The first rule: decide what one unit represents
Twenty-four labelled water samples are checked for three possible anions each. The laboratory sheet contains seventy-two individual test results, including twenty-one recorded positives. A student sees the number twenty-one and writes that twenty-one water samples were positive. This feels like ordinary arithmetic, but it is wrong. A single sample can contribute more than one positive test. The number of rows in a test log is not necessarily the number of separate parent samples.
This volume develops the Mathematics required to prevent that error. We will count overlapping categories, compare groups only with their proper denominators, distinguish an absolute difference from a percentage change, and express what remains unknown rather than inventing a missing cell. It follows the Chemistry investigation in Volume 0593, where chloride, sulfate and carbonate were treated as separate evidence questions. It does not repeat the chemical procedures. Its job is to turn a complex record into defensible numerical statements. By the end, you should be able to explain why a perfectly calculated percentage can still mislead when its unit has been chosen incorrectly.
The 2027 SEC Mathematics context
Singapore’s first Singapore–Cambridge Secondary Education Certificate examinations begin in 2027. The official school-candidate list identifies G3 Mathematics as K310, with 4052 given as the reference code for 2026 and earlier. K310 has two equally weighted papers, each lasting two hours fifteen minutes and carrying ninety marks. The final question of Paper 2 is explicitly described as focusing on a real-world application, so interpreting a word problem is part of the assessment rather than an optional extra.
This is a constructed revision case, not a forecast examination question, a specimen paper or a claim that every G3 student studies laboratory statistics as a separate official topic. The relevant mathematical skills include ratios, percentages, tabulation, set notation, data interpretation, and explanations of claims. Workings must show what is counted and why. The official syllabus also warns that omission of essential working loses marks. That practical instruction is worth remembering: show the total and the relevant subgroup before you write a proportion.
For the scientific meaning of a recorded positive, begin with Volume 0593 on controlled anion tests. For the previous water-project arithmetic, see Volume 0590 on volume and bottle feasibility. Earlier Volume 0578 develops two-way tables in a different context.
The full fictional record, clearly labelled
Our hypothetical school team has twelve untreated model samples labelled U01 to U12 and twelve recovered model samples labelled R01 to R12. The two groups consist of twenty-four distinct parent samples. Three tests were recorded for each: chloride, sulfate and carbonate. All seventy-two entries are marked positive or negative for this main example. The chloride column contains nine positives in U and two in R. The sulfate column contains five positives in U and one in R. The carbonate column contains three positives in U and one in R.
The totals are therefore eleven positive chloride checks, six positive sulfate checks and four positive carbonate checks. Adding those values gives twenty-one positive test outcomes, not twenty-one positive parent samples. In the main fictional table, the remaining fifty-one outcomes are recorded negative. The numbers are not measurements of actual Sengkang water, products, medical samples or drinking-water quality. Treat them as a numerical case in which the test targets were supplied by Chemistry and the challenge is to preserve exactly what each record counts.
For subject scope and format, consult SEAB’s 2027 G3 subject listing and the official K310 Mathematics syllabus. For a useful transition from Primary Mathematics, read the PSLE word-problem guide and Secondary Mathematics error prediction.
A complete two-way table for the first two tests
A second sheet describes whether each parent sample was positive in the chloride and sulfate checks. Among the twelve untreated samples, four were positive in both, five were positive in chloride only, one was positive in sulfate only and two were negative in both. The four cell counts add to twelve: 4 + 5 + 1 + 2 = 12. They also reproduce the margins: chloride positives are 4 + 5 = 9 and sulfate positives are 4 + 1 = 5.
Among the twelve recovered samples, one was positive in both, one was positive in chloride only, zero were positive in sulfate only and ten were negative in both. Again, 1 + 1 + 0 + 10 = 12. Chloride positives are 1 + 1 = 2; sulfate positives are 1 + 0 = 1. When you combine the groups, the four overall cells are five both, six chloride-only, one sulfate-only and twelve neither. That is a legitimate table because its rows, columns and grand total reconcile.
Use a table to check every margin
The strongest way to read a two-way table is to reconstruct the row and column totals yourself. Across both groups, the cell “both positive” contains five samples. The chloride-only cell contains six. The sulfate-only cell contains one. The neither-positive cell contains twelve. The grand total is 5 + 6 + 1 + 12 = 24. The chloride margin is 5 + 6 = 11, and the sulfate margin is 5 + 1 = 6. These match the separate test totals given in the original record.
If the table instead showed a chloride total of twelve, stop rather than forcing an answer. A mismatch can signal copying, transcription, category or arithmetic error. The calculations are not difficult; the discipline is to use every available relationship as a check. A student who verifies margins before solving an unfamiliar application question has a better chance of catching a mistaken denominator or a double-counted category before it affects the final explanation.
At least one of two positive results: count the union
How many distinct parent samples were positive for chloride or sulfate? The English word or usually signals the union in this context: all samples that meet at least one of the stated conditions, including those that meet both. The chloride total is eleven. The sulfate total is six. Five parent samples appear in both categories. Therefore the number in the union is 11 + 6 − 5 = 12. The subtraction removes the extra count of the five samples included twice.
This result also follows directly from the table: both five, chloride-only six and sulfate-only one, giving twelve. It does not mean twelve samples were necessarily positive for carbonate, and it does not say that the remaining twelve were chemically pure. The remaining twelve were negative for the specified chloride and sulfate tests under the fictional testing record. A correct numerical answer preserves that description. It counts membership in defined sets, not the health, safety or complete composition of a liquid.
The Venn diagram is a reasoning tool
Draw two overlapping circles within a rectangle representing all twenty-four parent samples. Label the circles C for chloride positive and S for sulfate positive. Place five in the overlap. Put six in the C-only region and one in the S-only region. Put twelve outside both circles. If you sum all four regions, you must recover the universal total twenty-four. That check is more informative than drawing attractive circles and writing random numbers in them.
The Venn diagram becomes valuable when the question changes wording. “Chloride but not sulfate” is the C-only region, six. “Sulfate and chloride” is the intersection, five. “Neither” is twelve. “At least one” is twelve. “Exactly one” is six plus one, or seven. “Not chloride” is sulfate-only one plus neither twelve, or thirteen. Translate each phrase into a set region before calculating. The diagram helps a reader see that one sample can occupy an overlap while remaining exactly one member of the universal set.
Why twenty-one divided by twenty-four is a wrong fraction
A student may compute 21 ÷ 24 × 100% and obtain 87.5%. The arithmetic is accurate. The statement “87.5% of samples were positive” is not supported. Twenty-one is the count of positive test outcomes across three checks per parent sample. Twenty-four is the count of parent samples. They are different units. Dividing them describes the number of positive test outcomes per parent sample multiplied by one hundred, not the percentage of samples meeting a single well-defined criterion.
If the question asks what fraction of all completed checks were recorded positive, the valid denominator is seventy-two. The fraction is 21/72 = 7/24, or approximately 29.2%. The label must be “percentage of the seventy-two recorded checks that were positive,” not “percentage of samples containing the ions.” The latter introduces an untested claim about individual samples. Before writing a percentage, ask whether the numerator and denominator refer to the same type of counted item.
Conditional percentages use the subgroup that was asked about
What percentage of untreated samples recorded a positive chloride check? The numerator is nine untreated samples with a positive chloride result. The denominator is twelve untreated samples. The calculation is 9/12 × 100% = 75%. For recovered samples, it is 2/12 × 100% = 16.666…, or approximately 16.7%. The first denominator is not twenty-four because the question explicitly asks about the untreated subgroup. The second denominator is not eleven because eleven is the total chloride-positive count across both groups.
A useful test is to say the sentence out loud before calculating: “Of the twelve untreated samples, nine recorded a positive chloride check.” The percentage then follows naturally. Change the wording to “Of all eleven chloride-positive samples, how many were untreated?” and the base changes to eleven: 9/11 × 100%, approximately 81.8%. Both figures are arithmetically correct for different questions. The most serious percentage errors often arise from answering a different question than the one asked.
Percentage points are not percentage change
The untreated chloride-positive rate is 75% and the recovered rate is approximately 16.7%. Their difference is approximately 58.3 percentage points. This is a subtraction of two percentages that share the same meaning but refer to different groups. It does not become “58.3% reduction” simply because the numbers include percentage signs. A relative decrease from the untreated rate, if the question explicitly asks for it, is calculated as (75 − 16.666…)/75 × 100%, approximately 77.8%.
The difference matters in public writing. “58.3 percentage points lower” and “77.8% lower relative to the initial rate” are not interchangeable. In this hypothetical case, even a correctly computed relative difference does not establish treatment effectiveness. The groups are not stated to be matched before-and-after versions of the same bottles, and no controlled assignment or uncertainty assessment is supplied. Mathematics can describe differences in recorded proportions; it cannot invent a causal relationship the sampling design does not provide.
Do not treat different groups as paired observations
Twelve untreated samples and twelve recovered samples may sound like a before-and-after study, particularly because one group name suggests an operation. But pairing is a property of how the observations were collected, not a feature of their labels. If the question does not identify U01 as the actual earlier state of R01, you cannot calculate twelve individual improvements by subtracting one reading from another. There is no stated pairing in our example.
This is a subtle examination skill. When a table appears to invite a comparison, first ask whether it is comparing independent groups, matched individuals, different times or different test types. A question might later supply the missing pairing, in which case paired differences become legitimate. Until then, a statement such as “the recovered group had fewer recorded chloride positives in these samples” is defensible. “The treatment removed chloride from ten of the original bottles” is not. The latter would require sample-level transition records that the fictional file does not contain.
Three-test totals do not reveal the triple overlap
We know eleven chloride-positive, six sulfate-positive and four carbonate-positive checks. We also know the exact overlap between chloride and sulfate. But we do not know which individual samples were positive for carbonate. Some carbonate-positive samples may already be in the chloride-or-sulfate union, and some may lie outside it. Therefore the number of distinct samples positive for at least one of the three checks cannot be calculated exactly from the margins given.
The correct response is to state that information is insufficient, then identify which missing information would resolve the question: the membership of the carbonate-positive cases relative to the other two tests. Do not assume independence, equal overlap or no overlap merely because a question has asked for a number. In a real-world application, discovering that a count is underdetermined is a valid mathematical conclusion. It is better than a confident but invented sum of twenty-one.
Find defensible bounds rather than invent a point estimate
Can we still find a useful range for the number of samples positive in at least one of the three checks? Yes. First, chloride or sulfate already covers twelve distinct parent samples. If all four carbonate-positive cases belong to those twelve, the three-test union remains twelve. At the opposite extreme, we might hope to add four new distinct samples, giving sixteen. But the group-level records let us improve that upper bound.
In the untreated group, chloride or sulfate already covers ten of twelve, leaving only two outside both; the three carbonate-positive cases can add at most two new untreated samples. In the recovered group, chloride or sulfate covers two of twelve, leaving ten outside; one carbonate-positive case can add at most one more. Therefore the total three-test union lies between 12 and 15 inclusive. This is a sharper bound than 12 to 16. The result follows from capacity constraints in the subgroups, not from guessing where a carbonate-positive sample might be. It is a rewarding advanced extension of ordinary Venn-diagram work.
Check that the extreme bounds are actually possible
A bound is useful only if it follows from the constraints; it is even better when you can demonstrate a configuration attaining each extreme. For the minimum of twelve, place all three untreated carbonate positives among the ten untreated samples already positive for chloride or sulfate. Place the one recovered carbonate positive among the two recovered samples already in the chloride-or-sulfate union. No new parent sample is added, so the union remains twelve.
For the maximum of fifteen, place two of the untreated carbonate positives among the two untreated samples outside the chloride-or-sulfate union, and the third among any of the ten already inside. Place the recovered carbonate positive among the ten recovered samples outside the first two-test union. This adds three new distinct parent samples. The resulting union is fifteen. Both configurations satisfy every group total. Constructing them is stronger than saying that the answer is somewhere in a range without checking feasibility.
A missing result is neither positive nor negative
The main case has seventy-two completed checks. Now change the question deliberately. Suppose three of those entries were not completed, and each is marked “not tested.” The number of completed checks is now sixty-nine. The unknown entries must be excluded from the count of positive and negative recorded outcomes unless an additional definition explicitly includes them. If the number of known positive checks is still twenty-one, the positive fraction among completed checks is 21/69, approximately 30.4%. If a question asks instead for the percentage of planned checks producing a recorded positive, the denominator might still be seventy-two, but the label must clearly state that this includes three uncompleted attempts.
The right denominator is defined by the question. Never silently convert “not tested” into a negative result to make the total seem tidy. This rule applies to school surveys, medical screening explanations, attendance lists and any real dataset with missing observations. Missingness is not a numerical value. It is a limitation on what is known.
An unresolved test can change the possible count
Imagine the recovered chloride column contains one unresolved entry instead of a definite negative. Ten other recovered chloride outcomes are negative and two are positive. The true count of positive outcomes, if the unresolved test were later resolved, could be two or three. The correct reported count today is two known positives, one unresolved and nine known negatives if these values sum to twelve. The percentage of all twelve samples known to be positive is 2/12, approximately 16.7%, while the maximum possible eventual fraction would be 3/12, or 25%.
A careful answer gives a range only if the question asks for possible eventual values. It does not choose the upper bound because it is safer, or the lower bound because it is more reassuring. There is no reason to conceal the unresolved record. This kind of question tests understanding of bounds, data categories and the meaning of a completed count. It is a useful bridge from ordinary arithmetic to responsible interpretation.
Recover a missing cell from known margins
Suppose the untreated table gives nine chloride positives, five sulfate positives and four positive in both. The chloride-only count must be nine minus four, giving five. The sulfate-only count must be five minus four, giving one. The union is four plus five plus one, or ten. Since twelve untreated samples exist, the neither count is twelve minus ten, giving two. Each step has a reason, and every final count is non-negative.
This is a good examination method even when a two-way table is not printed. Write the overlap first if it is given, then subtract to recover the exclusive categories. Use the total to recover the outside category last. Check the column totals after filling the cells. If a calculation produces a negative number of samples, the information has been copied incorrectly or the assumptions are inconsistent. A negative quantity is a warning, not something to round away.
Ratio questions: compare counts and rates accurately
The ratio of untreated to recovered chloride-positive checks is 9:2. That is a comparison of counts in two groups of equal size. The ratio of their positive rates is also 75%:16.666…, which simplifies to 9:2 here because both denominators are twelve. The equality of these two ratios depends on the subgroup sizes being equal. If one group had thirty samples and the other ten, comparing raw positive counts would not reveal their respective rates fairly.
A student should decide whether a question asks for the ratio of numbers of cases, the ratio of subgroup percentages, or a ratio of costs or measurement values. These may coincide in special examples but should not be assumed to do so. The words “in each group,” “of all positives,” and “out of all tests” are denominator instructions hidden inside ordinary language. Under time pressure, underline those phrases before doing any arithmetic.
Averages cannot repair a missing data definition
An average may sound like a simple way to summarise a long table. Here, however, we have binary positive-or-negative outcomes for different target ions rather than direct calibrated concentrations. The mean number of positive tests per parent sample is 21 ÷ 24 = 0.875 positive checks per sample, but it is not a mean concentration and not the percentage of samples positive. It tells us the number of positive outcomes distributed across twenty-four sample IDs, with each ID contributing up to three tests.
An average of 0.875 is not an impossible count for an individual sample; it is an average across the group. The important step is to label it correctly. If the question asked for average mass of chloride, these data would be insufficient because no quantitative chloride masses were supplied. This example shows why choosing a formula comes after identifying the measured variable. A numerical operation cannot create a physical quantity that was never observed.
Construct a fair comparison graph
If you graph the untreated and recovered chloride-positive proportions, put the category names clearly on the horizontal axis and percentage of parent samples testing positive on the vertical axis. Show the values as 75.0% and approximately 16.7%. Keep the vertical scale honest, and state that both groups have twelve parent samples. The graph tells the reader about recorded positive rates for a specific test target. It should not be titled “Water safety after treatment,” because no such safety assessment occurred.
It would also be misleading to put the twenty-one total positive test outcomes on the same axis as the twelve parent samples unless the units and categories were made explicit. A clear graph supports the true statement and resists a false one. Examiners can ask students to identify misleading scales, missing labels, poorly chosen representations or unsupported interpretations. The strongest defence is to name the quantity on the axis and check that every number means the same thing.
Interpret the larger statement: what can the data say?
A summary sentence says, “Recovered samples were much safer, because only two were positive.” This statement has several problems. Two positive for what? In the recorded example, two recovered parent samples had a positive chloride test. One also had a positive sulfate test; one had a positive carbonate test, and the overlap with carbonate is unknown. More importantly, none of these tests is a general drinking-water safety certificate. The statement also assumes causation from a comparison without documented pairing or controls.
A defensible replacement is: “In this fictional dataset, two of twelve recovered samples recorded a positive chloride check, compared with nine of twelve untreated samples. The collection method and other safety properties were not assessed by this comparison.” The sentence is longer, but it no longer changes category, denominator or meaning. That is mathematical communication: using numbers to describe the observed pattern without making them carry a stronger conclusion.
The distinction between difference and effect
We often want numbers to explain why something happened. A change in a table may tempt us to call it an effect, especially when the category names suggest a process. Yet an effect claim requires a design capable of distinguishing the process from other causes. In this study, the groups might differ because of their source material, the sampling procedure, a measurement error or actual treatment. The table alone does not isolate the cause. No statistical significance calculation or invented uncertainty margin would repair missing study design information.
This is not beyond Mathematics. It is a reminder that quantitative evidence belongs to a question. You can calculate a difference, report a proportion, test arithmetic consistency and describe a pattern. You should not relabel a comparison as a controlled experiment without evidence. Strong real-world problem solving includes knowing what mathematics cannot establish. That judgement is particularly valuable in examination questions where a persuasive-looking claim appears beside a data table.
A four-part checking routine for examination working
First, name the unit: sample, test, bottle, person, minute, litre or other quantity. Second, write the relevant group size beside every requested fraction. Third, check overlap whenever categories might co-occur. Fourth, translate the final numerical result back into a sentence using the same nouns as the problem. This prevents a common phenomenon in timed examinations: the student performs a correct fraction calculation and then writes an incorrect interpretation.
For a longer Paper 2 application, there may be many reasonable-looking starting points. The routine gives order to the work. Write the given totals; identify whether the categories are exclusive; calculate only what the stated relationships support; and mark unanswerable questions clearly with the missing information. If you cannot determine an exact count, perhaps you can find a bound. If you cannot justify a cause, report a comparison. The ability to limit an answer appropriately is part of mathematical maturity, not a refusal to solve.
Worked challenge: change the question without changing the table
Use the completed two-way table and answer six different questions. How many were chloride positive? Eleven. How many were sulfate positive? Six. How many were both? Five. How many were positive for exactly one of those two tests? Seven. How many were negative on both? Twelve. What percentage of recovered samples were negative on both tests? Ten out of twelve, approximately 83.3%. Notice that no new observations were needed; only the requested category changed.
The last answer illustrates why language matters. It is correct to say approximately 83.3% of recovered parent samples were negative on both the specified chloride and sulfate checks. It is incorrect to say 83.3% of them were free of every contaminant. The first is a defined mathematical classification. The second imports an unmeasured chemical conclusion. Practise saying each answer in complete words before simplifying it to a number. This helps prevent a result from losing its label in the final line.
Worked challenge: find possible three-test unions
The three-test union has a lower bound of twelve and an upper bound of fifteen. Can thirteen or fourteen also occur? Yes. Starting from the minimum configuration, move one carbonate-positive case from inside the chloride-or-sulfate union to a sample outside it, while preserving the total three untreated and one recovered carbonate positives. The union rises to thirteen. Move another available carbonate-positive case outside the earlier union, and it rises to fourteen. Move the recovered carbonate-positive case outside, and it reaches fifteen, subject to the group capacities described earlier.
This construction is a useful proof of possibility. It makes clear that the exact number is not hidden in the data. Several different individual-level arrangements produce the same published column totals. The answer is therefore a range, not a unique estimate. A strong student can explain this with a small Venn diagram or a carefully written sentence rather than trying to guess the missing sample IDs.
Worked challenge: an incorrect comparison on a school poster
A poster claims: “Seventeen positive findings were reduced to four, so contamination fell by thirteen percent.” The first two numbers are the counts of positive checks in the untreated and recovered groups, not percentages. Their difference is thirteen checks. Relative to seventeen positive checks, the count difference is 13/17 × 100%, approximately 76.5%, but that still does not prove the same material was purified or that a health risk fell. It compares recorded positive test counts across group categories.
An accurate label would be “The untreated set recorded seventeen positive checks across its thirty-six ion tests; the recovered set recorded four positive checks across its thirty-six tests.” These are 47.2% and 11.1% of the respective test entries. Those values describe checks, not sample-level safety. Their difference is about 36.1 percentage points. To improve an answer, calculate only after specifying the numerator, denominator and type of event. Then decide whether the evidence supports the claim the poster wants to make.
Ten-minute self-test and answer checks
Without notes, recover the four cells of the overall chloride-and-sulfate two-way table. Calculate the number positive on exactly one of the two checks. State the proportion positive for chloride among recovered samples. Find the proportion of all test entries that were positive, and give its denominator. State whether the number of samples positive on any of the three checks is exactly known. Finally, explain one reason the comparison does not prove a treatment effect.
The checks are: both five, chloride only six, sulfate only one, neither twelve; exactly one seven; recovered chloride positive 2/12 or about 16.7%; all test entries positive 21/72 or about 29.2%; the three-test union is not exact, with the feasible range twelve to fifteen; and the fictional groups are not documented as paired or controlled. Write a sentence for the last two answers. Arithmetic alone cannot show why a requested count is underdetermined. The sentence carries the reasoning.
The next two volumes: hearing the data, then judging the claim
Now that the table has a stable numerical meaning, the next problem is language. In an English Paper 3 listening task, a speaker may correct a planned sample count, qualify a claim or state a number only once. Students must listen for the final fact rather than preserving an abandoned early figure. The next volume will use this same fictional laboratory report to practise listening strategies. The final EMS case will examine how a report can be chemically plausible, mathematically correct and still misleading if it hides uncertainty or describes a negative test as proof of safe drinking water.
Those activities are not separate papers in an imaginary EMS qualification. They are practice across actual G3 subject skills. Mathematics has done its job when every figure has the correct base, overlapping groups have been handled once, and the final sentence says no more than the numbers permit. That standard is worth applying whether the question concerns water, transport, survey responses or everyday finance.
For more mathematics learning routes, use the post-PSLE subject-level planning guide and Secondary Mathematics mastery checkpoint. The following article will focus on English listening, not on repeating this table’s arithmetic.
