PSLE Mathematics: Fixed Fee vs Per-Use Cost Comparisons

PSLE Learner’s Guide · Vol 0105 · Mathematics · Primary 6

A cheaper price per item does not always mean a cheaper total

Two invented plans offer printing services for a school project. Plan A charges $2 per page and no membership fee. Plan B charges $10 to join and $1 per page. Which plan is cheaper? If you need two pages, Plan A costs $4 while Plan B costs $12. If you need twenty pages, Plan A costs $40 and Plan B costs $30. Both statements are correct because the quantity changes the comparison. You cannot rank the plans until you know how many pages the pupil needs and whether the fixed charge applies.

The first number you see on an advertisement is often the price per item. The second, sometimes written in a small note, is a fixed registration fee, delivery charge, deposit, base fare, subscription or minimum charge. A learner who compares only ‘$2 per page’ with ‘$1 per page’ has compared variable rates but ignored a cost that may matter more for a small order. The problem is not primarily multiplication; it is deciding which costs occur once and which repeat.

This guide teaches the Primary 6 examination job of separating a fixed fee from a quantity-dependent charge before comparing two plans. We will learn to calculate each total, identify the point where plans cost the same, account for what is included free, recognise whole-item restrictions, interpret a table correctly and make a final recommendation for a specified quantity. The examples are original classroom scenarios, not real current merchant prices or claims about a particular PSLE question.

Try a simple statement of structure before you calculate: total = fixed cost + (cost per unit × number of charged units). In the first plan the fixed cost is zero, and in the second it is ten dollars. That form is useful only when every used unit is charged at the stated rate and the problem describes no extra tiers, free allowances or minimums. As soon as the wording includes those conditions, you must adjust what counts as a charged unit.

The SEAB 2026 PSLE formats identify the current revised Mathematics assessment. For complete teaching on ratio, algebraic ideas, arithmetic and data, use the Mathematics Learning Hub. This performance guide concentrates on translating pricing language into a valid total and then selecting the suitable plan for the stated quantity.

1. Three questions before you compare a price

What cost occurs once? A registration fee of $10 is charged for a whole order or period if the question says it is. It is not automatically repeated for every page, hour or visit. A delivery fee may be charged for one delivery even when a parcel contains several items. Read the scope: ‘$6 per order’ is different from ‘$6 per parcel’.

What cost repeats? A rate such as ‘$2 per page’ is multiplied by the number of pages that are actually charged. A cost of ‘$3 per hour’ depends on hours; ‘$4 per visit’ depends on visits. ‘Per’ tells you the basis of the multiplication. Writing a dollar amount without its basis hides what you are counting.

How many charged units exist? If the question includes five free prints, and the pupil needs eighteen, the chargeable count is thirteen, assuming only prints beyond the five are billed. If the service sells whole tickets, the number of tickets may have to be a whole number. If an extra half-hour is billed as a full hour, the charged quantity is not always equal to exact time elapsed.

A useful first draft of working is not just numbers: ‘Plan A: fixed 0 + 2 × pages’ and ‘Plan B: fixed 10 + 1 × pages.’ This forces the fee and the rate to stay attached to their correct plans. A surprising number of errors are routing mistakes: using the $10 from B with the $2 per page from A, or adding the one-time fee repeatedly because it happens to stand in the same table as a per-use charge.

When the question asks which plan is better, decide what ‘better’ means. If it specifically asks for the lower price, compare total dollars for the same stated quantity and conditions. If it asks for a saving, subtract one total from the other. If it asks for the break-even number, determine the quantity where totals match. If it asks what happens at a maximum budget, solve for the number of units each plan can provide under the given rules. These are different jobs drawn from the same prices.

2. Construct a two-plan comparison without losing the fees

Move 1: Read the request and quantity first

Mark how many items, visits, pages or hours are needed. If the question does not supply one exact quantity, it might ask for a threshold or a condition instead. Do not assume the larger plan is always cheaper or that the cheaper rate wins at every use level.

After the step, say what each number means in a short phrase. The exercise succeeds when a new set of fees can be handled without repeating the memorised figures from the first example. The working should preserve the conditions even if the story changes from printing to transport or school activities.

Move 2: Name the fixed charge for each plan

Write ‘fixed A’ and ‘fixed B’ separately. The fixed amount may be zero, a registration fee or a delivery charge. Check whether it is charged once per order, once per month or once per day before adding it.

Move 3: Name the per-unit charge and unit

Write ‘$2 per page’, not just ‘2’. A rate may refer to a visit, hour, kilometre or pack. Mixing bases produces a seemingly ordinary arithmetic expression with meaningless units.

Move 4: Apply free allowances and minimum charges

If a plan includes three free uses, the number billed is the uses beyond three, not the total uses. If there is a minimum charge, check whether the calculated total falls below it. Some rules are piecewise and need separate cases.

Move 5: Compute the same target quantity for both plans

Compare two totals at one clearly stated number of uses. Calculating A for twelve visits and B for ten visits does not rank the plans fairly. Keep the quantity in view as you work.

Move 6: Check the equality or saving if requested

When both totals are equal, neither plan saves money at that quantity. For a saving, subtract smaller from larger and state the beneficiary. For a crossover, test one quantity below and one above the equality to verify the direction.

Move 7: Check the real-world count

If users cannot buy half a ticket or half a pack, do not leave a fractional required quantity as the final practical answer. If a time rate is billed in blocks, use the stated block rule rather than treating the cost as continuously proportional.

3. The break-even idea without guessing

Return to the original invented printing plans. Plan A costs $2 per page. Plan B costs $10 plus $1 per page. At ten pages, A costs $20 and B costs $20. They break even at ten pages. Below ten, A is cheaper. Above ten, B is cheaper. That is an explanation of a crossover, not merely the answer ‘ten’. The fixed charge makes B expensive to start, while its smaller repeated rate makes it cheaper after enough pages.

A small value table makes the relationship visible. At zero pages, A costs $0 while B costs $10. At five pages, the totals are $10 and $15. At ten pages, both are $20. At fifteen pages, the totals are $30 and $25. Each additional page increases A by $2 and B by $1. The difference in total cost changes by $1 in B’s favour whenever one more page is purchased.

You can reason to equality using the fixed disadvantage and per-unit saving. B starts $10 more expensive and saves $1 per page relative to A. It therefore needs ten pages to recover that $10 disadvantage. This shortcut works here because the per-unit rates are constant and there are no free allowances or tiers. Always verify any shortcut against the full expressions and the actual quantity.

The crossover quantity can be non-integer. If one plan breaks even at 7.5 tickets, that is a mathematical threshold but not a ticket count a pupil can purchase. With whole tickets, compare seven and eight to determine the practical choice. A fractional threshold is not an instruction to purchase half a ticket. Similarly, an hourly service billed by started hour needs the billing rule to turn elapsed time into charged hours.

Sometimes no crossover exists. If Plan A has both a lower fixed fee and a lower charge per item than B under identical conditions, A stays no more expensive at any nonnegative quantity. There is no need to force an equality out of every two-plan question. In other cases the plans are equal at every quantity because both fees and rates match. Read the data before assuming the exercise must contain a dramatic reversal.

4. Worked comparisons: different pricing conditions, different methods

Worked comparison 1: Printing an assignment

Situation: A prints at $2/page with no fee; B charges $10 once plus $1/page. The pupil needs 6 pages. First identify the cost paid once, the cost per charged unit and any allowance, minimum or billing constraint. Do not decide the better option from a headline price.

Calculation and result: A: 2×6=$12; B: 10+1×6=$16. A saves $4. Why this is the appropriate calculation: The lower repeated rate in B does not compensate for its fixed fee at only six pages. Saying B is cheaper because 1 is less than 2 omits a necessary cost.

Changed-detail rehearsal: Now calculate totals for 14 pages and explain why the recommendation changes. Recompute from the wording rather than adjusting the old final answer by instinct. The result should make sense both as money and as a count of available whole units.

Worked comparison 2: A classroom magazine

Situation: A print shop charges $5 setup plus $0.50/page; another charges $1/page with no setup. A class needs 20 pages. First identify the cost paid once, the cost per charged unit and any allowance, minimum or billing constraint. Do not decide the better option from a headline price.

Calculation and result: First: 5+0.5×20=$15. Second: 1×20=$20. The first saves $5. Why this is the appropriate calculation: The setup is added once, not twenty times. The per-page price applies to twenty pages. Units must stay consistent throughout.

Changed-detail rehearsal: Change the order to 8 pages and find which plan is cheaper. Recompute from the wording rather than adjusting the old final answer by instinct. The result should make sense both as money and as a count of available whole units.

Worked comparison 3: A transport pass

Situation: A fictional school shuttle costs $4 per ride or a $12 card plus $2 per ride. A pupil takes 10 rides. First identify the cost paid once, the cost per charged unit and any allowance, minimum or billing constraint. Do not decide the better option from a headline price.

Calculation and result: Pay-as-you-go=$40. Card=$12+$20=$32. The card saves $8. Why this is the appropriate calculation: The card fee is a fixed cost. A tempting shortcut comparing only $4 with $2 would choose the card even for one ride.

Changed-detail rehearsal: Find the number of rides at which the two totals are equal. Recompute from the wording rather than adjusting the old final answer by instinct. The result should make sense both as money and as a count of available whole units.

Worked comparison 4: A museum activity

Situation: Option A has no entry fee but costs $3 for each activity. Option B costs $8 for entry plus $1 per activity. The pupil wants 4 activities. First identify the cost paid once, the cost per charged unit and any allowance, minimum or billing constraint. Do not decide the better option from a headline price.

Calculation and result: A=4×3=$12. B=$8+4=$12. Neither option saves money. Why this is the appropriate calculation: Break-even is a tie, not evidence that B is better because its per-activity rate is smaller. The question’s quantity matters.

Changed-detail rehearsal: Compare 3 activities and 5 activities, and explain which side of equality each lies on. Recompute from the wording rather than adjusting the old final answer by instinct. The result should make sense both as money and as a count of available whole units.

Worked comparison 5: A free-use allowance

Situation: A service charges a $6 joining fee, includes the first 5 prints, then charges $2 for each additional print. The pupil prints 9. First identify the cost paid once, the cost per charged unit and any allowance, minimum or billing constraint. Do not decide the better option from a headline price.

Calculation and result: There are 9−5=4 chargeable prints, so total=$6+4×$2=$14. Why this is the appropriate calculation: It would be wrong to bill all nine prints because the first five are included. Equally wrong is subtracting the five free prints from the joining fee.

Changed-detail rehearsal: Calculate for only 3 prints and decide whether the per-print charge applies at all. Recompute from the wording rather than adjusting the old final answer by instinct. The result should make sense both as money and as a count of available whole units.

Worked comparison 6: Two plans with different free items

Situation: Plan A costs $1 per print. Plan B costs $5 and includes 5 prints, then costs $0.50 per additional print. The order has 13 prints. First identify the cost paid once, the cost per charged unit and any allowance, minimum or billing constraint. Do not decide the better option from a headline price.

Calculation and result: A=$13. B=$5+8×$0.50=$9. B is $4 cheaper. Why this is the appropriate calculation: The included five prints reduce the charged count, but the fixed $5 remains. Calculating $5+13×$0.50 would ignore the allowance.

Changed-detail rehearsal: Change the order to 4 prints. State how B’s fixed cost is applied. Recompute from the wording rather than adjusting the old final answer by instinct. The result should make sense both as money and as a count of available whole units.

Worked comparison 7: Per-order delivery

Situation: A shop charges $2 per item plus $5 per order for delivery. Six items are delivered together. First identify the cost paid once, the cost per charged unit and any allowance, minimum or billing constraint. Do not decide the better option from a headline price.

Calculation and result: Total=6×$2+$5=$17. Why this is the appropriate calculation: The phrase ‘per order’ means the $5 is charged once for this single delivery. Multiplying it by six would treat each item as a separate order.

Changed-detail rehearsal: Change the scenario to two separate deliveries with three items in each, each attracting the fee. Recompute from the wording rather than adjusting the old final answer by instinct. The result should make sense both as money and as a count of available whole units.

Worked comparison 8: A monthly membership

Situation: A club charges $15 once each month plus $3 for each lesson attended. A learner attends five lessons that month. First identify the cost paid once, the cost per charged unit and any allowance, minimum or billing constraint. Do not decide the better option from a headline price.

Calculation and result: Total=$15+5×$3=$30. Why this is the appropriate calculation: The fee belongs to the month, not to each lesson. A two-month comparison must account for whether the membership renews.

Changed-detail rehearsal: Compute the cost if the learner attends three lessons in each of two months. Recompute from the wording rather than adjusting the old final answer by instinct. The result should make sense both as money and as a count of available whole units.

Worked comparison 9: Fixed costs that never cross

Situation: Plan A charges $3 fixed plus $2 per ticket; B charges $6 fixed plus $3 per ticket. Compare 1, 5 and 10 tickets. First identify the cost paid once, the cost per charged unit and any allowance, minimum or billing constraint. Do not decide the better option from a headline price.

Calculation and result: A costs $5,$13,$23; B costs $9,$21,$36. A is lower for every nonnegative ticket count. Why this is the appropriate calculation: A starts cheaper and grows more slowly. There is no positive break-even quantity. Do not invent a crossover because other worksheets had one.

Changed-detail rehearsal: Explain the conclusion without doing a table for every possible ticket count. Recompute from the wording rather than adjusting the old final answer by instinct. The result should make sense both as money and as a count of available whole units.

Worked comparison 10: Two equal pricing rules

Situation: Both plans charge $4 joining fee plus $2 per use, but the brochures use different layouts. First identify the cost paid once, the cost per charged unit and any allowance, minimum or billing constraint. Do not decide the better option from a headline price.

Calculation and result: They cost the same for any equal number of uses under the stated conditions. Why this is the appropriate calculation: Different advertising styles are not different mathematical costs. The expressions are identical and remain so for every nonnegative use count.

Changed-detail rehearsal: Change only B’s joining fee to $6. Can equality still occur if its per-use rate stays $2? Recompute from the wording rather than adjusting the old final answer by instinct. The result should make sense both as money and as a count of available whole units.

Worked comparison 11: One dollar saving per ride

Situation: A costs $5 per ride; B costs $9 fixed plus $4 per ride. Find the break-even count. First identify the cost paid once, the cost per charged unit and any allowance, minimum or billing constraint. Do not decide the better option from a headline price.

Calculation and result: B starts $9 dearer and saves $1 each ride, so at 9 rides both cost $45. Why this is the appropriate calculation: The fixed disadvantage is recovered by the repeated saving only after nine rides. The answer is a number of rides, not nine dollars.

Changed-detail rehearsal: Find totals at 8 and 10 rides to verify which plan wins. Recompute from the wording rather than adjusting the old final answer by instinct. The result should make sense both as money and as a count of available whole units.

Worked comparison 12: An inconvenient noninteger equality

Situation: A costs $3/use with no fee; B costs $5 once plus $2/use. Suppose the question changes A to $3.50/use. First identify the cost paid once, the cost per charged unit and any allowance, minimum or billing constraint. Do not decide the better option from a headline price.

Calculation and result: Equality: 3.5n=5+2n gives 1.5n=5, so n=3⅓ uses; for whole uses A cheaper at 3 and B cheaper at 4. Why this is the appropriate calculation: A practical recommendation should test available whole counts around the mathematical boundary. Rounding the equality to three and saying they tie is wrong.

Changed-detail rehearsal: Calculate both totals at n=3 and n=4 to check the changeover. Recompute from the wording rather than adjusting the old final answer by instinct. The result should make sense both as money and as a count of available whole units.

Worked comparison 13: Pack versus loose items

Situation: Loose pencils cost $1 each. A 6-pencil pack costs $5. A pupil needs 13 pencils, and a pack cannot be split for billing. First identify the cost paid once, the cost per charged unit and any allowance, minimum or billing constraint. Do not decide the better option from a headline price.

Calculation and result: Two packs provide 12 pencils for $10 plus one loose pencil for $1, total $11; three packs cost $15, so the mixed option is cheaper under these choices. Why this is the appropriate calculation: The ‘per pencil in a pack’ average is not an actual price for buying exactly 13 from packs. Whole pack constraints matter.

Changed-detail rehearsal: Change the requirement to 17 pencils and compare combinations of packs and loose pencils. Recompute from the wording rather than adjusting the old final answer by instinct. The result should make sense both as money and as a count of available whole units.

Worked comparison 14: Minimum charge

Situation: A service charges $1 per item with a minimum bill of $8. The pupil needs five items. First identify the cost paid once, the cost per charged unit and any allowance, minimum or billing constraint. Do not decide the better option from a headline price.

Calculation and result: The calculated variable amount is $5, but the minimum bill makes the amount payable $8. Why this is the appropriate calculation: A minimum is a lower bound on the bill, not a one-time fee added to every variable total. Charging $8+$5=$13 would misunderstand the term.

Changed-detail rehearsal: Find the cost for 10 items and explain why the minimum no longer changes the bill. Recompute from the wording rather than adjusting the old final answer by instinct. The result should make sense both as money and as a count of available whole units.

Worked comparison 15: A fixed admission and extra events

Situation: A venue’s ticket is $7 and each optional activity costs $2. A pupil attends three activities. First identify the cost paid once, the cost per charged unit and any allowance, minimum or billing constraint. Do not decide the better option from a headline price.

Calculation and result: Total=$7+3×$2=$13. Why this is the appropriate calculation: The base ticket is paid once, while activities are counted separately. If the question asks for optional activity cost alone, $6 rather than $13 is relevant.

Changed-detail rehearsal: Calculate only the additional cost of choosing two more activities after admission is already paid. Recompute from the wording rather than adjusting the old final answer by instinct. The result should make sense both as money and as a count of available whole units.

Worked comparison 16: A refundable deposit

Situation: A club asks for a refundable $10 deposit plus a non-refundable $4 fee. The deposit will be returned in full after the equipment is safely handed back. First identify the cost paid once, the cost per charged unit and any allowance, minimum or billing constraint. Do not decide the better option from a headline price.

Calculation and result: Cash initially paid is $14; net cost after the full stated refund is $4. Why this is the appropriate calculation: The answer depends on whether the question asks for upfront payment or final nonrefundable expense. A refundable deposit is not automatically a permanent charge.

Changed-detail rehearsal: Change the question to how much money must be brought initially, then to final cost after return. Recompute from the wording rather than adjusting the old final answer by instinct. The result should make sense both as money and as a count of available whole units.

Worked comparison 17: A discount on variable charges

Situation: Plan A charges $8 joining and $4 per lesson; the school provides a 25% discount on lesson fees only. The pupil takes 6 lessons. First identify the cost paid once, the cost per charged unit and any allowance, minimum or billing constraint. Do not decide the better option from a headline price.

Calculation and result: Discounted lesson rate=$3; total=$8+6×$3=$26. Why this is the appropriate calculation: Applying 25% to the joining fee would change the condition. The discount applies to the repeated lesson portion only.

Changed-detail rehearsal: Recalculate if the discount applies to the entire bill instead, and compare. Recompute from the wording rather than adjusting the old final answer by instinct. The result should make sense both as money and as a count of available whole units.

Worked comparison 18: A late delivery surcharge

Situation: A courier charges $5 per parcel plus $9 fixed surcharge for one urgent delivery. Four parcels travel together. First identify the cost paid once, the cost per charged unit and any allowance, minimum or billing constraint. Do not decide the better option from a headline price.

Calculation and result: Total=4×$5+$9=$29. Why this is the appropriate calculation: Urgent delivery adds a fixed fee per order, not per parcel, in this described scenario. Count delivered parcels and deliveries independently.

Changed-detail rehearsal: Split the four parcels into two urgent deliveries and compute the total surcharge. Recompute from the wording rather than adjusting the old final answer by instinct. The result should make sense both as money and as a count of available whole units.

Worked comparison 19: First three hours included

Situation: A rental service charges $12 fixed for up to 3 hours and $4 for each further whole hour. A pupil’s event lasts 6 hours. First identify the cost paid once, the cost per charged unit and any allowance, minimum or billing constraint. Do not decide the better option from a headline price.

Calculation and result: Three additional hours are billed, so total=$12+3×$4=$24. Why this is the appropriate calculation: The included three hours are not charged again. ‘Up to 3 hours’ is a piecewise condition, not the same as ‘$12 per 3 hours’ repeated without limit.

Changed-detail rehearsal: Compute totals for 2, 3 and 4 hours, and state when the extra fee first applies. Recompute from the wording rather than adjusting the old final answer by instinct. The result should make sense both as money and as a count of available whole units.

Worked comparison 20: One started hour counts as whole

Situation: A supervised room costs $7 for each started hour plus $5 fixed booking fee. A group uses it for 2 hours 10 minutes. First identify the cost paid once, the cost per charged unit and any allowance, minimum or billing constraint. Do not decide the better option from a headline price.

Calculation and result: Three started hours are billed: total=$5+3×$7=$26. Why this is the appropriate calculation: The billing rule is not proportional to the exact fraction of an hour. Charging 2.166… hours ignores the word ‘started’.

Changed-detail rehearsal: Calculate the bill for exactly 2 hours and compare with 2 hours 1 minute. Recompute from the wording rather than adjusting the old final answer by instinct. The result should make sense both as money and as a count of available whole units.

Worked comparison 21: Budget as the constraint

Situation: A hobby station charges $8 membership plus $3/use. A pupil has $29. How many complete uses can the pupil afford? First identify the cost paid once, the cost per charged unit and any allowance, minimum or billing constraint. Do not decide the better option from a headline price.

Calculation and result: After the fee, $21 remains; 21÷3=7 complete uses. Why this is the appropriate calculation: The fee must be removed once from the budget before counting uses. Eight uses would cost $32 and exceed it.

Changed-detail rehearsal: If the budget is $31, determine the largest complete-use count and leftover money. Recompute from the wording rather than adjusting the old final answer by instinct. The result should make sense both as money and as a count of available whole units.

Worked comparison 22: Membership already paid

Situation: A pupil already paid a non-refundable $10 monthly fee. Each further club session costs $2; a separate pay-as-you-go room costs $3 per session. The choice concerns future sessions only. First identify the cost paid once, the cost per charged unit and any allowance, minimum or billing constraint. Do not decide the better option from a headline price.

Calculation and result: For additional sessions, $2 per session is the lower extra cost when the membership cannot be recovered and no other conditions differ. Why this is the appropriate calculation: The historic non-refundable fee has already been incurred for this decision. If the question instead asks total monthly expenditure, include it. Define the comparison window.

Changed-detail rehearsal: State both totals for four sessions under a full-month accounting question. Recompute from the wording rather than adjusting the old final answer by instinct. The result should make sense both as money and as a count of available whole units.

Worked comparison 23: Different rates but same total for a specified order

Situation: Plan A costs $5 fixed plus $1/item, Plan B costs $1 fixed plus $2/item. Compare four items. First identify the cost paid once, the cost per charged unit and any allowance, minimum or billing constraint. Do not decide the better option from a headline price.

Calculation and result: A=$9 and B=$9; both cost the same. Why this is the appropriate calculation: The fixed difference of $4 is exactly offset by the $1 per-item rate difference across four items. A lower fixed fee does not determine the winner alone.

Changed-detail rehearsal: Compare three and five items to test which plan becomes cheaper. Recompute from the wording rather than adjusting the old final answer by instinct. The result should make sense both as money and as a count of available whole units.

Worked comparison 24: A charge per person versus per group

Situation: A venue charges $6 per visitor plus a $10 booking fee for the entire group. Seven visitors attend. First identify the cost paid once, the cost per charged unit and any allowance, minimum or billing constraint. Do not decide the better option from a headline price.

Calculation and result: Total=7×$6+$10=$52. Why this is the appropriate calculation: The group fee occurs once for the booking, not once for each person. If two separate groups book independently, the fee may occur twice.

Changed-detail rehearsal: Split into two independent bookings of four and three visitors; compute total under the stated per-booking rule. Recompute from the wording rather than adjusting the old final answer by instinct. The result should make sense both as money and as a count of available whole units.

5. Five misconception repairs before an examination

Mistake 1: Using the cheaper unit price as the whole answer

A lower rate only becomes a lower total after you account for fixed fees and use count. Compare totals at the exact target quantity before recommending a plan.

For a useful correction, create another problem that keeps the same mathematical issue but changes the story. If a pupil can explain why the fee is counted once or why whole units matter without seeing the earlier numbers, the new method is more likely to survive the examination.

Mistake 2: Charging a one-time fee repeatedly

‘Per order’, ‘per month’ and ‘per visit’ name different bases. Multiply only by the number of occurrences of that basis. Do not use the number of items for a per-order fee.

Mistake 3: Charging free units twice

When a plan includes five free uses, subtract five from the charged count only when the total exceeds five. Keep any required fixed charge separate from the allowance.

Mistake 4: Treating a break-even threshold as a whole-number purchase

An equality such as 7.5 uses may be mathematically meaningful, but a customer cannot necessarily buy a fraction of a ticket. Test neighbouring whole quantities using the actual terms.

Mistake 5: Comparing incompatible quantities

Do not compare Plan A for ten uses with Plan B for twelve. Fix the same order, time horizon, number of people and units before judging which plan costs less.

Mistake 6: Mistaking a refundable deposit for a permanent cost

A deposit changes upfront cash requirement; it may not change the net amount spent if fully returned under the described conditions. Read exactly what ‘cost’ means in the question.

Mistake 7: Ignoring billing intervals

A started-hour policy can charge three hours for two hours and one minute. Do not apply continuous proportional rates when the problem states billing in whole blocks.

Mistake 8: Assuming every pair of plans crosses once

One plan can stay cheaper for all allowed quantities, or totals can be identical everywhere. Check both the fixed amounts and the slopes of the variable charges.

Mistake 9: Overlooking the decision’s time frame

A fee already paid may still belong in full-period accounting but not change the incremental cost of an additional use. Distinguish total expenditure from the next purchase decision.

Mistake 10: Forgetting the unit in the final answer

A break-even quantity is a number of visits, items or hours, not dollars. A saving is an amount of money. Use the correct label.

6. Practice from familiar numbers to unfamiliar pricing rules

First pass: classify fees without calculating. Read a short table of ten charges and mark each as fixed per order, repeated per item, monthly, conditional or refundable. The point is to learn the meaning of the words before starting multiplication. A wrong classification can produce a confidently calculated but incorrect bill.

Second pass: one plan at a time. Find totals for three quantities under one simple fixed-plus-variable rule. Then do the same for another plan. Ask why a fee remains unchanged when the quantity rises, while the variable cost changes. This isolates the mechanism before you compare plans.

Third pass: compare at the same quantity. Make a four-row table for quantities 0, 5, 10 and 15. Calculate both totals at each row. Circle the equality if it exists and describe which plan costs less to each side. A pupil who can explain the pattern will not need to guess from the first price listed.

Fourth pass: use a changed condition. Add free units, a minimum charge or a discount that applies only to the repeated part. Do not combine all three on the first difficult question. Teach one condition, check one contrast and then mix them. The learner must update the number of charged units or the rate while keeping the other costs unchanged.

Fifth pass: select a practical quantity. Introduce whole packs, started hours and a budget. A result may produce a noninteger theoretical threshold that must be interpreted using the real count rules. Ask the child whether the final answer describes a quantity that can actually be purchased.

Sixth pass: delay and transfer. Leave the topic for two days, then compare a fresh story about school clubs or library printing with changed fees. After another interval, place a simple cost comparison among ratio, fraction and graph questions without labelling it a two-plan exercise. The learner should independently recognise the fixed and repeated quantities.

7. Efficient examination control: one line that prevents the main error

Write each plan on its own line: ‘A total = …’, ‘B total = …’. Identify the fixed amounts first and keep them at the beginning. Next write the repeated rate multiplied by its charged quantity. The final subtraction should use the two resulting totals, not the unit prices. If one plan includes something free, compute the number of units that remain chargeable before multiplication.

If the question asks which plan is cheaper for a given order, you do not need to solve for the full break-even threshold first. Calculate the two required totals directly. If it asks ‘from what number of uses does B become cheaper?’, the threshold matters; use the difference in fixed fees and per-unit charges, then check whole-number quantities if needed.

For a quick final check, choose a very small and a larger quantity mentally. The plan with the lower joining fee often starts with an advantage; the plan with the lower per-use cost may gain later. That rough behaviour should be consistent with your table, although special free allowances and minimums can change the pattern. A prediction is a reason to verify, not a substitute for calculation.

This task is distinct from the Vol 0094 survey-overlap guide, which counts people in two categories, and the Vol 0101 interval guide, which counts locations and gaps. Here the student counts chargeable uses and attaches the right fee to each. Use the Mathematics Learning Hub for deeper arithmetic and algebra.

8. Parent and tutor guidance: diagnose the cost structure

When a learner makes a wrong cost comparison, first ask which fee is paid once and which repeats. If the answer is wrong, teach the wording and the idea of a repeated charge before rehearsing subtraction. If the classification is correct but multiplication fails, the repair is numerical. If both totals are correct but the wrong plan is selected, the issue is the comparison or the requested goal.

Use a two-column sheet with headings ‘once’ and ‘each time’. Give ordinary invented classroom contexts: a club joining fee, a per-visit activity, a single delivery charge or a print allowance. Ask the child to sort the costs without seeing a worked total. Move to realistic quantities only when each fee has a clear basis.

For stronger students, pose a pair of plans that are equally expensive at a certain number of uses. Ask them to prove the equality with both totals, then change the quantity by one. This demands reasoning about the difference in rates rather than blind calculation. Later introduce the practical complication that only whole tickets can be purchased.

Keep the feedback specific: ‘You charged the registration fee for every visit’ is useful; ‘careless with money’ is not. After a correction, remove the model and ask for a fresh scenario after a delay. A pupil who recognises the fixed-versus-variable split under new wording has gained a transferable performance skill.

9. Frequently asked questions

Is the plan with the lower price per unit always cheaper?

No. A higher fixed fee can outweigh the per-unit saving when few units are purchased. Compare total costs at the stated quantity. Construct one fresh example and see how the answer changes for a small order and a large order. The goal is to justify the plan from the total, not to repeat the attractive headline rate.

What is a fixed charge?

It is a charge applied once over the scope described, such as per order or per month. You must read that scope carefully. Construct one fresh example and see how the answer changes for a small order and a large order. The goal is to justify the plan from the total, not to repeat the attractive headline rate.

How do I calculate a break-even number?

Find the quantity at which the two total-cost expressions are equal, then check that the quantity makes sense for the way units are sold. Construct one fresh example and see how the answer changes for a small order and a large order. The goal is to justify the plan from the total, not to repeat the attractive headline rate.

What if equality occurs at a fraction of a ticket?

That is a mathematical threshold, but whole-ticket recommendations must compare legal integer quantities around it. Construct one fresh example and see how the answer changes for a small order and a large order. The goal is to justify the plan from the total, not to repeat the attractive headline rate.

Does a free-use allowance remove the membership fee?

Not unless the wording says so. Free uses usually reduce the number billed at the repeated rate. Construct one fresh example and see how the answer changes for a small order and a large order. The goal is to justify the plan from the total, not to repeat the attractive headline rate.

Is a deposit part of the cost?

It may be part of the amount paid upfront but not the final expense if the full deposit is returned. Read what the question asks. Construct one fresh example and see how the answer changes for a small order and a large order. The goal is to justify the plan from the total, not to repeat the attractive headline rate.

What if a plan has a minimum fee?

Calculate the usual charge and compare it with the stated minimum. Follow the policy described instead of adding the minimum as a separate fee. Construct one fresh example and see how the answer changes for a small order and a large order. The goal is to justify the plan from the total, not to repeat the attractive headline rate.

How do I check a two-plan answer quickly?

Verify the same quantity was used, each fixed fee was counted at the right frequency and each per-unit rate had the correct unit. Construct one fresh example and see how the answer changes for a small order and a large order. The goal is to justify the plan from the total, not to repeat the attractive headline rate.

Must I use algebra?

Not always. A small table or simple difference reasoning may solve the task. Use the method that fits the numbers and the level of the question. Construct one fresh example and see how the answer changes for a small order and a large order. The goal is to justify the plan from the total, not to repeat the attractive headline rate.

Can I compare plans before the required number of uses is known?

You can describe which plan wins at different quantities or identify a threshold, but one universally cheaper plan is not guaranteed. Construct one fresh example and see how the answer changes for a small order and a large order. The goal is to justify the plan from the total, not to repeat the attractive headline rate.

10. Your exit test and next learning route

Invent two plans of your own with a different fixed fee and per-unit charge. Compute their totals at three, eight and fifteen uses, then determine whether and where they cost the same. Add an allowance of two free uses to one plan and solve again. Make sure you can name which values represent dollars, which represent uses and which are total charges. Then test a whole-item requirement so the mathematical break-even and the practical recommendation do not get confused.

Return to the Mathematics Learning Hub for the broader topic foundations, or the PSLE Learning Guide for the next performance skill. The earlier Vol 0099 table-reading guide helps if the cost values are in an unfamiliar table. Your aim is a decision you can repeat when the next problem hides its fixed and repeated charges in new wording.

How to Perform in PSLE | Learner’s Guide Vol 0105 | Mathematics: Separate the Fixed Fee From the Per-Use Cost Before Comparing Two Plans