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Build accurate cost models from tiered tables: separate marginal rates from all-unit rates, test thresholds and check inverse solutions.
1. Start here · 2. Model · 3. Thresholds · 4. Inverse problem · 5. Compare models · 6. Rounding units · 7. Practise
This original G3 K310 application workshop uses fictional charges, not real tariffs or current prices. Its interval notation is a teaching aid for organising rules; it does not claim an additional standalone syllabus topic.
Chapter index
Chapters 1–3
Chapters 4–6
Chapters 7–7
For a tiered-charge problem, circle the threshold and underline which units each rate covers. “The next units cost” means a marginal band. “All units cost” means the selected rate applies to the whole quantity. The same numbers can produce different bills under those two wordings.
Calculate a stated quantity using the correct rule, then check whether it belongs to the interval you selected. When working backwards from a bill, test the proposed answer against the interval as well as the arithmetic. A solved equation can be correct while the selected pricing rule is wrong.
This workshop follows the comparison and feasibility guides with a more specific reading task. The aim is to make the relationship visible before the calculator begins. All fees below are fictional and apply only under the written assumptions.
A fictional copying service charges a fixed $4 order fee, $0.20 per page for the first 50 pages, and $0.12 per page for each page above 50. There is no minimum charge, no tax and no rounding beyond ordinary money notation. Let n be a non-negative integer page count.
For 0 ≤ n ≤ 50, C = 4 + 0.20n. For n > 50, C = 4 + 0.20 × 50 + 0.12(n − 50), so C = 14 + 0.12(n − 50). The expression n − 50 counts only the pages in the second band.
At 80 pages, C = 4 + 10 + 0.12 × 30 = $17.60. The tempting expression 4 + 0.12 × 80 = $13.60 applies the lower rate to every page, contrary to the stated rule.
At n = 0, the literal mathematical model still gives $4 because an order fee was specified. A real business might not accept a zero-page order, but you must not invent a real-world policy inside the fictional problem. If a task states n ≥ 1, use that restricted domain.
For the copying service, C(50) = 4 + 0.20 × 50 = $14.00. At 51 pages, C(51) = 14 + 0.12 = $14.12. The first page above the threshold adds twelve cents; the earlier fifty pages keep their original charge.
This neighbouring-value check catches a common off-by-one mistake. The second band includes pages 51 onward. It does not include page 50 again, and it does not begin at page 52.
A correct model joins at the threshold in this particular problem: writing the second expression at 50 would also produce $14. This is a consequence of its marginal pricing rule, not a universal requirement that every tariff be continuous. An all-unit discount can create a jump.
Sketching a cost-versus-page-count graph can show the change in slope, but page counts are integers. A continuous line is a useful model picture; it does not imply that a customer orders a fraction of a copied page. Preserve the practical domain when interpreting an inverse result.
A bill under the fictional copying rule is $20. How many pages were ordered? The maximum bill in the first band is $14 at 50 pages, so $20 must belong to the second-band calculation.
Write 14 + 0.12(n − 50) = 20. Then 0.12(n − 50) = 6, so n − 50 = 50 and n = 100. Check forwards: $4 + $10 + $6 = $20. The result is an integer above 50, as required.
Now suppose the bill is $15.10 with all the same exact assumptions. The equation gives n − 50 = 1.10/0.12 = 55/6, so n = 59⅙. That is not an admissible page count. Do not round to 59 or 60 and call the original bill exact: those counts produce $15.08 and $15.20.
The defensible conclusion is that $15.10 cannot arise from this exact model and an integer page count. A task with discounts, tax or a rounded measurement might require a different interpretation, but no such features were supplied here.
A different fictional printer charges $0.20 per page for orders of at most 50 pages. For orders of more than 50 pages, every page costs $0.12. The same fixed $4 fee applies.
At 80 pages, the bill is 4 + 0.12 × 80 = $13.60. Here the earlier tempting expression is correct because the wording now says every page receives the lower rate. At 50 pages the bill is $14.00; at 51 it is $10.12. That fall is consistent with the fictional all-unit discount.
Do not repair this discontinuity by silently changing the rule to marginal pricing. You may note that the discount creates an unusual incentive to order an extra page, but the model must first represent the supplied information faithfully.
When comparing the two printers, use the same page count and include the fixed fee in both totals. A lower displayed rate does not by itself settle the complete bill. The earlier comparison-basis workshop supplies that habit; this task adds the interval condition.
A fictional locker charges $3 for the first hour and $2 for each additional started hour. A stay of 2 hours 10 minutes uses the first hour plus two started additional hours, so the charge is 3 + 2 × 2 = $7.
For a stay of exactly 2 hours, only one additional hour is used: the charge is $5. The phrase “started hour” matters at the boundary. The extra ten minutes are not priced as ten sixtieths of $2 under this rule.
Draw a timeline if the intervals are confusing. Separate total elapsed time from billable blocks. The same reasoning appears in deliveries charged per started kilogram or materials sold in whole packs. Always use the particular task’s words; some services prorate continuously and others charge by blocks.
If the rate is per minute, convert the elapsed time into minutes before multiplying. Do not mix decimal-hour arithmetic with a started-hour rule simply because both involve time.
A: Using the marginal copying rule, find the bill for 65 pages. B: Under that rule, how many pages produce a $16.40 bill? C: Using the all-unit printer, find the bill for 65 pages. D: Under the locker rule, find the bill for exactly 3 hours and for 3 hours 1 minute.
Checks: A = 14 + 0.12 × 15 = $15.80. B: 14 + 0.12(n − 50) = 16.40 gives n = 70. C = 4 + 0.12 × 65 = $11.80. D is $7 at exactly 3 hours and $9 at 3 hours 1 minute.
If A and C match, reread which units receive the second rate. If B is wrong, inspect both the equation and its interval. If D has the same price twice, check whether you treated a started additional block as a completed one.
Retest later with a fictional water bill or delivery table. Record the interval first, the model second and the calculation third. A reliable solution explains the bill from the rule and can reverse-check it without looking at the original worked example.
Continue learning and official sources
Official K310 Mathematics syllabus · Complete G3 EMS Learner’s Guide · Constraints and feasibility workshop
