An inventory is a story about change, not a pile of numbers
A school library, a science laboratory and an assembly hall begin the day with different numbers of folders and adhesive labels. Each location receives supplies; each also uses some. By the end of the day, someone must explain how the opening quantities, deliveries and withdrawals fit together. A spreadsheet can show the answer, but mathematics must first decide what each row and column means. Without that decision, even a perfectly performed addition can report the wrong stock.
This chapter is about matrices as a disciplined way to track several related quantities. A matrix is a rectangular arrangement of values whose positions carry meaning. If the rows mean places and the columns mean supplies, every entry describes one supply at one place. That structure lets us update all the stock figures together. Yet it also creates a responsibility: never add numbers that occupy similar-looking positions unless they represent the same kind of quantity.
The previous Mathematics guide explored products of matrices, including how compatible dimensions permit row-by-column multiplication. This chapter owns a different task. We add and subtract aligned stock tables, interpret signed changes, diagnose negative balances and use conservation checks to repair accounting mistakes. The goal is not merely correct arithmetic. It is being able to defend an inventory statement when quantities, dates and labels become complicated.
A careful stock audit is also a useful examination skill because it forces three habits. You must write a model that matches the story, retain the units in every calculation and examine whether the result is possible. These habits also support practical STEM investigations and clear English reporting. If a result says there are minus two physical folders on a shelf, that should prompt a question about the model rather than an attempt to hide the minus sign.
Check the exact G3 SEC Mathematics syllabus and paper format
The first Singapore-Cambridge Secondary Education Certificate examination is in 2027. The official SEAB G3 2027 school-candidate syllabus list identifies Mathematics as code K310, succeeding the old O-Level Mathematics code 4052. The 2027 K310 Mathematics syllabus includes matrix notation, interpretation of matrices of any order, scalar multiplication and problems involving the calculation of matrix sums and products where appropriate. It is the current syllabus, rather than an older code or a tuition worksheet title, that determines the official assessed scope.
The K310 assessment consists of two 2-hour-15-minute written papers, each worth 90 marks and 50 per cent of the subject result. Paper 1 contains short-answer questions, while Paper 2 contains questions of varying length and includes a real-world scenario question. The syllabus may ask learners to interpret a table in context, identify an appropriate operation and show an answer that makes sense. This article uses an original inventory scenario to practise these skills, not an alleged past question or a prediction of the next examination.
Matrix subtraction can be understood by adding the negative of a corresponding matrix. This is a natural extension of matrix sums and scalar multiplication: A minus B is the same as A plus negative one times B when A and B have compatible dimensions. We use that interpretation here, without extending into determinants, inverses or other topics that should not be assumed to belong to this particular syllabus. Students must also check whether their own class is taking G3 Mathematics K310 or a different subject code.
Every numerical example in this chapter is fictional. The school locations and item counts are invented to illustrate operations and checks. The correct mathematical method should survive a completely different story, but the invented figures should never be quoted as real school inventory data.
Start with one labelled stock table
Imagine three school locations arranged as rows in this order: Library, Laboratory, Hall. Every row has two entries. The first column counts folders; the second counts adhesive labels. This is a 3 by 2 matrix, because there are three rows and two columns. The location names are not present in the matrix arithmetic itself, so we must preserve the labels in a written key.
The opening stock matrix S is represented by three row pairs: Library (25 folders, 40 labels), Laboratory (18 folders, 30 labels), Hall (12 folders, 21 labels). The deliveries matrix D uses exactly the same ordering: Library (10, 15), Laboratory (8, 20), Hall (6, 9). The usage matrix U, again in that order, is Library (14, 18), Laboratory (11, 22), Hall (8, 10). Every quantity in these three tables is a count of individual objects; no units of mass or money are mixed in.
The closing stock is F = S + D − U. We combine corresponding positions, not whichever numbers happen to be near one another in a sentence. The Library folders entry is 25 + 10 − 14 = 21. Laboratory labels are 30 + 20 − 22 = 28. Hall folders are 12 + 6 − 8 = 10. Computing all six positions produces Library (21, 37), Laboratory (15, 28), Hall (10, 20).
Before using the six entries, inspect the story. Supplies available at the end must equal what was initially present, plus arrivals, minus documented withdrawals. There is no new quantity created by the algebra. A matrix is simply a convenient way to perform six related conservation statements without losing the location and item categories. If a question includes an item in transit or a delayed delivery, its timing must be stated before it can enter the balance.
Two conditions for legitimate matrix addition
Two matrices can be added entry by entry only when they have the same dimensions. A 3 by 2 matrix can be added to another 3 by 2 matrix, producing a 3 by 2 result. A 3 by 2 matrix cannot be added directly to a 2 by 3 matrix, because their positions cannot be paired one to one in the required order. This is a mathematical rule and should be checked before any arithmetic.
Matching shape is necessary but not always sufficient for a sensible real-world model. Suppose one 3 by 2 table lists folders and labels, while another equally shaped table lists notebooks and rulers. Adding their first columns may be numerically possible, but the total would not mean total folders. The same issue arises when one table places Laboratory first and another places Library first. Mathematical compatibility does not repair mismatched categories or labels.
Time is part of the meaning. Opening stock on Monday plus Monday deliveries produces Monday availability, provided the time definitions are consistent. Adding a final-stock report from Thursday as if it were a Monday delivery double-counts items or mixes periods. A clear question usually provides the needed timing. When a word problem is less clear, state the assumption that makes the operation valid rather than silently inventing a timetable.
Units matter even when every entry is a number. Twenty labels cannot be added to twenty metres of adhesive tape as if they were the same quantity. Ten boxes of folders cannot be treated as ten individual folders unless the number of folders per box is specified. Express both matrices in the same units first or explain why the proposed addition cannot yet be evaluated.
Subtraction means a change, not a forbidden operation
Suppose a location begins with 25 folders and uses 14. The net reduction is 14; the remaining amount is 11 before deliveries. In matrix form, subtracting the usage table means subtracting each matching entry. We can also write the operation as adding a matrix whose entries are the negatives of those usage figures. Either way, the arithmetic must retain the original location and item labels.
The sign of a change has a meaning. A positive net change means an increase in the relevant quantity during the specified period; a negative net change means a decrease. A negative change is not the same as negative stock. If a shelf begins with 25 folders and ends with 21, the change is negative four. The shelf still holds a positive count of 21 folders. Confusing changes with stock levels can make a sound balance sheet appear impossible.
Addition of aligned matrices is commutative: S + D and D + S produce the same numerical totals, although the real-world language still calls one opening stock and the other deliveries. Subtraction is not commutative: S − U and U − S generally produce opposite signs. Ordering matters because it reflects the direction of change. A practical answer should name the quantities before calculating so the sign is not determined by guesswork.
An inventory audit also introduces a useful distinction between recorded, predicted and counted final stock. The predicted figure follows the algebra from the records. The counted figure comes from an actual physical check. Their difference may flag a missing transaction, counting error, damage, delayed delivery or inaccurate record. The difference alone does not prove which event occurred.
Check totals from two directions
The six entries of the closing matrix can be checked by column totals. At opening, the locations together hold 25 + 18 + 12 = 55 folders and 40 + 30 + 21 = 91 labels. Deliveries add 10 + 8 + 6 = 24 folders and 15 + 20 + 9 = 44 labels. Usage removes 14 + 11 + 8 = 33 folders and 18 + 22 + 10 = 50 labels. The predicted closing totals are 55 + 24 − 33 = 46 folders and 91 + 44 − 50 = 85 labels.
The entry-by-entry final matrix confirms this: 21 + 15 + 10 = 46 folders; 37 + 28 + 20 = 85 labels. The agreement between row-level calculation and column-level totals is a reconciliation check. It does not prove every record is true, because two matching errors could still cancel, but it catches many ordinary mistakes and identifies where to inspect when totals disagree.
Rows can also be checked, but interpret any combined total carefully. Library has 21 folders and 37 labels; there are 58 individual items if both types are counted as separate objects. That combined count is not 58 folders, 58 sets or 58 equally useful supplies. If the question asks for a particular resource, report the relevant column. The ease of adding quantities must not erase the categories that give the answer meaning.
Transfers offer another test. If four folders move from Laboratory to Library, Library increases by four and Laboratory decreases by four. The total number of folders across all three locations stays 46. A transfer matrix has a positive entry at the receiving location and a matching negative entry at the sending location. A report in which the overall count rises by four from a mere transfer has treated movement as creation.
Reverse the balance to find what is missing
Forward questions give opening stock, arrivals and withdrawals, then ask for the finish. Reverse questions hide one part of that chain. If the Laboratory finished with 15 folders, began with 18 and received eight, it must have recorded 18 + 8 − 15 = 11 folders used. The same conclusion follows from rearranging the identity F = S + D − U to U = S + D − F. Each entry remains tied to its corresponding location and supply.
To find an unknown delivery, use D = F − S + U. If a storeroom began with 18 folders, used 11 and finished with 22, then deliveries must have been 22 − 18 + 11 = 15 folders. Notice how the rearrangement expresses a real story: to finish with 22 after using 11, there must have been 33 available; 18 were already present, so 15 arrived. An algebraic solution becomes easier to trust when it can be retold without algebra.
Some calculations produce a negative missing delivery or negative usage. That may mean the assumed transactions cannot all be nonnegative, that a return or transfer was omitted, or that the given figures are inconsistent. A well-written answer distinguishes an impossible interpretation from a valid signed adjustment. In a forecast, negative ending stock describes an unmet requirement or shortage rather than physical negative objects.
Reverse checks do not authorise a student to invent the cause of a discrepancy. If the predicted closing stock is 15 and the physical count is 13, the difference is minus two. The record indicates two fewer items than expected, but it does not show whether two were used without logging, misplaced, miscounted or stolen. A responsible report records the discrepancy and identifies a follow-up check.
Forty original worked decisions: from matrix foundations to examination judgement
1. Read the axes before the entries
Question. A 3 by 2 table has locations as rows and item categories as columns. Which entry describes Laboratory labels?
Reasoned answer. The second row, second column is Laboratory labels. In the opening stock table, its value is 30. The first row belongs to Library and the second column to labels, so neither coordinate can be omitted when describing a number.
Examination check. An answer of simply 30 is incomplete if the question asks for interpretation. Say 30 adhesive labels in the Laboratory at opening.
2. Check the order
Question. A student calls the opening table a 2 by 3 matrix because it contains two kinds of supplies and three locations.
Reasoned answer. The order is rows by columns. It has three rows, one for each location, and two columns, one for each supply, so its order is 3 by 2. The count of entries, six, is not itself the matrix order.
Examination check. When looking at an unfamiliar arrangement, count horizontal rows first and vertical columns second.
3. Add the Library folders
Question. Library starts with 25 folders and receives 10. What is the intermediate figure before any usage?
Reasoned answer. The available quantity is 25 + 10 = 35 folders. This is the first entry of S + D, and it still belongs to the Library and the folders category. It is not the closing balance because documented withdrawals have not yet been subtracted.
Examination check. Write the stage beside the answer: opening, available or closing. All three can be correct numbers for different questions.
4. Subtract Library usage
Question. The Library has 35 folders available and uses 14. Find the closing figure.
Reasoned answer. The closing quantity is 35 − 14 = 21 folders. In matrix terms the first entry of S + D − U is 21. A final check reconstructs the available stock: 21 left plus 14 used equals 35 available.
Examination check. A subtraction question should be checked against the conservation story, not merely with a calculator.
5. The Laboratory labels entry
Question. Laboratory labels begin at 30, receive 20 and use 22. Find the closing number.
Reasoned answer. The result is 30 + 20 − 22 = 28 labels. This is the second-row, second-column entry of the closing 3 by 2 matrix. It must not be placed under folders because 28 happens to sit near an unrelated folder value.
Examination check. After arithmetic, preserve the row and column label in the reported result.
6. Hall folders
Question. Hall folders begin at 12, receive six and use eight. What remains?
Reasoned answer. The balance is 12 + 6 − 8 = 10 folders. Although the other locations have larger opening stocks, the same element-by-element rule applies. No special calculation is needed because a number is small.
Examination check. Use one general model across all entries. Changing the operation from row to row without a stated reason is a modelling error.
7. Hall labels
Question. Hall labels begin at 21, receive nine and use ten. Calculate the balance.
Reasoned answer. The Hall has 21 + 9 − 10 = 20 labels at closing. Check 20 + 10 = 30, which matches 21 + 9. The check is independent of the wording used to introduce the question.
Examination check. Backward arithmetic is a quick way to detect transposed entries and accidental sign changes.
8. A full predicted table
Question. List the closing figures for all three locations after deliveries and usage.
Reasoned answer. The closing rows in order are Library (21 folders, 37 labels), Laboratory (15 folders, 28 labels), Hall (10 folders, 20 labels). This is a 3 by 2 matrix. The answer should retain a heading or key for its two supply columns.
Examination check. A matrix of six unlabeled numbers may be arithmetically correct yet difficult for a reader to use.
9. A 3 by 2 plus a 2 by 3
Question. Someone wants to add a three-location, two-item table to a two-location, three-item table.
Reasoned answer. The addition is undefined as written because the orders differ. A valid matrix sum needs matching row and column counts. Rearranging a table may make two shapes appear equal, but the category alignment must then be checked separately.
Examination check. Do not add entries simply because both tables contain six numbers.
10. A 3 by 2 plus a 3 by 2
Question. Opening stock and deliveries each have three location rows and two supply columns in the same order.
Reasoned answer. The sum is defined and meaningful. Its rows are Library (35, 55), Laboratory (26, 50), Hall (18, 30). Each pair counts the number available before usage at that specific location.
Examination check. Shape, matching labels and a common point in time make this addition defensible.
11. Same shape, wrong items
Question. A second 3 by 2 table lists three locations but columns are rulers and notebooks instead of folders and labels.
Reasoned answer. Its shape matches the stock table, but adding corresponding columns would mix unlike supplies. The result cannot be interpreted as a folders-and-labels inventory. You would need a common category scheme or a different reporting question.
Examination check. Algebraic permission does not create a physical unit that the context has not defined.
12. Same categories, reversed positions
Question. A supplier submits two columns labelled labels, then folders, while the school’s table uses folders, then labels.
Reasoned answer. Before addition, reorder the supplier values to match the school columns, keeping each value attached to its original label. Otherwise, the first-column numbers look addable but combine folders with labels. Record the corrected order explicitly.
Examination check. Matrix positions are not meaning-free. The column key is part of the data.
13. A missing row
Question. The deliveries table omits Hall entirely. May the two visible rows simply be added to the three-row stock table?
Reasoned answer. Not directly: the shapes are 2 by 2 and 3 by 2. First determine whether Hall received zero items or whether the record is incomplete. A missing row is not automatically a zero-delivery row. Only after resolving the factual issue can the tables be aligned.
Examination check. Zero is a known quantity; blank or missing is unknown. They should not be silently interchanged.
14. One delivery reported twice
Question. Library’s ten delivered folders were entered twice in D. What would happen?
Reasoned answer. The recorded closing stock would be overestimated by ten folders if all other figures stayed unchanged. The predicted Library folder count would become 31 instead of 21. A physical count could reveal the discrepancy but would not independently identify its cause.
Examination check. Correct records, not the arithmetic rule, when a transaction has been duplicated.
15. A negative change is ordinary
Question. Library folders move from 25 at opening to 21 at closing. Express the net change.
Reasoned answer. The net change is 21 − 25 = −4 folders. That means a decrease of four during the period, not a shelf containing negative four folders. It also matches ten delivered minus fourteen used, which is minus four.
Examination check. Always attach a word such as change, difference or balance to a signed number.
16. Subtraction order
Question. Compare final minus opening with opening minus final for Library folders.
Reasoned answer. Final minus opening is 21 − 25 = −4, showing a decrease. Opening minus final is 25 − 21 = +4, showing how much the opening exceeds the finish. Both arithmetic statements are true but answer different questions.
Examination check. A subtraction sign does not have a fixed real-world meaning until the order of the terms is specified.
17. Add a negative matrix
Question. A student is familiar with addition and scalar multiplication but not a separate rule for matrix subtraction.
Reasoned answer. Write S + D − U as S + D + (−1)U. Multiplying every usage entry by negative one and adding corresponding entries yields the same closing stock. No matrix inverse is involved in this calculation.
Examination check. This connects a new-looking operation with the assessed ideas of matrix sums and scalar multiplication.
18. Compare two totals
Question. Opening folders across all locations total 55; 24 are delivered and 33 used. What should the final total be?
Reasoned answer. The final total must be 55 + 24 − 33 = 46 folders. Summing the corresponding final matrix column yields 21 + 15 + 10 = 46. Agreement between these methods is a useful reconciliation check.
Examination check. When totals disagree, inspect the data entries and signs rather than choosing whichever total seems more plausible.
19. A different column total
Question. There are 91 labels at opening, 44 arrive and 50 are used. Find the closing total.
Reasoned answer. The final total is 91 + 44 − 50 = 85 labels. The closing matrix gives 37 + 28 + 20 = 85. The two independently organised calculations support the arithmetic but do not guarantee that the underlying transaction records are true.
Examination check. A balance check tests consistency; it does not certify evidence that was never verified.
20. Adding unlike counts
Question. The Library has 21 folders and 37 labels after closing. Is the total 58 folders?
Reasoned answer. No. The combined count is 58 individual objects only if we deliberately count both categories as objects. The number of folders remains 21. In an order for folders, adding labels would lead to the wrong purchase requirement.
Examination check. The arithmetic sum of unlike items can have a limited meaning but must not be renamed as one category.
21. A location transfer
Question. Four folders move from Laboratory to Library after the ordinary stock updates. What are the new location counts?
Reasoned answer. Library folders become 21 + 4 = 25; Laboratory folders become 15 − 4 = 11; Hall folders remain 10. The combined total is still 25 + 11 + 10 = 46. The transfer changes where supplies are stored, not how many exist.
Examination check. A valid transfer adds and removes the same number of the same item in the same accounting interval.
22. The transfer that creates stock
Question. A record adds four folders to Library but fails to subtract them from Laboratory.
Reasoned answer. The overall folder count appears to rise from 46 to 50, even though no delivery occurred. The issue is missing transfer bookkeeping. The record should show a positive four in the receiving row and negative four in the sending row.
Examination check. A conservation check can expose the missing half of a movement even without knowing who made the error.
23. The return that reverses usage
Question. Hall previously recorded six folders as used, but two unused folders are later returned during the same interval.
Reasoned answer. If the returns legitimately reverse two of the recorded withdrawals, net usage is four folders for the interval, not six. The stock balance rises by two compared with the previous record. Document the return separately rather than pretending that the original event never happened.
Examination check. A change to a record should preserve a traceable reason and its timing.
24. From finish back to usage
Question. Laboratory begins with 18 folders, receives eight and finishes with 15. How many were used?
Reasoned answer. Usage is opening plus deliveries minus closing: 18 + 8 − 15 = 11 folders. The result is nonnegative and can be verified by 18 + 8 − 11 = 15. It is the reverse of the same stock identity used in the forward question.
Examination check. When one term is missing, rearrange the entire balance before entering numbers.
25. From finish back to deliveries
Question. A storeroom starts with 18 folders, uses eleven and ends with 22. Find deliveries.
Reasoned answer. Deliveries are finish minus opening plus usage: 22 − 18 + 11 = 15 folders. The reconstructed supply is 18 + 15 = 33, of which 11 were used, leaving 22. A one-line reverse explanation can make the result easy to audit.
Examination check. Reverse reasoning is more reliable when it tells the same physical story as the forward equation.
26. A counted difference
Question. Laboratory’s predicted final stock is 15 folders, but a physical count finds 13. State the discrepancy.
Reasoned answer. Define discrepancy as counted minus predicted. It equals 13 − 15 = −2 folders, so the shelf has two fewer folders than the record predicts. The number does not prove whether usage was missed, delivery was delayed or counting was inaccurate.
Examination check. A discrepancy identifies a question to investigate; it does not identify an offender or cause.
27. Another discrepancy definition
Question. A second student computes predicted minus counted and obtains positive two. Has the arithmetic become wrong?
Reasoned answer. Not necessarily. The difference is +2 under that stated subtraction order. It still means the predicted count exceeds the physical count by two. The answer should define its sign convention so readers do not confuse a shortage with a surplus.
Examination check. Sign conventions may differ; physical interpretation must remain consistent.
28. A forecast below zero
Question. A store has five labels, expects two deliveries and plans to use nine. The forecast closing entry is minus two.
Reasoned answer. The arithmetic 5 + 2 − 9 = −2 signals a shortage of two labels if the plan is followed without changing arrivals or usage. A negative forecast is useful as a warning, but no physical shelf can contain minus two labels. Supply or demand must be adjusted.
Examination check. Do not round a negative balance up to zero and pretend the original plan is feasible.
29. An impossible completed record
Question. The same store states it physically used nine labels after having seven available and no transfers or extra arrivals.
Reasoned answer. The three claims cannot all be true under the defined system. There is a missing inflow, timing difference, measurement error or incorrect usage record. The appropriate conclusion is inconsistency, not a story about which person made a mistake.
Examination check. Mathematics can diagnose contradiction without supplying an unsupported explanation.
30. A reorder threshold
Question. The school requires at least twelve folders at each location at closing. Which location misses this rule?
Reasoned answer. Library has 21 and Laboratory has 15, both meeting the threshold. Hall has 10 folders and falls short by two. If a delivery before the next check is allowed, at least two more folders would meet the numerical threshold, assuming no further use.
Examination check. A threshold comparison is element-specific; the combined total of 46 cannot repair Hall’s local shortage.
31. Threshold applied to wrong column
Question. A student checks Hall’s 20 labels against the twelve-folder minimum and declares the threshold satisfied.
Reasoned answer. The threshold specifically concerns folders. Hall has only ten folders, so the condition fails. Checking the labels column is a category error even though the displayed number 20 is accurate.
Examination check. This is why the units and row-column key must stay visible until the decision is made.
32. A box is not one folder
Question. One entry records three boxes of folders; another records twelve individual folders. May they be added as fifteen folders?
Reasoned answer. Not without knowing how many folders are in each box. If each box held 20 folders, the first quantity would represent 60 folders and the combined total would be 72. The conversion is conditional, not an assumption hidden inside a matrix.
Examination check. Convert units before aligning entries. Equal shape does not mean equal unit size.
33. A count versus a mass
Question. One table counts labels; another measures adhesive material in grams. Both have a column headed labels.
Reasoned answer. The heading is insufficient. Counts and masses are different units, and a value of 30 labels cannot simply be added to 30 grams. The question would need a mass per label or a different modelling decision.
Examination check. The unit label belongs to the number. Do not drop it when moving data into a table.
34. A changing reporting date
Question. An opening matrix comes from Monday morning while usage entries cover Monday and Tuesday. Can it be combined with Monday-only deliveries?
Reasoned answer. That combination spans inconsistent periods and can omit Tuesday arrivals. First define a consistent start and end time for all recorded flows and adjust the transactions accordingly. The arithmetic of aligned matrices cannot correct a missing date condition.
Examination check. Date alignment is as important in real reporting as row and column alignment.
35. An order of operations that works
Question. Compare (S + D) − U with S + (D − U) for the same stock scenario.
Reasoned answer. Both yield the same closing matrix because each entry follows ordinary addition and subtraction. Grouping helps calculate but does not change the interpretation of the flow, provided every matrix has the same order and categories. Write the signs carefully when regrouping.
Examination check. This is a legitimate way to organise a longer calculation, unlike changing which entries are paired.
36. Addition can commute
Question. Compare S + D with D + S as mathematical expressions.
Reasoned answer. They have the same numerical result because corresponding entries are added as real numbers, and addition is commutative. Still, a narrative should describe which matrix is opening stock and which represents deliveries; swapping words carelessly can confuse the accounting period.
Examination check. Mathematical symmetry does not erase contextual roles or time labels.
37. Do not infer multiplication
Question. A learner sees two matrices and automatically tries row-by-column multiplication because the previous volume covered it.
Reasoned answer. The question asks for combined stock at matching locations, so addition is the relevant operation. Multiplication is useful when quantities are paired with unit prices or other transformations, but it would not represent adding two batches of the same supplies here.
Examination check. Choose an operation from the relationship in the question, not from the most recently memorised technique.
38. A shortage hidden by a total
Question. The final folder total is 46, but Hall has only ten against a twelve-folder minimum.
Reasoned answer. A healthy total does not imply that every location meets the local requirement. The matrix preserves distribution information that a single total hides. Transfer two folders from a location with a surplus if permitted, while updating both source and destination entries.
Examination check. A global average or total cannot certify every local constraint.
39. Two simultaneous changes
Question. A revised plan delivers four more Hall folders but also records three extra Hall folder withdrawals.
Reasoned answer. Starting from the predicted Hall closing 10, adding four and subtracting three gives 11, still one below the required twelve. The revised plan changes the net balance by positive one, not positive four. Both changes must be included.
Examination check. When scenarios change, apply every named adjustment and retest the original condition.
40. Misplaced negative sign
Question. A candidate writes S − D + U for a stock balance and gets Library folders 25 − 10 + 14 = 29.
Reasoned answer. This counts arrivals as if they were withdrawals and treats usage as an addition. The correct expression is S + D − U, yielding 21 folders. To repair the mistake, retell the story: deliveries increase available stock; usage decreases it.
Examination check. A sign check tied to ordinary language is more reliable than visual inspection of the formula alone.
41. An error in one cell
Question. The final table shows Library labels as 73 even though the original entries are 40, 15 and 18.
Reasoned answer. The correct result is 40 + 15 − 18 = 37 labels. A transposed digit can be caught by reconstructing the available 55 and checking the used 18. The other five closing entries need not be changed if they already reconcile.
Examination check. Locate the precise incorrect cell rather than recalculating every unrelated entry blindly.
42. A final written explanation
Question. A Paper 2 response asks why the inventory model reports the Hall as needing attention.
Reasoned answer. A complete explanation might read: ‘The predicted Hall closing stock is ten folders after starting with twelve, receiving six and using eight. Because the requirement is at least twelve folders at every location, Hall is two below the minimum. The other locations meet it, so Hall requires a local supply adjustment.’
Examination check. An examinable answer combines calculation, the relevant rule and a conclusion about the stated context.
43. A two-minute checking routine
Question. A learner has finished a multi-row matrix stock problem and has little time left.
Reasoned answer. Check the order of each matrix, verify row and column labels, inspect whether operations match the story, recompute one edge entry, compare column totals and test any negative or threshold result. These checks are short enough to perform deliberately and cover most conceptual error categories.
Examination check. The aim is not to redo every line. It is to find the kind of mistake that ordinary arithmetic checking often misses.
A complete audit you can redo without notes
Read the original 3 by 2 tables again and rewrite them in your notebook with locations down the left and item names above the columns. Do not calculate immediately. First underline which figures are stock levels and which represent flows. Then form the total available matrix S + D. Next subtract U to find closing stock, and write the closing table with units. Use the column totals as a separate check rather than copying the total shown earlier.
Now imagine the school performs a physical count and finds Laboratory labels at 27 rather than the predicted 28, with all other counts exactly as expected. Define discrepancy as observed minus predicted. Only the Laboratory labels position in the discrepancy matrix should read negative one; other positions should read zero. The entry identifies a one-label discrepancy, not an established explanation for that discrepancy. A good mathematical report does not invent a delivery, accusation or loss event without evidence.
Finally, apply the rule that each location must retain at least twelve folders and at least twenty labels. Library finishes with 21 folders and 37 labels, meeting both; Laboratory finishes with 15 and 28, also meeting both; Hall finishes with 10 folders and 20 labels, missing the folder threshold by two while meeting the label threshold exactly. The answer is not ‘the school has enough supplies overall’. It is ‘the Hall folder condition is not met’.
If a teacher changes the data, do not memorise the conclusion. Repeat the modelling process. A new delivery changes one or more cells, not the entire logic. A switch from pieces to boxes adds a unit-conversion step. A new location adds a row. A new supply category adds a column. These are predictable adjustments because the model is grounded in what rows, columns and signs mean.
Practise progressively rather than repeatedly
Begin with six single-entry questions in which one opening quantity, one delivery and one usage figure are given. Practise saying the unit and location in every answer. Next, move to two complete matrices whose rows and columns are already aligned. Only then introduce a table with reordered labels, an omitted entry or a new accounting period. If the easy stage is still error-prone, the harder stage will mostly test your ability to survive confusing notation rather than your mathematical understanding.
Once the entries are reliable, practise one reverse question and one discrepancy question. For reverse questions, isolate the unknown term symbolically before substituting values. For discrepancies, define which figure is subtracted from which. Then practise a threshold scenario: identify which location violates a requirement and by how many units. This sequence builds from basic operations to interpretation, evidence and decisions.
Timed practice should reflect the examination demand, not become a race to perform the most arithmetic. Give yourself eight minutes to model, compute, reconcile and explain the school inventory. Afterward, mark each error as an arithmetic slip, matrix-shape error, category mismatch, sign error, timing problem or unsupported conclusion. The repair is different for each cause. Read the reason aloud once, write a new example and try again without a worked answer.
If your answer is correct but you cannot explain which location and item the number describes, revisit the introductory matrix representation. If you understand the representation but your sums are unreliable, practise column totals. If your numbers are right but your conclusions overstate what a discrepancy proves, revisit the distinction between computed and observed counts. The best way to improve is to solve the cause of the error you actually make.
Connected learning: earlier volumes, PSLE transfer and official references
For matrix product reasoning, dimensions and non-commutative multiplication, return to Volume 0302. The foundational owner of reading rows, columns and scalar multiplication is Volume 0166. For a real-world calculation with prices, quantities and product claims, see Volume 0304 on a fictional sports-bag audit. It uses multiplication for component costs; this volume uses sums and changes for stock reconciliation.
For clear reporting of quantifiers such as all, some and at least, Volume 0305 explains why a mathematically correct number can still be misstated. The earlier PSLE guide on keeping every per attached to its dividing quantity trains the same respect for units that makes a stock matrix meaningful. The PSLE-to-secondary learning route provides a bridge for newly promoted learners.
For subject-specific support, use the G3 Mathematics learning owner or the Secondary 1 Mathematics foundation reference. Examination facts come from the official 2027 SEC G3 syllabus list and K310 Mathematics syllabus PDF. Check those sources when planning the subject and paper structure; the invented stock case is teaching material rather than an official examination question.
