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In a progressive pricing table, the rate for a later band applies only to the units inside that band. Crossing the boundary does not normally change the price of all the earlier units. The word progressive matters: an all-units pricing rule works differently, so read the actual instructions before multiplying.
For G1 Mathematics, the reliable method is to split the quantity across the stated bands, calculate each band’s charge, and add the charges. If the first ten units cost two dollars each and additional units cost three dollars each, twelve units cost 10 × 2 + 2 × 3 = 26 dollars.
If your child calculates 12 × 3 = 36 dollars, the multiplication itself is fine. The question is which units the three-dollar rate covers. Repairing that reading decision is more useful than repeating arithmetic the student already understands.
Contents
Sections 1–5
Sections 6–10
Sections 11–15
Sections 16–20
Sections 21–25
Sections 26–28
Our opening rule is fictional. It is not a current electricity tariff, tax rule, transport fare or business offer. It simply says: the first ten units cost two dollars per unit; every unit beyond ten costs three dollars per unit; there is no fixed fee, tax or rounding complication.
At twelve units, ten belong to the first band and two belong to the second. The band quantities add back to twelve. Their charges are twenty and six dollars, giving a total of twenty-six. Only the two additional units receive the higher rate.
At exactly ten units, the charge is twenty dollars. At eleven units, it is twenty-three. The extra unit adds three dollars; it does not cause the first ten units to be charged again at a different price.
The central habit is to ask what each rate applies to. A rate without its quantity range is incomplete information. Once the ranges are understood, the calculation is usually a short sequence of multiplication and addition rather than a difficult new technique.
The SEAB G1 Mathematics syllabus for 2027 includes rates and applications involving personal and household quantities. Reading a pricing table is one way to connect arithmetic with a clearly defined everyday-style model.
This guide teaches the mathematics of invented pricing rules. Actual bills and offers can use different bands, fixed charges, minimum charges, rounding procedures and eligibility conditions. The authoritative rule for a real purchase is the current provider’s stated terms, not the examples on this page.
A learner does not need advanced algebra to understand progressive bands. Dividing a quantity into parts and checking that the parts add back to the whole is enough for the initial examples. Later sections show how a simple expression can compress the same reasoning.
G1 describes a subject level, not a child’s ability to make sense of the world. A student who can multiply correctly may need only a clearer picture of where one rate stops and another begins. Start with that decision and build confidence through a nearby independent example.
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The unit is what is being counted or measured. The quantity is how many such units are used. The rate is the charge per unit within a stated part of the pricing rule. The charge is the resulting amount of money.
In the opening example, twelve is the total quantity. Two dollars per unit is the first-band rate, and three dollars per unit is the later-band rate. Twenty-six dollars is the total charge. These numbers play different roles even though they appear together in one short question.
Writing unit labels helps keep those roles stable. Ten units multiplied by two dollars per unit gives twenty dollars. The unit count cancels in the rate calculation, leaving a monetary charge. Adding ten units directly to three dollars would combine incompatible quantities.
The quantity-reading guide develops this foundation. Before choosing an operation, name what each number measures. A learner who identifies the quantities accurately has already completed an important part of the problem, even before pressing any calculator keys.
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A progressive table divides the total quantity into successive portions. The first portion fills the first band up to its capacity. Only the remaining quantity can enter the next band. Each portion is then charged at its own rate.
For twelve units with a first-band capacity of ten, the allocation is ten plus two. For eight units, the allocation is eight plus zero. For twenty units, the allocation is ten plus ten. The first band does not contain more than ten units in this model.
The rate attached to the final band reached is sometimes called the rate on the next or additional unit. It is not automatically the average rate across the complete quantity. The complete charge still contains contributions calculated at earlier rates.
Avoid the vague instruction “use the rate for twelve.” Instead say, “Allocate twelve across the bands, then apply each rate to its portion.” That wording makes the process inspectable. It also adapts when there are three bands rather than two, without inventing a new calculation rule.
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A small table makes the allocation and the charge visible at the same time. Keep the band quantity separate from the total quantity and from the unit rate.
| Portion of twelve units | Units in this band | Rate in dollars per unit | Charge in dollars |
|---|---|---|---|
| First ten units | 10 | 2 | 20 |
| Units beyond ten | 2 | 3 | 6 |
| Total | 12 | No single band rate | 26 |
Two independent totals can now be checked. The allocated units add to twelve, matching the stated quantity. The charges add to twenty-six, producing the final bill in the fictional model. Both checks are useful because an incorrect allocation can still lead to neat arithmetic.
Do not add the two rates and call the result five dollars per unit. They apply to different portions. Likewise, do not multiply twelve by both rates and add, because that would charge the same twelve units twice. The table’s purpose is to keep every unit assigned exactly once.
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“The first ten units” identifies a portion starting at zero usage and ending after ten units. “The next ten units” identifies a further portion with its own capacity. “Every additional unit” identifies the quantity remaining after the earlier bands have been filled.
These phrases are instructions about allocation. They are not decorative wording around a rate. If a student ignores them, they can turn a progressive rule into an all-units rule without noticing that the mathematical model has changed.
For fifteen units under a rule charging the first ten at two dollars and the next ten at three dollars, only five units occupy the second band. The word next does not require buying all ten units in that band. It states the maximum size of that portion before another rule might begin.
Read whether partial units are permitted. Some invented examples measure a continuous quantity, while others count whole items. A rate per unit can be multiplied by a fractional quantity only if the model allows fractional units and does not specify a rounding or whole-block rule.
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At exactly ten units in the opening model, all ten units belong to the first band. There are no units beyond ten, so the second-band contribution is zero. The total is twenty dollars, not thirty.
At eleven units, one unit belongs to the second band and the charge is twenty-three dollars. At nine units, all nine remain in the first band and the charge is eighteen dollars. These neighbouring cases reveal which part of the rule changes at the boundary.
For whole-number items, a phrase such as “up to ten units” commonly includes ten when the rule states that the next band begins beyond ten. Do not rely on a general habit if the table uses explicit inequalities or a different convention; follow the exact wording supplied.
A useful student check is to calculate the boundary quantity, one below and one above. If moving from ten to eleven suddenly reprices all earlier units in a progressive model, the allocation has probably been misread. The charge increase should reflect the one new unit at its stated rate.
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Now consider a different fictional rule: quantities up to ten are charged at two dollars per unit, but if more than ten units are bought, every unit is charged at three dollars. Under that explicit all-units rule, twelve units cost thirty-six dollars.
That is the answer which was incorrect for the progressive example. It becomes correct here because the instructions have changed. The difference is not a disagreement about multiplication. It is a difference in what the later rate applies to.
An all-units rule can also offer a lower rate above a threshold. For example, a fictional seller might charge every unit at two dollars when more than ten units are bought, rather than three dollars below that threshold. Such a rule must be stated; do not assume it from the presence of a table.
When comparing two solutions, check the model before arguing about their final totals. Ask whether the rate applies only to additional units or retrospectively to every unit in the purchase. Two calculations can each be correct for different rules while only one matches the actual question.
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Suppose a fictional service charges the first twenty units at 0.50 dollars per unit, the next thirty units at 0.80 dollars per unit, and any units beyond fifty at 1.20 dollars per unit. Find the charge for sixty-five units, with no other fees.
The first band contains twenty units, costing ten dollars. The second contains thirty, costing twenty-four dollars. Fifty units have now been allocated, leaving fifteen units for the third band. Those cost eighteen dollars.
The total is 10 + 24 + 18 = 52 dollars. The allocation check is 20 + 30 + 15 = 65. Both the quantity and the money columns reconcile with the stated model.
A common mistake is to treat “the next thirty” as “up to thirty in total,” leaving the wrong remainder for the third band. Write cumulative boundaries separately: twenty after the first band, fifty after the second. The width of a band and its upper cumulative boundary are different quantities.
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The third-band quantity in the previous example is 65 − 20 − 30 = 15. It is also 65 − 50 because fifty is the total capacity of the first two bands. Both calculations express the same allocation.
The expression 65 − 30 = 35 would be wrong because it removes only the second band’s capacity, leaving first-band units inside the alleged remainder. A remainder must be measured after every earlier portion that has already been assigned.
For a total of forty units, the first band contains twenty and the second contains twenty. The third band contains zero. Do not subtract fifty and use a negative third-band quantity. A negative remainder indicates that the later band has not been reached, not that a negative charge should be added.
This is another reason a table can be safer than an unexplained formula. It asks how much of each band has actually been occupied. The learner can stop allocating when the total quantity has been fully assigned, rather than continuing through every printed row as though every band must be used.
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Add a fixed service fee of four dollars to the original two-band rule. Twelve units still produce a usage charge of twenty-six dollars, but the complete charge becomes thirty dollars. The fixed fee is added once, not once per band or once per unit.
At zero units, the usage charge is zero. If the fictional rule says the fixed fee applies even with no usage, the total is four dollars. If the question specifies a different zero-usage rule, follow that instead. The formula must represent the actual terms.
The fixed-starting-amount guide explains why a fixed charge breaks a simple direct-proportion shortcut. Pricing bands introduce a separate complication: even the usage rate may change as the quantity enters later portions.
Keep these two features distinct. One asks whether a charge exists independently of usage. The other asks which per-unit rate applies to each portion. A correct solution can handle both by calculating the allocated usage charges first and then adding the single stated fixed amount.
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For twelve units costing twenty-six dollars, the average charge per unit is 26/12 = 13/6 dollars, approximately 2.17 dollars. The next whole unit would cost three dollars under the opening rule, but that does not make every existing unit cost three.
The average rate reflects the mixture of ten units at two dollars and two at three dollars. It lies between those two rates because both portions are positive and there are no other fees in this example.
If the four-dollar fixed fee is included, the complete average becomes 30/12 = 2.50 dollars per unit. At a very small quantity, a fixed fee can push the overall average above every usage-band rate. Therefore the between-rates check applies only when its assumptions are satisfied.
Ask which rate a question requests: the rate for an additional unit, the average usage charge per unit, or the average complete charge per unit. These values can differ without any contradiction. The numerator and the included charges determine what the average means.
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Under the opening rule, twelve units cost twenty-six dollars and fifteen units cost thirty-five dollars. The difference is nine dollars for three extra units, all in the later band. This equals three units multiplied by three dollars per unit.
Comparing totals this way is a useful independent check. It avoids recalculating the first-band charge unnecessarily and confirms that the earlier ten units remain charged at their original rate in both cases.
If the quantities straddle the boundary, split the increase too. Moving from eight to twelve units adds two units at two dollars and two at three dollars, increasing the charge by ten dollars. The complete charges are sixteen and twenty-six dollars, which agree with that difference.
This technique reinforces the band interpretation. The rate depends on where the additional quantity lies, not simply on the final total. It is especially useful when a question asks how much a bill increases, because the increase may occupy more than one band even though the final quantity occupies only one named final band.
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Under the opening rule with no fixed fee, suppose the charge is thirty-two dollars. The first ten units cost twenty dollars, leaving twelve dollars of charge for units beyond ten. At three dollars each, that remaining charge represents four units.
The total quantity is therefore fourteen units. Check forward: 10 × 2 + 4 × 3 = 32. The backward calculation is not thirty-two divided by three, because the first portion was charged at a different rate.
The band assumption must be checked. A charge below twenty dollars would belong entirely to the first band in this simple model. A charge above twenty is consistent with reaching the second band, because all rates are positive and no rebates or special adjustments are present.
For a three-band rule, find the charges accumulated at the cumulative boundaries before selecting the remaining rate. Backward reasoning uses the same structure as forward reasoning: identify which earlier portions are complete, remove their charges, and convert the remaining charge into the remaining quantity.
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Suppose the opening rule applies to whole items and the budget is thirty-one dollars. The first ten cost twenty dollars, leaving eleven dollars. At three dollars per additional item, that remainder can buy three more whole items, with two dollars left.
The maximum is thirteen items, costing twenty-nine dollars. Fourteen would cost thirty-two dollars and exceed the budget. The final whole-item answer is thirteen, not the fractional quantity 13 and 2/3 suggested by dividing every last dollar continuously.
If the quantity were instead continuously measurable and the rule permitted fractional units, the maximum could be 13 and 2/3 units. The arithmetic model must match what is being purchased or measured. The same numerical budget can therefore lead to a different permitted answer type.
Check the next excluded quantity as well as the proposed maximum. Showing that thirteen fits is not enough to prove it is maximal; showing that fourteen fails under an increasing whole-item charge completes the practical check. State the domain before rounding a quantity down by habit.
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Use the original progressive rule: first ten units at two dollars each, additional units at three dollars each, with no fixed fee. Find the charges for A: eight units, B: ten units, C: thirteen units, and D: twenty units.
For E, find the quantity when the charge is thirty-eight dollars. For F, find the maximum whole-number quantity affordable with thirty dollars. For G, add a fixed fee of five dollars and calculate the complete charge for twelve units.
For H, use the three-band rule from section eight and calculate the charge for forty-five units. For I, explain why thirteen units would cost thirty-nine dollars under an explicitly stated all-units rate of three dollars, but not under the original progressive rule.
Finally, J asks for the increase in charge when usage rises from nine to twelve units under the original rule. Show which part of the increase falls in each band. A clear allocation is part of the answer, even when the numerical calculation is short.
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For A, eight units cost sixteen dollars. For B, ten cost twenty dollars. For C, the allocation is ten plus three, so the charge is twenty plus nine, or twenty-nine dollars. For D, ten plus ten gives twenty plus thirty, or fifty dollars.
For E, remove the first-band charge of twenty from thirty-eight. The remaining eighteen dollars buys six later-band units, giving sixteen units altogether. For F, thirteen units cost twenty-nine dollars while fourteen cost thirty-two, so thirteen is the maximum whole-number quantity.
For G, twelve units have a usage charge of twenty-six dollars; adding the fixed fee gives thirty-one. For H, twenty units cost ten dollars and the remaining twenty-five cost twenty dollars, giving thirty dollars. The third band is not reached.
For I, an all-units rule applies three dollars to all thirteen units, while the progressive rule applies it only to the three additional units. For J, one extra unit is charged at two dollars and two at three dollars, so the increase is eight dollars, from eighteen to twenty-six.
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When reviewing a solution, check the band quantities before the products. If twelve has been placed in the second-band row, the error precedes multiplication. The student needs to identify the first ten units and the remaining two, not practise multiplying twelve by three.
If the allocation is correct but the charges are wrong, repair the arithmetic. If the charge total is correct but the answer is labelled dollars per unit, repair the quantity description. If a fixed fee has been added once for each band, repair where that fee belongs.
A useful annotation is to write the allocation as a sum beside the question: 12 = 10 + 2. For the three-band example, write 65 = 20 + 30 + 15. This simple conservation check catches omitted and double-counted quantities.
Feedback should identify the decision that changes the next attempt. “Only the additional two units get the later rate” is actionable. “Be careful with bills” is too broad to tell the learner how to correct the model. Keep the explanation focused on the information the student has actually misread.
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For a quantity q between zero and ten inclusive, the opening charge is 2q. For q greater than ten, it is 20 + 3(q − 10). The second expression contains the completed first-band charge and the charge for the remaining quantity.
Do not use the second expression at q = 4 and accept its result of two dollars. That input does not satisfy the expression’s stated branch condition. A formula associated with a band is not automatically valid for every possible quantity.
For q greater than ten, the expression can be simplified to 3q − 10. The simpler form is mathematically equivalent on that domain, but the original allocated form often explains the pricing structure more clearly to a learner.
An algebraic expression is useful when it compresses reasoning already understood. It should not conceal which band applies. Keep the domain beside the expression and check a boundary value to confirm that the formula represents the intended model rather than merely producing convenient arithmetic.
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Suppose fractional units are permitted under the original progressive model. A quantity of 10.5 units costs twenty dollars for the first ten and 0.5 × 3 = 1.50 dollars for the remainder, giving 21.50 dollars.
A different fictional rule might charge every started additional unit as a whole block. Under that explicitly stated rule, the same measured quantity could require payment for one additional whole unit, producing twenty-three dollars. The calculation changes because the billing convention changes.
Do not insert whole-block rounding simply because a quantity has a decimal part. Equally, do not charge proportionally for a partial block when the question says every started block is charged in full. Both procedures can be valid under different instructions.
Write the measured quantity and the billable quantity separately if they differ. This prevents a learner from saying that 10.5 has somehow become eleven as a physical measurement. The measurement remains 10.5; the fictional rule determines how many chargeable blocks are associated with it.
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An invented meter might show 140 units at the start of a period and 152 at the end. The period usage is twelve units, found by subtraction. Under the opening rule, the usage charge is twenty-six dollars.
Applying the pricing bands directly to the ending reading of 152 would charge for quantities outside the stated period. The ending cumulative reading and the period increase are different quantities. Both can appear on a page, but only the relevant usage belongs in this model.
The broader electricity-bill mathematics guide connects measurements, rates and bill interpretation. Actual meters and bills may include adjustments or estimated readings, so the real document’s definitions remain essential.
For the classroom example, no adjustment is present. The calculation has two stages: find the period quantity from the readings, then allocate that quantity across the bands. Keeping those stages separate makes it easier to identify whether an incorrect answer came from the measurement difference or the pricing rule.
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The examples so far use numbers that give exact currency amounts without awkward fractions of a cent. If a question supplies decimal rates that create longer products, retain the calculation accurately and follow the stated rounding procedure.
Rounding each band’s charge before adding can differ from adding unrounded contributions and rounding the final total. Neither procedure should be silently assumed when a task specifies the other. The pricing allocation and the reporting precision are independent decisions.
The rounding-items-and-totals guide explains that distinction. Here the priority is to allocate the correct quantities first. Accurate rounding cannot repair a charge calculated from the wrong band quantities.
In a written answer, show enough working to make both decisions visible. A band table can retain unrounded products, followed by one stated final rounding step. If the problem requires line-by-line rounding, show those rounded lines and then add them. Describe the procedure actually used rather than treating every small difference as a mysterious calculator error.
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Imagine that the original two-band rule resets for each separate day. Eight units on Monday and eight on Tuesday would cost sixteen dollars each day, giving thirty-two dollars across the two days.
If instead one sixteen-unit quantity were billed under a single period with the same first-ten band, the charge would be twenty plus eighteen, or thirty-eight dollars. The total physical quantity is sixteen in both descriptions, but the band-reset rule changes the allocation.
This example is entirely fictional and does not describe an actual provider’s policy. Its purpose is to show that “first ten” needs a scope: first ten per purchase, per day, per account, per period or under another explicitly defined unit of billing.
Do not split a real or classroom quantity into separate periods merely to obtain a lower result unless the rules genuinely reset that way. Conversely, do not combine separate allocations if the stated model requires independent resets. The period definition is part of the mathematical input, just as the numerical rates are.
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In the progressive example, moving from ten units to just over ten changes only the charge on the small additional portion, provided continuous quantities are allowed. The earlier twenty-dollar contribution stays fixed. The charge grows from the boundary rather than being recalculated at the later rate for every unit.
This behaviour differs from some explicitly defined all-units rules, which can create a jump at a threshold. The existence of a jump is therefore not something to infer from a higher rate label alone. It follows from how the rule applies that rate.
For whole items, compare ten and eleven rather than imagining fractional quantities the model does not permit. The charges are twenty and twenty-three dollars. The three-dollar difference matches the additional whole item.
A learner who understands this can predict the approximate scale of an answer before calculating. If a progressive model’s total suddenly increases by the cost of repricing all earlier units, inspect the allocation. Prediction is a useful check, while the exact band calculation still supplies the answer.
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Suppose fictional Plan A uses the opening progressive rule with no fixed fee. Fictional Plan B charges 2.40 dollars per unit at every quantity with no bands. At twelve units, A costs twenty-six dollars and B costs 28.80 dollars.
At twenty units, A costs fifty dollars while B costs forty-eight. A is cheaper in the first comparison and B in the second. A single unit-rate label does not settle every comparison when the pricing structures differ.
This is a classroom comparison, not a recommendation of a real purchase. Keep the quantity, period and included charges identical across both calculations. Otherwise a lower total might reflect a different amount or an omitted fixed fee rather than a genuinely lower charge for the same stated service.
The guide to comparing offers and conditions develops this wider reading habit. Calculate each complete model for the same scenario, then compare the relevant totals. The question is not which rate looks smallest in isolation, but what the stated rules charge for the requested quantity.
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Start with a quantity just above a boundary, such as twelve with a first band of ten. Ask the learner to separate twelve counters into ten and two before calculating any money. The physical grouping makes the meaning of additional units immediate.
Then use a quantity below the boundary and one exactly at it. These contrasts test whether the student is applying the allocation rule or merely repeating the ten-plus-two pattern. Finally, add a third band only after the two-band decision is secure.
The Clementi Secondary 1 Mathematics small-group guide provides the wider teaching context. Useful support identifies which reading or calculation decision is missing and checks that decision on a fresh example.
A student who explains the allocation clearly but makes one multiplication error needs different practice from a student who multiplies perfectly using the wrong quantity. Listen to the explanation as well as checking the total. The aim is a learner who can interpret a new table independently, not simply recognise a familiar bill layout.
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Keep the opening twelve-unit usage charge of twenty-six dollars. Suppose a fictional instruction reduces only the later-band charge by ten percent. The first-band charge remains twenty dollars. The later-band charge of six dollars becomes 5.40 dollars, giving a revised total of 25.40 dollars.
If the instruction instead reduces the complete usage charge by ten percent, the result is 26 × 0.9 = 23.40 dollars. These are different rules. The same discount percentage does not produce the same monetary reduction when it applies to a different base amount.
Now include a fixed fee of four dollars, making the undiscounted complete charge thirty dollars. If the discount applies to usage only, the revised charge is 23.40 + 4 = 27.40 dollars. If it explicitly applies to the complete thirty-dollar charge, the result is twenty-seven dollars.
All four calculations can be correct under their own stated instructions. The question is which charge the discount names: one band’s contribution, the complete usage charge, or the final charge including the fixed fee. Do not infer that scope from the percentage alone.
This is an extension of the same allocation habit used throughout the guide. A per-unit rate needs the quantity it applies to, and a percentage adjustment needs the charge it applies to. Label that base before calculating. The numerical operation is then straightforward and its meaning remains visible.
The examples are invented arithmetic rules, not claims about how actual businesses apply discounts or fees. If a real document is unclear, its terms need clarification. A learner should not resolve missing information by choosing whichever interpretation produces the lower total or a rounder number.
For practice, ask the student to explain why the reductions are sixty cents, 2.60 dollars or three dollars in the relevant examples. The difference comes from the base charge, not from a change in the meaning of ten percent. A correct explanation should point to that base explicitly.
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Is the highest band rate the rate for every unit?
Only if the rule explicitly says so. In a progressive rule it applies to the quantity inside that band. In an all-units rule, reaching a threshold can change the rate applied to the complete quantity. Read the scope before choosing a calculation.
Can a later band have a lower rate?
Yes, a fictional model can state lower rates for additional units. The allocation method still works: calculate each portion at its stated rate. A lower later rate does not automatically mean every earlier unit receives that rate. Do not confuse a progressive discount with an all-units discount.
Does every table row have to contribute a charge?
No. A band not reached has zero allocated units. Its printed rate does not create usage. Stop when the entire stated quantity has been assigned, and check that the nonzero portions add back to the total.
What should a clear final answer show?
Show the band quantities, their rates, the separate charges and the complete total with its currency unit. Add fixed fees or rounding only as instructed. Return to the Mathematics explanation directory when the next reading decision needs a focused explanation.
Continue the G1/G2/G3 Mathematics series
Browse the Mathematics explanation directory
What Is G2 Mathematics | Why Two Conditions Need a Shared Solution When Both Must Hold
What Is G3 Mathematics | Why At Least One Includes Both in a Probability Question
What Is G1/G2/G3 Mathematics | Why a Rise from 20% to 30% Is Not a 10% Increase