearest integer can exceed the budget. Round down for an upper bound, and check the original total.
Check and transfer
Verify both the proposed boundary and the next integer beyond it. This shows why a maximum is feasible and why a larger choice is not.
Worked comparison: why the alternative fails
A $20 budget buys items costing $6 each after a $3 fee. (20−3)/6 = 17/6, so the maximum is 2, not 3. Two cost $15; three cost $21. Checking the next integer proves the proposed maximum.
Transfer and explain
To earn at least $20 at $6 per whole hour, 3 hours give only $18 and 4 give $24. The minimum therefore rounds up; the direction comes from the constraint.
With a $59 budget, a club pays a $12 setup fee and $9 per badge. What is the greatest whole number of badges it can buy?
- Subtract the fixed setup fee before allocating money to badges.
- The whole-number maximum is the floor of the remaining budget divided by 9.
- The requested value is 5. Check its units and meaning against the original question.
Compare the other answer choices
Treat the setup fee as a fee for every badge and divide by 12 + 9.
Subtract the setup fee twice before allocating the remaining budget.
Round the allowable fraction upward, exceeding the budget.
Explain your thinking.
Optional written extension. Self-assessed only; it does not change your automatic practice result. These are teaching criteria, not official awarded marks.
Explain your model, identify a tempting alternative and show why it fails. This extension is self-assessed and does not affect automatic scoring.
This draft stays on this page only. Use paper if you want to keep your working.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
GED Mathematical Reasoning overviewSiyavula Grade 10 mathematics foundation reference