Strong G3 SEC Mathematics answers begin by deciding what each number measures. In this fictional habitat food pantry, two packet sizes have different prices and labelled quantities. The useful mathematical journey runs from fractions and rates to simultaneous equations, inequalities, infeasibility and a checked whole-number order. Each stage prepares the next; doing arithmetic without deciding what is being compared is the first error to avoid.
This is a teaching case for 2027 SEC G3 Mathematics K310, with 4052 shown by SEAB as the pre-2027 reference code. It does not prescribe meals or make nutritional claims. The fictional pantry data measure packet labels for mathematical modelling only. Biology explains digestion and absorption in Volume 1353. Here we deliberately keep the owner of the maths distinct.
Quick route: record units; normalise unequal packet sizes; define whole-number variables; translate exactly, at least and at most; derive two bounds; detect when they conflict; revise the stated budget rather than the evidence; check a new solution and explain its limitations. For earlier cross-subject quantitative thinking, compare Volume 1350’s unit rates and feasible schedules; use Mathematics Hub to revisit prerequisite skills.
Official examination facts
SEAB’s current 2027 G3 Mathematics K310 syllabus has Paper 1 and Paper 2, each 2 hours 15 minutes, 90 marks and 50% weighting. Paper 1 contains about 26 short-answer questions; Paper 2 has nine to ten questions of varying lengths, with the last focusing on a real-world context. Both require all questions to be answered. An approved calculator may be used in both, relevant formulae are provided, and omitting essential working may cost marks. Consult SEAB’s K310 syllabus and the 2027 G3 subject list for the authoritative format.
1. First read the unit under every number
The fictional pantry offers packet A of mass 80 g for $2.40 and packet B of mass 120 g for $3.00. A has 12 g carbohydrate, 4 g protein and 2 g fat per packet; B has 18 g, 5 g and 3 g respectively. These are invented teaching figures.
Worked pantry checkpoint. If a student treats 12 g carbohydrate as 12 g per 100 g for A, every later comparison is wrong. Identify the denominator printed in the question before calculating.
Write the model in ordinary words before using symbols. At this stage the key skill is interpretation: a plausible computation is only useful if it answers the stated question with the correct unit.
Independent check. What does ‘per packet’ mean when packets have unequal masses? Write the quantity or equation that answers the question, perform the calculation if necessary, and state the unit or interpretation. If the result depends on an unstated assumption, name that assumption rather than hiding it.
Now transfer the same idea to a different packet size or price without changing the logical method. This is how a student learns a reusable skill: the surface details can change, but definitions, relationships and verification remain. If the transferred version fails, return to the first incorrect step and correct that step before practising speed.
2. Distinguish quantity from rate
A packet mass is a quantity. A price per gram, amount per 100 g, or grams per dollar is a rate or ratio derived from quantities. The same numbers can support different questions, but the units keep those questions separate.
Worked pantry checkpoint. A shopper buys one A and one B. Their combined mass is 200 g, but the purchase price per gram requires total cost divided by total mass, not addition of the two rates.
Verify the direction of each comparison. The number alone does not carry the meaning of ‘at least’, ‘exactly’ or ‘no more than’. Translating those words faithfully is part of the solution.
Independent check. Which units should appear in the final result of a price comparison? Write the quantity or equation that answers the question, perform the calculation if necessary, and state the unit or interpretation. If the result depends on an unstated assumption, name that assumption rather than hiding it.
Now transfer the same idea to a different packet size or price without changing the logical method. This is how a student learns a reusable skill: the surface details can change, but definitions, relationships and verification remain. If the transferred version fails, return to the first incorrect step and correct that step before practising speed.
3. Use a common denominator
A has 12/80 × 100 = 15 g of carbohydrate per 100 g. B has 18/120 × 100 = 15 g per 100 g. A larger packet contains more carbohydrate in total, yet the carbohydrate concentration by mass is equal in this model.
Worked pantry checkpoint. A candidate claims B is richer because 18 is greater than 12. That conclusion changes when the mass of each pack is considered.
Check your intermediate result with the original quantities. Substitution can reveal a sign error, missed multiplier or mistaken denominator before it affects the final conclusion.
Independent check. How could equal per-100-g quantities coexist with unequal per-packet totals? Write the quantity or equation that answers the question, perform the calculation if necessary, and state the unit or interpretation. If the result depends on an unstated assumption, name that assumption rather than hiding it.
Now transfer the same idea to a different packet size or price without changing the logical method. This is how a student learns a reusable skill: the surface details can change, but definitions, relationships and verification remain. If the transferred version fails, return to the first incorrect step and correct that step before practising speed.
4. Compare protein without changing the question
A contains 4/80 × 100 = 5 g protein per 100 g; B contains 5/120 × 100 = 4.1666… g per 100 g. By total grams per pack B has more; by grams per 100 g A has more.
Worked pantry checkpoint. A question asks for the higher concentration, but the answer compares only the totals printed on the packets.
A competent answer contains more than a calculator display. Show the relevant relationship, a readable step of algebra and the contextual meaning of the answer.
Independent check. Which packet has the higher protein amount per 100 g under the invented label? Write the quantity or equation that answers the question, perform the calculation if necessary, and state the unit or interpretation. If the result depends on an unstated assumption, name that assumption rather than hiding it.
Now transfer the same idea to a different packet size or price without changing the logical method. This is how a student learns a reusable skill: the surface details can change, but definitions, relationships and verification remain. If the transferred version fails, return to the first incorrect step and correct that step before practising speed.
5. Compare price fairly
A costs 2.40/80 × 100 = $3.00 per 100 g, whereas B costs 3.00/120 × 100 = $2.50 per 100 g. Price per packet and price per unit mass are different criteria; the cheaper packet need not be better value per gram.
Worked pantry checkpoint. The learner says $2.40 is lower than $3.00 and ends the explanation. The specified question is about value for mass.
Do not repair a contradictory model by inventing extra freedoms. If a given condition makes a plan impossible, explain that impossibility; only a stated changed assumption can create a new problem.
Independent check. What changes when the criterion is price per 100 g? Write the quantity or equation that answers the question, perform the calculation if necessary, and state the unit or interpretation. If the result depends on an unstated assumption, name that assumption rather than hiding it.
Now transfer the same idea to a different packet size or price without changing the logical method. This is how a student learns a reusable skill: the surface details can change, but definitions, relationships and verification remain. If the transferred version fails, return to the first incorrect step and correct that step before practising speed.
6. Choose your variables deliberately
Let x be the number of A packets and y the number of B packets. Both must be whole numbers at least zero. Writing the meanings of x and y first prevents algebra from becoming disconnected from the packet counts.
Worked pantry checkpoint. A student finds x = 4.5 and interprets this as a permissible purchase without a rule allowing half packets.
Ask whether another student could reproduce the calculation from your working. Precision, variable definitions and proper units matter because Mathematics is a written argument, not a private mental guess.
Independent check. Why does a real-world packet decision require non-negative integers? Write the quantity or equation that answers the question, perform the calculation if necessary, and state the unit or interpretation. If the result depends on an unstated assumption, name that assumption rather than hiding it.
Now transfer the same idea to a different packet size or price without changing the logical method. This is how a student learns a reusable skill: the surface details can change, but definitions, relationships and verification remain. If the transferred version fails, return to the first incorrect step and correct that step before practising speed.
7. Translate an exact total
If the pantry needs exactly 12 packets, write x + y = 12. Do not replace equality with an inequality because the sentence says exactly, and do not write x + y = 12 g because packet number is dimensionless.
Worked pantry checkpoint. A fictional order form specifies 12 sealed packets to fit a delivery box. There is no permission to add a thirteenth.
Write the model in ordinary words before using symbols. At this stage the key skill is interpretation: a plausible computation is only useful if it answers the stated question with the correct unit.
Independent check. What would the statement become if the order specified at least 12 packets? Write the quantity or equation that answers the question, perform the calculation if necessary, and state the unit or interpretation. If the result depends on an unstated assumption, name that assumption rather than hiding it.
Now transfer the same idea to a different packet size or price without changing the logical method. This is how a student learns a reusable skill: the surface details can change, but definitions, relationships and verification remain. If the transferred version fails, return to the first incorrect step and correct that step before practising speed.
8. Build the cost expression
The total is C = 2.40x + 3.00y dollars. A ceiling of $33 becomes 2.40x + 3.00y ≤ 33. That does not mean each packet costs at most $33; it constrains the combined order.
Worked pantry checkpoint. The candidate forgets to multiply a price by the number of packets or adds masses to prices.
Verify the direction of each comparison. The number alone does not carry the meaning of ‘at least’, ‘exactly’ or ‘no more than’. Translating those words faithfully is part of the solution.
Independent check. How does a cost ceiling differ from a minimum expenditure? Write the quantity or equation that answers the question, perform the calculation if necessary, and state the unit or interpretation. If the result depends on an unstated assumption, name that assumption rather than hiding it.
Now transfer the same idea to a different packet size or price without changing the logical method. This is how a student learns a reusable skill: the surface details can change, but definitions, relationships and verification remain. If the transferred version fails, return to the first incorrect step and correct that step before practising speed.
9. Build the protein expression
The total labelled protein is P = 4x + 5y grams. The fictional mathematical requirement ‘at least 56 g’ becomes 4x + 5y ≥ 56. It is a constraint in a textbook model, not a personalised dietary recommendation.
Worked pantry checkpoint. The symbol is reversed to ≤ because a student reads ‘at least’ as ‘no more than’.
Check your intermediate result with the original quantities. Substitution can reveal a sign error, missed multiplier or mistaken denominator before it affects the final conclusion.
Independent check. What inequality symbol means a minimum is acceptable even when it is met exactly? Write the quantity or equation that answers the question, perform the calculation if necessary, and state the unit or interpretation. If the result depends on an unstated assumption, name that assumption rather than hiding it.
Now transfer the same idea to a different packet size or price without changing the logical method. This is how a student learns a reusable skill: the surface details can change, but definitions, relationships and verification remain. If the transferred version fails, return to the first incorrect step and correct that step before practising speed.
10. Build the fat expression
The label amount is F = 2x + 3y grams. If an invented display plan sets a mathematical maximum of 33 g, then 2x + 3y ≤ 33. Do not confuse this model constraint with health guidance.
Worked pantry checkpoint. A student replaces 2x + 3y with 2 + 3, overlooking the number of each packet.
A competent answer contains more than a calculator display. Show the relevant relationship, a readable step of algebra and the contextual meaning of the answer.
Independent check. What would change if the cap applied to average fat per packet rather than total fat? Write the quantity or equation that answers the question, perform the calculation if necessary, and state the unit or interpretation. If the result depends on an unstated assumption, name that assumption rather than hiding it.
Now transfer the same idea to a different packet size or price without changing the logical method. This is how a student learns a reusable skill: the surface details can change, but definitions, relationships and verification remain. If the transferred version fails, return to the first incorrect step and correct that step before practising speed.
11. Substitute to reduce unknowns
From x + y = 12, obtain x = 12 − y. Cost becomes 2.40(12 − y) + 3y = 28.80 + 0.60y. Algebra makes the trade-off visible: each replacement of A with B raises total cost by $0.60.
Worked pantry checkpoint. The candidate writes 2.40(12 − y) = 28.80 − y, losing the factor 2.40 multiplying y.
Do not repair a contradictory model by inventing extra freedoms. If a given condition makes a plan impossible, explain that impossibility; only a stated changed assumption can create a new problem.
Independent check. How does each extra B packet affect cost when the total packet count stays fixed? Write the quantity or equation that answers the question, perform the calculation if necessary, and state the unit or interpretation. If the result depends on an unstated assumption, name that assumption rather than hiding it.
Now transfer the same idea to a different packet size or price without changing the logical method. This is how a student learns a reusable skill: the surface details can change, but definitions, relationships and verification remain. If the transferred version fails, return to the first incorrect step and correct that step before practising speed.
12. Derive the cost bound
With the $33 ceiling, 28.80 + 0.60y ≤ 33, so 0.60y ≤ 4.20 and y ≤ 7. Since y is an integer, y can be 0 through 7 under that condition before checking other constraints.
Worked pantry checkpoint. A candidate reaches y ≤ 7 but decides the final order immediately. One inequality never replaces the rest of the question.
Ask whether another student could reproduce the calculation from your working. Precision, variable definitions and proper units matter because Mathematics is a written argument, not a private mental guess.
Independent check. What must be checked before declaring a feasible order? Write the quantity or equation that answers the question, perform the calculation if necessary, and state the unit or interpretation. If the result depends on an unstated assumption, name that assumption rather than hiding it.
Now transfer the same idea to a different packet size or price without changing the logical method. This is how a student learns a reusable skill: the surface details can change, but definitions, relationships and verification remain. If the transferred version fails, return to the first incorrect step and correct that step before practising speed.
13. Derive the protein bound
With x = 12 − y, P = 4(12 − y) + 5y = 48 + y. The invented requirement P ≥ 56 gives y ≥ 8. This conflicts with the cost result y ≤ 7.
Worked pantry checkpoint. The student chooses y = 8 because it meets the protein requirement and forgets that the $33 budget would be breached.
Write the model in ordinary words before using symbols. At this stage the key skill is interpretation: a plausible computation is only useful if it answers the stated question with the correct unit.
Independent check. What does the pair y ≥ 8 and y ≤ 7 prove? Write the quantity or equation that answers the question, perform the calculation if necessary, and state the unit or interpretation. If the result depends on an unstated assumption, name that assumption rather than hiding it.
Now transfer the same idea to a different packet size or price without changing the logical method. This is how a student learns a reusable skill: the surface details can change, but definitions, relationships and verification remain. If the transferred version fails, return to the first incorrect step and correct that step before practising speed.
14. Recognise infeasibility
No real number, and therefore no non-negative integer, can satisfy y ≥ 8 and y ≤ 7 simultaneously. The original requirements are inconsistent. A correct final answer is that the pantry cannot meet both under the given prices and packet total.
Worked pantry checkpoint. Some candidates expect every word problem to have a feasible numerical answer and force y = 7.5. The contradictions are the mathematical result.
Verify the direction of each comparison. The number alone does not carry the meaning of ‘at least’, ‘exactly’ or ‘no more than’. Translating those words faithfully is part of the solution.
Independent check. Why is ‘no feasible solution’ a stronger answer than a guessed compromise? Write the quantity or equation that answers the question, perform the calculation if necessary, and state the unit or interpretation. If the result depends on an unstated assumption, name that assumption rather than hiding it.
Now transfer the same idea to a different packet size or price without changing the logical method. This is how a student learns a reusable skill: the surface details can change, but definitions, relationships and verification remain. If the transferred version fails, return to the first incorrect step and correct that step before practising speed.
15. Revise one constraint without rewriting facts
Suppose the fictional committee raises the allowed cost ceiling from $33 to $33.60, keeping the food labels and total of 12 packets unchanged. Then 28.80 + 0.60y ≤ 33.60 yields y ≤ 8.
Worked pantry checkpoint. It is legitimate to change a hypothetical planning condition, but not to quietly change packet prices or nutrient amounts to rescue the result.
Check your intermediate result with the original quantities. Substitution can reveal a sign error, missed multiplier or mistaken denominator before it affects the final conclusion.
Independent check. Which assumption changed and which measurements stayed constant? Write the quantity or equation that answers the question, perform the calculation if necessary, and state the unit or interpretation. If the result depends on an unstated assumption, name that assumption rather than hiding it.
Now transfer the same idea to a different packet size or price without changing the logical method. This is how a student learns a reusable skill: the surface details can change, but definitions, relationships and verification remain. If the transferred version fails, return to the first incorrect step and correct that step before practising speed.
16. Find the boundary solution
The protein condition still requires y ≥ 8, while the revised budget requires y ≤ 8. Hence y = 8 and x = 4. This is the only possible pair under those two conditions and the exact-12 rule.
Worked pantry checkpoint. A learner writes (8,4) but has not said whether the ordered pair is (x,y) or (y,x). Define the variables when reporting.
A competent answer contains more than a calculator display. Show the relevant relationship, a readable step of algebra and the contextual meaning of the answer.
Independent check. How can the feasibility proof be checked by substitution into both inequalities? Write the quantity or equation that answers the question, perform the calculation if necessary, and state the unit or interpretation. If the result depends on an unstated assumption, name that assumption rather than hiding it.
Now transfer the same idea to a different packet size or price without changing the logical method. This is how a student learns a reusable skill: the surface details can change, but definitions, relationships and verification remain. If the transferred version fails, return to the first incorrect step and correct that step before practising speed.
17. Check the actual total cost
Four A packets cost 4 × $2.40 = $9.60. Eight B packets cost 8 × $3.00 = $24.00. Combined cost is $33.60, exactly the revised ceiling. Equality counts as meeting a ≤ condition.
Worked pantry checkpoint. A careless calculation turns $9.60 + $24.00 into $32.60 and incorrectly creates extra budget.
Do not repair a contradictory model by inventing extra freedoms. If a given condition makes a plan impossible, explain that impossibility; only a stated changed assumption can create a new problem.
Independent check. Why should you recompute the result directly rather than rely only on transformed algebra? Write the quantity or equation that answers the question, perform the calculation if necessary, and state the unit or interpretation. If the result depends on an unstated assumption, name that assumption rather than hiding it.
Now transfer the same idea to a different packet size or price without changing the logical method. This is how a student learns a reusable skill: the surface details can change, but definitions, relationships and verification remain. If the transferred version fails, return to the first incorrect step and correct that step before practising speed.
18. Check the labelled amounts
For (x,y) = (4,8), total protein is 4×4 + 8×5 = 56 g. Total fat is 4×2 + 8×3 = 32 g, which satisfies an illustrative maximum of 33 g. These are labelled amounts, not absorption predictions.
Worked pantry checkpoint. The candidate assumes that meeting a mathematical total establishes that a human diet is safe or balanced. The task supports no such conclusion.
Ask whether another student could reproduce the calculation from your working. Precision, variable definitions and proper units matter because Mathematics is a written argument, not a private mental guess.
Independent check. What does the calculation show, and what does it not show? Write the quantity or equation that answers the question, perform the calculation if necessary, and state the unit or interpretation. If the result depends on an unstated assumption, name that assumption rather than hiding it.
Now transfer the same idea to a different packet size or price without changing the logical method. This is how a student learns a reusable skill: the surface details can change, but definitions, relationships and verification remain. If the transferred version fails, return to the first incorrect step and correct that step before practising speed.
19. Check mass with different packet sizes
The total mass of four A and eight B packets is 4×80 + 8×120 = 320 + 960 = 1,280 g. This is 1.28 kg. It would be an error to multiply 12 packets by one packet size because the sizes differ.
Worked pantry checkpoint. A real-world question might add a delivery mass constraint, but the required upper bound must be stated before using it.
Write the model in ordinary words before using symbols. At this stage the key skill is interpretation: a plausible computation is only useful if it answers the stated question with the correct unit.
Independent check. What is the difference between 12 packets and 1.28 kg? Write the quantity or equation that answers the question, perform the calculation if necessary, and state the unit or interpretation. If the result depends on an unstated assumption, name that assumption rather than hiding it.
Now transfer the same idea to a different packet size or price without changing the logical method. This is how a student learns a reusable skill: the surface details can change, but definitions, relationships and verification remain. If the transferred version fails, return to the first incorrect step and correct that step before practising speed.
20. Check average rather than total
For the chosen order, the average protein per packet is 56/12 ≈ 4.67 g. The protein per 100 g across the combined order is 56/1280×100 = 4.375 g. Different denominators produce different correct averages.
Worked pantry checkpoint. A learner gives 4.67 g per packet in answer to a per-100-g question. The arithmetic is sound, the answer unit is wrong.
Verify the direction of each comparison. The number alone does not carry the meaning of ‘at least’, ‘exactly’ or ‘no more than’. Translating those words faithfully is part of the solution.
Independent check. Which denominator represents the question’s requested comparison? Write the quantity or equation that answers the question, perform the calculation if necessary, and state the unit or interpretation. If the result depends on an unstated assumption, name that assumption rather than hiding it.
Now transfer the same idea to a different packet size or price without changing the logical method. This is how a student learns a reusable skill: the surface details can change, but definitions, relationships and verification remain. If the transferred version fails, return to the first incorrect step and correct that step before practising speed.
21. Use weighted rates correctly
To find a combined per-100-g amount, weight packets by their masses. Do not average 5 and 4.1667 directly unless the quantities being averaged have equal masses. The combined 4.375 g/100 g is based on 1,280 g of actual product.
Worked pantry checkpoint. The unweighted mean of two label concentrations answers a different question: the mean of two product-category rates, not the concentration of the purchased mixture.
Check your intermediate result with the original quantities. Substitution can reveal a sign error, missed multiplier or mistaken denominator before it affects the final conclusion.
Independent check. What makes an average ‘weighted’ in this example? Write the quantity or equation that answers the question, perform the calculation if necessary, and state the unit or interpretation. If the result depends on an unstated assumption, name that assumption rather than hiding it.
Now transfer the same idea to a different packet size or price without changing the logical method. This is how a student learns a reusable skill: the surface details can change, but definitions, relationships and verification remain. If the transferred version fails, return to the first incorrect step and correct that step before practising speed.
22. Represent constraints in a table
A table with columns x, y, total count, cost, protein, fat and feasibility makes checking easy. For y=7, x=5: cost $33.00 and protein 55 g, so the original protein threshold fails. For y=8, x=4: protein 56 g and cost $33.60.
Worked pantry checkpoint. The student uses an incomplete table containing only rows that satisfy one condition and then claims all requirements are met.
A competent answer contains more than a calculator display. Show the relevant relationship, a readable step of algebra and the contextual meaning of the answer.
Independent check. Which column exposes why each tempting order fails? Write the quantity or equation that answers the question, perform the calculation if necessary, and state the unit or interpretation. If the result depends on an unstated assumption, name that assumption rather than hiding it.
Now transfer the same idea to a different packet size or price without changing the logical method. This is how a student learns a reusable skill: the surface details can change, but definitions, relationships and verification remain. If the transferred version fails, return to the first incorrect step and correct that step before practising speed.
23. Graph without forgetting integers
On a continuous graph, a feasible region may be bounded by lines. For sealed packets, only integer grid points have physical meaning. A line intersection at a fractional packet count is not automatically a valid order.
Worked pantry checkpoint. A graph suggests y = 7.5 as a compromise between conditions; neither the integer rule nor the two original inequalities permits that value.
Do not repair a contradictory model by inventing extra freedoms. If a given condition makes a plan impossible, explain that impossibility; only a stated changed assumption can create a new problem.
Independent check. How do real-world restrictions change the interpretation of a continuous solution region? Write the quantity or equation that answers the question, perform the calculation if necessary, and state the unit or interpretation. If the result depends on an unstated assumption, name that assumption rather than hiding it.
Now transfer the same idea to a different packet size or price without changing the logical method. This is how a student learns a reusable skill: the surface details can change, but definitions, relationships and verification remain. If the transferred version fails, return to the first incorrect step and correct that step before practising speed.
24. Interpret strict and non-strict inequality
The words ‘less than’ correspond to <, whereas ‘at most’ corresponds to ≤. Similarly, ‘more than’ is > and ‘at least’ is ≥. At a tight boundary, the distinction can determine whether the one remaining solution is legal.
Worked pantry checkpoint. The revised cost of $33.60 meets a ceiling of $33.60 but would violate a rule saying cost must be below $33.60.
Ask whether another student could reproduce the calculation from your working. Precision, variable definitions and proper units matter because Mathematics is a written argument, not a private mental guess.
Independent check. Why is the equality line part of some constraints but not others? Write the quantity or equation that answers the question, perform the calculation if necessary, and state the unit or interpretation. If the result depends on an unstated assumption, name that assumption rather than hiding it.
Now transfer the same idea to a different packet size or price without changing the logical method. This is how a student learns a reusable skill: the surface details can change, but definitions, relationships and verification remain. If the transferred version fails, return to the first incorrect step and correct that step before practising speed.
25. Sensitivity to a price increase
If B rises from $3.00 to $3.10, an order of four A and eight B would cost $9.60 + $24.80 = $34.40. The old $33.60 budget would no longer be enough; the student should not present an obsolete plan as still feasible.
Worked pantry checkpoint. A small rate change can invalidate a boundary solution because the original plan had no slack.
Write the model in ordinary words before using symbols. At this stage the key skill is interpretation: a plausible computation is only useful if it answers the stated question with the correct unit.
Independent check. How much would the ceiling have to increase to retain this exact order? Write the quantity or equation that answers the question, perform the calculation if necessary, and state the unit or interpretation. If the result depends on an unstated assumption, name that assumption rather than hiding it.
Now transfer the same idea to a different packet size or price without changing the logical method. This is how a student learns a reusable skill: the surface details can change, but definitions, relationships and verification remain. If the transferred version fails, return to the first incorrect step and correct that step before practising speed.
26. Read the command before writing
‘Calculate’ calls for workings and a numerical answer. ‘Show’ calls for a demonstration. ‘Explain why no solution is possible’ calls for the contradiction y ≥ 8 and y ≤ 7 and a contextual conclusion.
Worked pantry checkpoint. A candidate produces a page of arithmetic but never states that the constraints cannot all be satisfied.
Verify the direction of each comparison. The number alone does not carry the meaning of ‘at least’, ‘exactly’ or ‘no more than’. Translating those words faithfully is part of the solution.
Independent check. What final sentence translates the symbolic result into a decision? Write the quantity or equation that answers the question, perform the calculation if necessary, and state the unit or interpretation. If the result depends on an unstated assumption, name that assumption rather than hiding it.
Now transfer the same idea to a different packet size or price without changing the logical method. This is how a student learns a reusable skill: the surface details can change, but definitions, relationships and verification remain. If the transferred version fails, return to the first incorrect step and correct that step before practising speed.
27. Keep enough precision
In 2027 G3 Mathematics K310, use non-exact answers to three significant figures unless specified otherwise, and angle answers to one decimal place. Exact packet numbers and exact dollar totals should not be rounded into a change of meaning.
Worked pantry checkpoint. Rounding 7.999 to 8 without showing its source can hide whether the exact inequality actually allowed an integer.
Check your intermediate result with the original quantities. Substitution can reveal a sign error, missed multiplier or mistaken denominator before it affects the final conclusion.
Independent check. When should the working show more precision than the reported answer? Write the quantity or equation that answers the question, perform the calculation if necessary, and state the unit or interpretation. If the result depends on an unstated assumption, name that assumption rather than hiding it.
Now transfer the same idea to a different packet size or price without changing the logical method. This is how a student learns a reusable skill: the surface details can change, but definitions, relationships and verification remain. If the transferred version fails, return to the first incorrect step and correct that step before practising speed.
28. Use the approved calculator thoughtfully
SEAB allows an approved calculator in both Mathematics papers, but it does not replace the interpretation of the question. Use it to check products and divisions while keeping the algebra and logical restrictions visible.
Worked pantry checkpoint. A calculator displays 4.375 but cannot tell whether the unit is g/100 g, g/packet, dollars or kilograms.
A competent answer contains more than a calculator display. Show the relevant relationship, a readable step of algebra and the contextual meaning of the answer.
Independent check. What part of the work must a student still explain in words? Write the quantity or equation that answers the question, perform the calculation if necessary, and state the unit or interpretation. If the result depends on an unstated assumption, name that assumption rather than hiding it.
Now transfer the same idea to a different packet size or price without changing the logical method. This is how a student learns a reusable skill: the surface details can change, but definitions, relationships and verification remain. If the transferred version fails, return to the first incorrect step and correct that step before practising speed.
29. Prepare for the extended Paper 2 context
The final question in K310 Paper 2 focuses on an application of Mathematics to a real-world scenario and may combine several syllabus strands. This pantry is only a practice context; the actual examination could use travel, finance or other everyday settings.
Worked pantry checkpoint. A student memorises the pantry numbers instead of learning to define variables and test feasibility in any context.
Do not repair a contradictory model by inventing extra freedoms. If a given condition makes a plan impossible, explain that impossibility; only a stated changed assumption can create a new problem.
Independent check. How could the same method solve an excursion, transport or purchase problem? Write the quantity or equation that answers the question, perform the calculation if necessary, and state the unit or interpretation. If the result depends on an unstated assumption, name that assumption rather than hiding it.
Now transfer the same idea to a different packet size or price without changing the logical method. This is how a student learns a reusable skill: the surface details can change, but definitions, relationships and verification remain. If the transferred version fails, return to the first incorrect step and correct that step before practising speed.
Full worked problem: is the original order possible?
The fictional pantry must choose exactly twelve packets from A and B. Write x+y=12. A costs $2.40 and provides 4 g labelled protein; B costs $3.00 and provides 5 g. The order must have at least 56 g total labelled protein and cost no more than $33. The label figures are only invented numbers for mathematical practice, not diet recommendations.
From x=12−y, the protein inequality becomes 4(12−y)+5y≥56, giving y≥8. The cost inequality becomes 2.40(12−y)+3y≤33, giving y≤7. Both are required simultaneously. No y can satisfy them. Therefore, under the original constraints the order is impossible. This explanation is superior to proposing an order that violates one condition. The next task is not to apologise for the mathematics but to identify which condition a hypothetical planner is authorised to change.
If the cost ceiling is deliberately raised to $33.60 while every packet figure stays fixed, cost now requires y≤8. Together with y≥8, this gives y=8 and x=4. Verify all constraints: the counts total 12; cost is $33.60; labelled protein is 56 g; the combined mass is 1,280 g; and the illustrative fat total is 32 g. A student should report the change in budget alongside the answer. Saying merely ‘the order works’ would conceal that it did not work under the original $33 ceiling.
Five exam-style questions with solution notes
Question 1
Prompt. A buyer compares 12 g in 80 g against 18 g in 120 g. Which product has more carbohydrate per 100 g?
Solution. Neither: 12÷80×100 and 18÷120×100 are both 15 g per 100 g. A larger total in a larger packet does not establish a larger concentration.
Compare your line of reasoning with the solution, not only your final value. If the answer is numerically the same but the units, variable meanings or inequality directions are missing, improve the working before moving on. An examiner can award marks for visible correct mathematical steps but cannot infer reasoning you never wrote.
Question 2
Prompt. What is the minimum number of B packets if exactly 12 packets must contain at least 56 g labelled protein?
Solution. Substitute x=12−y into 4x+5y≥56. This simplifies to 48+y≥56 and therefore y≥8. Eight B packets is the minimum under the stated model.
Compare your line of reasoning with the solution, not only your final value. If the answer is numerically the same but the units, variable meanings or inequality directions are missing, improve the working before moving on. An examiner can award marks for visible correct mathematical steps but cannot infer reasoning you never wrote.
Question 3
Prompt. Explain why an original budget of $33 is incompatible with that minimum.
Solution. The budget gives 28.80+0.60y≤33, so y≤7. Protein gives y≥8. Their intersection is empty. The contradiction should be stated, not concealed with rounding.
Compare your line of reasoning with the solution, not only your final value. If the answer is numerically the same but the units, variable meanings or inequality directions are missing, improve the working before moving on. An examiner can award marks for visible correct mathematical steps but cannot infer reasoning you never wrote.
Question 4
Prompt. Verify a revised order of four A and eight B at a $33.60 ceiling.
Solution. It contains twelve packets, costs 4×2.40+8×3.00=$33.60, provides 4×4+8×5=56 g labelled protein, and weighs 4×80+8×120=1,280 g. It meets the specified constraints exactly.
Compare your line of reasoning with the solution, not only your final value. If the answer is numerically the same but the units, variable meanings or inequality directions are missing, improve the working before moving on. An examiner can award marks for visible correct mathematical steps but cannot infer reasoning you never wrote.
Question 5
Prompt. B increases in price by ten cents. How does that affect the revised order?
Solution. Eight B packets add 8×$0.10=$0.80, so the order costs $34.40. The $33.60 ceiling fails. An accurate conclusion records the sensitivity to price instead of treating a previous solution as permanent.
Compare your line of reasoning with the solution, not only your final value. If the answer is numerically the same but the units, variable meanings or inequality directions are missing, improve the working before moving on. An examiner can award marks for visible correct mathematical steps but cannot infer reasoning you never wrote.
The seven-day Mathematics route
Day one: convert all figures into the same units. Day two: compare totals and rates and explain why they can rank products differently. Day three: define variables, write the exact-total equation and form each inequality. Day four: eliminate one unknown and derive the upper and lower bounds. Day five: explain an infeasible result without guessing. Day six: revise one authorised constraint and check the new boundary solution. Day seven: sit a timed problem using different products and prices, then inspect your working for the first mistaken decision. Use the Voyage Series for the PSLE-to-Secondary transition and the Clementi first-principles Mathematics reference for how algebra grows from arithmetic.
What this volume owns
This Mathematics owner teaches the skill of translating label quantities into comparable rates, equations, inequalities and integer decisions. Volume 1353 supplies the biological meaning of digestion; a following English volume will explain how a written notice should disclose conditions and uncertainty; a final EMS review will test whether the three kinds of evidence can be combined without inventing claims. None of those other articles replaces the detailed mathematics here.
