Which denominator makes a classroom percentage fair?
The answer depends on what the question counts. If a class collects forty sample records, obtains thirty-six valid results and confirms twenty-one positives, twenty-one over thirty-six is the positive share of valid outcomes. Twenty-one over forty is the confirmed-positive share of all submissions. These two fractions measure different populations. This Mathematics learner’s guide explains how to construct frequency tables, handle unresolved results and compare percentages without hiding the denominator.
All data below are fictional teaching numbers. They do not measure real foods or recommend diets. The Biology meaning of a positive food test belongs to the previous volume. Here, the examination task is to report counts accurately, explain the choice of denominator and use the correct mathematical language for a conclusion.
Quick route: name total submissions, valid records, positive results, negatives and unresolved outcomes; check each row total; calculate the requested proportion; compare like groups; read chart scales; understand limits on missing records; and attach the correct population and units to the final answer.
Official 2027 G3 SEC Mathematics context
The 2027 SEAB Mathematics K310 syllabus includes statistics, probability, data interpretation and real-world problem solving. Two 90-mark papers each last two hours and fifteen minutes, with each paper contributing fifty percent. The official G3 catalogue gives K310 as the new code and 4052 as the earlier reference. This is original practice, not an official paper or an independent EMS subject.
Why the denominator matters
A school demonstration may collect forty test records and classify thirty-six as valid. If twenty-one of those valid tests are positive, the positive rate among valid results is twenty-one divided by thirty-six, approximately 58.3%. Yet twenty-one divided by all forty collected records is only 52.5%. Neither answer is a mathematical error. They answer two different questions. The first measures a fraction of the valid subset. The second counts the confirmed positives among every submission. Every examination answer must tell its reader which group is being described.
A learner should distinguish a recorded positive from a negative, an invalid test and a sample that was never tested. The last two categories do not create a genuine biological negative result. Mathematics can calculate a fraction from records accurately without having enough scientific evidence to identify the true status of missing records. The strongest solution both performs the arithmetic and names the limitation.
This is a follow-up to the food-test Biology guide. That guide explained what iodine, Benedict’s, biuret and ethanol tests can indicate. This chapter teaches another skill: organising classifications as data, choosing fair denominators, and resisting claims that the observed records cannot support. These are separate academic jobs even when one school activity supplies the context for both.
The official examination framework
The 2027 Singapore–Cambridge Secondary Education Certificate G3 Mathematics syllabus has code K310. It builds on the prior O-Level Mathematics 4052 specification and covers number, algebra, geometry, statistics, probability and mathematical reasoning, including real-world problem solving. The assessment uses two papers worth ninety marks each, with each paper lasting two hours and fifteen minutes and contributing half the overall subject assessment.
The fictional sample-testing figures below practise fractions, percentages, frequency tables, probability, comparisons and reporting. There is no separate EMS examination paper. An integrated example is a study tool, not a change to official subject requirements. Students should always read the instructions and conventions on the actual examination paper, especially precision requirements and the demand for essential working.
For readability, percentage examples below show one decimal place while retaining the exact fractional form. Where a question specifies a different number of significant figures or decimal places, that instruction controls the final answer. Rounding early can spoil a later calculation; keep a full fraction or unrounded calculator value until the final reporting step.
Build the data ledger before calculating
Let T represent all submitted records, V the records with valid classified outcomes, and P the confirmed positives among those valid outcomes. Then N = V − P represents classified negatives and U = T − V represents unresolved records. Check that P + N + U = T. If a category is not mutually exclusive with the others, this simple model cannot be applied as written and you must reformulate it.
The fraction P/V measures the positive proportion among valid results, provided V is not zero. P/T measures the proportion of all submissions with confirmed positives, provided T is not zero. U/T measures the unresolved fraction among submissions. None measures the concentration of a nutrient or the health effect of a food. Those claims would require a different measurement and different evidence.
A quick table with the five letters is often enough to prevent a denominator error. If the class submitted sixty records and only fifty were classified, the answer to a question about valid outcomes must not use sixty by habit. Conversely, if the question asks what fraction of all collected records had confirmed positive results, fifty is not the relevant whole.
Two groups with equal rates but unequal counts
Imagine Group A obtained twelve positive outcomes from twenty valid tests and Group B obtained eighteen positives from thirty valid tests. Group B has more positives as an absolute count, yet both groups have a sixty-percent positive share among valid tests. The statement that Group B has a higher positive rate would be false. It confuses size of the group with frequency of the outcome.
Now suppose Group A also had five unresolved records, while Group B had none. Among all collected records, the confirmed-positive fractions become twelve out of twenty-five and eighteen out of thirty. The first is forty-eight percent, the second sixty percent. Both are correctly computed; the different treatment of unresolved records explains why the new comparison is not identical to the valid-only comparison.
Students should not treat the second comparison as proof that the samples had different true underlying compositions. A batch of observations can be affected by what was or was not classified. Report the chosen denominator, the observed quantity and the uncertainty. This is an honest application of mathematical reasoning rather than an attempt to avoid a difficult conclusion.
Percentage points versus relative percentage change
Suppose a valid positive percentage changes from forty to fifty-five percent. The increase is fifteen percentage points, because 55 minus 40 equals 15. The relative percentage increase is fifteen divided by forty multiplied by one hundred, which equals 37.5%. These quantities serve different purposes, so writing simply fifteen percent for the relative increase is inaccurate.
Reversing a change does not preserve the same relative percentage. The relative decrease from fifty-five to forty percent is fifteen divided by fifty-five times one hundred, about 27.3%. The denominator is the starting value in each direction. A correct final sentence makes clear whether the figure reports a change in percentage points or a percentage of the initial rate.
One exam technique is to write a tiny phrase beside the computation: original value equals forty percent. It prevents using the new figure as the base for an increase calculation. This same denominator discipline applies to prices, growth, participation statistics and many unfamiliar word problems.
Multiple reagents and possible overlap
A single physical sample can produce positive results with more than one food-test reagent. If ten iodine tests and nine biuret tests are positive, adding ten and nine counts positive test outcomes, not necessarily nineteen distinct samples. Some of the same samples may appear in both groups. To find the number of distinct samples positive in at least one of the two tests, the overlapping count is required.
For example, if four of the samples tested positive in both categories, the distinct count is ten plus nine minus four, or fifteen. Without the overlapping count, no single distinct total follows. A question may supply a two-way table so that the intersection is visible. Read its headings before adding anything.
The broader lesson is that category totals cannot always be added. A person may use several transport services, and a library book may belong to several thematic lists. Before summing counts in an examination, ask whether the units are mutually exclusive, whether the same unit might be recorded twice, and what the question actually asks to count.
Bounds for missing outcomes
When twelve records are positive, twenty-four are negative and four are unresolved, forty total records exist and thirty-six are valid. The known positive fraction among all records is twelve out of forty, or thirty percent. If every unresolved item eventually tested positive, the final positive share could reach sixteen out of forty, or forty percent. If all were negative, the final share would remain thirty percent.
The range from thirty to forty percent is a logical bound under a stated assumption that each unresolved record will ultimately be classified positive or negative. It is not a confidence interval or a prediction of what the unresolved records will show. The current records cannot determine an exact ultimate percentage. The proper response to an exact-value question might be that additional classification results are needed.
A bound is useful because it states what information allows without inventing what information does not contain. Students who practise this can recognise when a conclusion is underdetermined and still give a quantitative, well-justified answer.
Probability and the named sample space
If a test record is uniformly randomly selected from a group of thirty valid outcomes, nine of which are positive, the probability that the chosen valid record is positive is nine over thirty, or 0.3. If the selection comes from all thirty-five submissions, including five unresolved records, the probability that the chosen submission already has a confirmed positive outcome is nine over thirty-five. The sample spaces differ.
Do not automatically treat the observed proportion as the likelihood that any future food sample will give the same result. That requires assumptions about how future samples compare with the study records. In ordinary G3 questions, a probability fraction comes from the particular equally likely outcomes specified by the question.
An answer should make the experiment and event clear. It is better to say a randomly selected valid record is positive than to announce that thirty percent of all foods are positive. The latter conclusion is much larger and has not been supported by the classroom data.
Graph scales and missing categories
A bar chart is a useful way to compare counts, but a vertical axis starting at forty-four can make fifty look dramatically greater than forty-five. The difference is five, whatever the visual height of the bars. A pie chart works best when the categories form distinct parts of one whole; positive tests from overlapping nutrient categories do not automatically form such a whole.
Look for axis titles, measurement units, category definitions and totals. A graph with an unlabeled missing-data category can conceal unresolved outcomes. A chart that mixes grams per sample with number of positive results puts different units side by side and risks misleading the reader. The solution is usually a clearer table or separate graphs, not more decoration.
When interpreting a graph under time pressure, first extract the numbers and decide which mathematical operation the question requires. Describe what is shown rather than what the chart’s colours seem to suggest. If the needed numerical values are absent, identify the insufficiency instead of imagining them.
Check the answer as a mathematical object
A classified positive count cannot exceed the valid count. The classified negative count cannot be negative. P + N + U must equal T when the three categories exhaust all submissions and each record belongs to exactly one category. Fractions representing subset proportions must lie from zero to one inclusive. These checks are quick and catch surprisingly many errors.
If a student computes 120% as a share of a valid subset, the chosen denominator, numerator or category definition must be reconsidered. If the row totals in a frequency table do not match the stated grand total, something is missing or duplicated. A mistake discovered through checking is easier to correct than an unsupported numerical answer left in the script.
Finish by attaching the correct unit and population to every answer. A percentage of valid test records is not a percentage concentration by mass. A probability of a randomly selected record being positive is not a prediction about a person. The number is only as good as the event, group and unit that define it.
Forty original worked frequency-table exercises
Each situation below is an independent invented data set. Use T for collected records, V for valid outcomes and P for confirmed positives. Derive negative outcomes by V minus P and unresolved outcomes by T minus V. The rotating questions test different interpretations of the same type of table: a vital examination skill is knowing when changing the question changes the correct denominator.
Practice 1: Canteen cards
The fictional teaching group recorded 40 submitted records, 36 valid classifications and 21 positive outcomes. Before using a calculator, identify the population described by the question. The numbers concern records, not the nutrient content of any particular item. The derived classified negative count is 15 and unresolved count is 4. Check the table explicitly: 21 + 15 + 4 = 40. Every entry has a meaning and the categories are mutually exclusive for the particular test in this exercise.
Core worked answer: The positive fraction of valid classifications is 21/36, giving 58.3% when expressed as a percentage to one decimal place. The share of all submitted records that already have a confirmed positive outcome is 21/40, giving 52.5%. Both use the same numerator, but the populations differ. When writing a result, add the words among valid records or among all submissions; otherwise the calculated figure may be misleading despite correct arithmetic.
Extension 1: Calculate the negative share among valid results: 15/36 = 41.7%. This counts classified negatives and uses only the valid-result population. An unresolved record cannot silently become negative. To check, the valid positive share 58.3% and valid negative share 41.7% should total about one hundred percent, allowing for rounding. Explain why the population of interest excludes the 4 unresolved submissions. The important checking habit is to retain the fraction and its definitions until the final sentence, rather than making an unqualified statement about the entire set of real-world samples.
Practice 2: Lab trays
The fictional teaching group recorded 60 submitted records, 50 valid classifications and 20 positive outcomes. Before using a calculator, identify the population described by the question. The numbers concern records, not the nutrient content of any particular item. The derived classified negative count is 30 and unresolved count is 10. Check the table explicitly: 20 + 30 + 10 = 60. Every entry has a meaning and the categories are mutually exclusive for the particular test in this exercise.
Core worked answer: The positive fraction of valid classifications is 20/50, giving 40.0% when expressed as a percentage to one decimal place. The share of all submitted records that already have a confirmed positive outcome is 20/60, giving 33.3%. Both use the same numerator, but the populations differ. When writing a result, add the words among valid records or among all submissions; otherwise the calculated figure may be misleading despite correct arithmetic.
Extension 2: Calculate the unresolved share of all submissions: 10/60 = 16.7%. The denominator is the full set of submissions because unresolved items still belong to that set. The percentage is not a rate of negative tests, nor a rate of any food substance. It describes only how much of the data remains unclassified, an important limitation on stronger conclusions. The important checking habit is to retain the fraction and its definitions until the final sentence, rather than making an unqualified statement about the entire set of real-world samples.
Practice 3: Learning kits
The fictional teaching group recorded 48 submitted records, 42 valid classifications and 15 positive outcomes. Before using a calculator, identify the population described by the question. The numbers concern records, not the nutrient content of any particular item. The derived classified negative count is 27 and unresolved count is 6. Check the table explicitly: 15 + 27 + 6 = 48. Every entry has a meaning and the categories are mutually exclusive for the particular test in this exercise.
Core worked answer: The positive fraction of valid classifications is 15/42, giving 35.7% when expressed as a percentage to one decimal place. The share of all submitted records that already have a confirmed positive outcome is 15/48, giving 31.3%. Both use the same numerator, but the populations differ. When writing a result, add the words among valid records or among all submissions; otherwise the calculated figure may be misleading despite correct arithmetic.
Extension 3: Calculate the possible range of final positive shares if every unresolved result eventually becomes positive or negative. The minimum is 15/48 = 31.3%; the maximum is (15+6)/48 = 43.8%. The interval is a logical bound under the stated assumption, not a prediction. There is not enough information to name one exact final rate. The important checking habit is to retain the fraction and its definitions until the final sentence, rather than making an unqualified statement about the entire set of real-world samples.
Practice 4: Workshop stations
The fictional teaching group recorded 75 submitted records, 60 valid classifications and 39 positive outcomes. Before using a calculator, identify the population described by the question. The numbers concern records, not the nutrient content of any particular item. The derived classified negative count is 21 and unresolved count is 15. Check the table explicitly: 39 + 21 + 15 = 75. Every entry has a meaning and the categories are mutually exclusive for the particular test in this exercise.
Core worked answer: The positive fraction of valid classifications is 39/60, giving 65.0% when expressed as a percentage to one decimal place. The share of all submitted records that already have a confirmed positive outcome is 39/75, giving 52.0%. Both use the same numerator, but the populations differ. When writing a result, add the words among valid records or among all submissions; otherwise the calculated figure may be misleading despite correct arithmetic.
Extension 4: Suppose a valid record is chosen uniformly at random. The probability it is positive is 39/60 = 65.0%. If instead a submission is chosen at random from all records, the probability that it already has a confirmed positive result is 39/75 = 52.0%. These are two different events and sample spaces. Name the sample space in the answer. The important checking habit is to retain the fraction and its definitions until the final sentence, rather than making an unqualified statement about the entire set of real-world samples.
Practice 5: Sample envelopes
The fictional teaching group recorded 28 submitted records, 25 valid classifications and 10 positive outcomes. Before using a calculator, identify the population described by the question. The numbers concern records, not the nutrient content of any particular item. The derived classified negative count is 15 and unresolved count is 3. Check the table explicitly: 10 + 15 + 3 = 28. Every entry has a meaning and the categories are mutually exclusive for the particular test in this exercise.
Core worked answer: The positive fraction of valid classifications is 10/25, giving 40.0% when expressed as a percentage to one decimal place. The share of all submitted records that already have a confirmed positive outcome is 10/28, giving 35.7%. Both use the same numerator, but the populations differ. When writing a result, add the words among valid records or among all submissions; otherwise the calculated figure may be misleading despite correct arithmetic.
Extension 5: Use the table to compare count and rate. The absolute positive count is 10, but the valid positive rate is 10/25 = 40.0%. A group with a larger positive count may have a smaller rate if its valid denominator is sufficiently large. A fair comparison with a second group requires that second group’s positive count and relevant valid total. Do not invent them. The important checking habit is to retain the fraction and its definitions until the final sentence, rather than making an unqualified statement about the entire set of real-world samples.
Practice 6: Class registers
The fictional teaching group recorded 80 submitted records, 72 valid classifications and 36 positive outcomes. Before using a calculator, identify the population described by the question. The numbers concern records, not the nutrient content of any particular item. The derived classified negative count is 36 and unresolved count is 8. Check the table explicitly: 36 + 36 + 8 = 80. Every entry has a meaning and the categories are mutually exclusive for the particular test in this exercise.
Core worked answer: The positive fraction of valid classifications is 36/72, giving 50.0% when expressed as a percentage to one decimal place. The share of all submitted records that already have a confirmed positive outcome is 36/80, giving 45.0%. Both use the same numerator, but the populations differ. When writing a result, add the words among valid records or among all submissions; otherwise the calculated figure may be misleading despite correct arithmetic.
Extension 6: Calculate the negative share among valid results: 36/72 = 50.0%. This counts classified negatives and uses only the valid-result population. An unresolved record cannot silently become negative. To check, the valid positive share 50.0% and valid negative share 50.0% should total about one hundred percent, allowing for rounding. Explain why the population of interest excludes the 8 unresolved submissions. The important checking habit is to retain the fraction and its definitions until the final sentence, rather than making an unqualified statement about the entire set of real-world samples.
Practice 7: Display notes
The fictional teaching group recorded 36 submitted records, 30 valid classifications and 18 positive outcomes. Before using a calculator, identify the population described by the question. The numbers concern records, not the nutrient content of any particular item. The derived classified negative count is 12 and unresolved count is 6. Check the table explicitly: 18 + 12 + 6 = 36. Every entry has a meaning and the categories are mutually exclusive for the particular test in this exercise.
Core worked answer: The positive fraction of valid classifications is 18/30, giving 60.0% when expressed as a percentage to one decimal place. The share of all submitted records that already have a confirmed positive outcome is 18/36, giving 50.0%. Both use the same numerator, but the populations differ. When writing a result, add the words among valid records or among all submissions; otherwise the calculated figure may be misleading despite correct arithmetic.
Extension 7: Calculate the unresolved share of all submissions: 6/36 = 16.7%. The denominator is the full set of submissions because unresolved items still belong to that set. The percentage is not a rate of negative tests, nor a rate of any food substance. It describes only how much of the data remains unclassified, an important limitation on stronger conclusions. The important checking habit is to retain the fraction and its definitions until the final sentence, rather than making an unqualified statement about the entire set of real-world samples.
Practice 8: Bottle labels
The fictional teaching group recorded 90 submitted records, 81 valid classifications and 27 positive outcomes. Before using a calculator, identify the population described by the question. The numbers concern records, not the nutrient content of any particular item. The derived classified negative count is 54 and unresolved count is 9. Check the table explicitly: 27 + 54 + 9 = 90. Every entry has a meaning and the categories are mutually exclusive for the particular test in this exercise.
Core worked answer: The positive fraction of valid classifications is 27/81, giving 33.3% when expressed as a percentage to one decimal place. The share of all submitted records that already have a confirmed positive outcome is 27/90, giving 30.0%. Both use the same numerator, but the populations differ. When writing a result, add the words among valid records or among all submissions; otherwise the calculated figure may be misleading despite correct arithmetic.
Extension 8: Calculate the possible range of final positive shares if every unresolved result eventually becomes positive or negative. The minimum is 27/90 = 30.0%; the maximum is (27+9)/90 = 40.0%. The interval is a logical bound under the stated assumption, not a prediction. There is not enough information to name one exact final rate. The important checking habit is to retain the fraction and its definitions until the final sentence, rather than making an unqualified statement about the entire set of real-world samples.
Practice 9: Practice tubes
The fictional teaching group recorded 55 submitted records, 44 valid classifications and 11 positive outcomes. Before using a calculator, identify the population described by the question. The numbers concern records, not the nutrient content of any particular item. The derived classified negative count is 33 and unresolved count is 11. Check the table explicitly: 11 + 33 + 11 = 55. Every entry has a meaning and the categories are mutually exclusive for the particular test in this exercise.
Core worked answer: The positive fraction of valid classifications is 11/44, giving 25.0% when expressed as a percentage to one decimal place. The share of all submitted records that already have a confirmed positive outcome is 11/55, giving 20.0%. Both use the same numerator, but the populations differ. When writing a result, add the words among valid records or among all submissions; otherwise the calculated figure may be misleading despite correct arithmetic.
Extension 9: Suppose a valid record is chosen uniformly at random. The probability it is positive is 11/44 = 25.0%. If instead a submission is chosen at random from all records, the probability that it already has a confirmed positive result is 11/55 = 20.0%. These are two different events and sample spaces. Name the sample space in the answer. The important checking habit is to retain the fraction and its definitions until the final sentence, rather than making an unqualified statement about the entire set of real-world samples.
Practice 10: Visitor examples
The fictional teaching group recorded 32 submitted records, 24 valid classifications and 12 positive outcomes. Before using a calculator, identify the population described by the question. The numbers concern records, not the nutrient content of any particular item. The derived classified negative count is 12 and unresolved count is 8. Check the table explicitly: 12 + 12 + 8 = 32. Every entry has a meaning and the categories are mutually exclusive for the particular test in this exercise.
Core worked answer: The positive fraction of valid classifications is 12/24, giving 50.0% when expressed as a percentage to one decimal place. The share of all submitted records that already have a confirmed positive outcome is 12/32, giving 37.5%. Both use the same numerator, but the populations differ. When writing a result, add the words among valid records or among all submissions; otherwise the calculated figure may be misleading despite correct arithmetic.
Extension 10: Use the table to compare count and rate. The absolute positive count is 12, but the valid positive rate is 12/24 = 50.0%. A group with a larger positive count may have a smaller rate if its valid denominator is sufficiently large. A fair comparison with a second group requires that second group’s positive count and relevant valid total. Do not invent them. The important checking habit is to retain the fraction and its definitions until the final sentence, rather than making an unqualified statement about the entire set of real-world samples.
Practice 11: Science fair
The fictional teaching group recorded 100 submitted records, 92 valid classifications and 46 positive outcomes. Before using a calculator, identify the population described by the question. The numbers concern records, not the nutrient content of any particular item. The derived classified negative count is 46 and unresolved count is 8. Check the table explicitly: 46 + 46 + 8 = 100. Every entry has a meaning and the categories are mutually exclusive for the particular test in this exercise.
Core worked answer: The positive fraction of valid classifications is 46/92, giving 50.0% when expressed as a percentage to one decimal place. The share of all submitted records that already have a confirmed positive outcome is 46/100, giving 46.0%. Both use the same numerator, but the populations differ. When writing a result, add the words among valid records or among all submissions; otherwise the calculated figure may be misleading despite correct arithmetic.
Extension 11: Calculate the negative share among valid results: 46/92 = 50.0%. This counts classified negatives and uses only the valid-result population. An unresolved record cannot silently become negative. To check, the valid positive share 50.0% and valid negative share 50.0% should total about one hundred percent, allowing for rounding. Explain why the population of interest excludes the 8 unresolved submissions. The important checking habit is to retain the fraction and its definitions until the final sentence, rather than making an unqualified statement about the entire set of real-world samples.
Practice 12: Demonstration session
The fictional teaching group recorded 45 submitted records, 40 valid classifications and 28 positive outcomes. Before using a calculator, identify the population described by the question. The numbers concern records, not the nutrient content of any particular item. The derived classified negative count is 12 and unresolved count is 5. Check the table explicitly: 28 + 12 + 5 = 45. Every entry has a meaning and the categories are mutually exclusive for the particular test in this exercise.
Core worked answer: The positive fraction of valid classifications is 28/40, giving 70.0% when expressed as a percentage to one decimal place. The share of all submitted records that already have a confirmed positive outcome is 28/45, giving 62.2%. Both use the same numerator, but the populations differ. When writing a result, add the words among valid records or among all submissions; otherwise the calculated figure may be misleading despite correct arithmetic.
Extension 12: Calculate the unresolved share of all submissions: 5/45 = 11.1%. The denominator is the full set of submissions because unresolved items still belong to that set. The percentage is not a rate of negative tests, nor a rate of any food substance. It describes only how much of the data remains unclassified, an important limitation on stronger conclusions. The important checking habit is to retain the fraction and its definitions until the final sentence, rather than making an unqualified statement about the entire set of real-world samples.
Practice 13: Fictional rice samples
The fictional teaching group recorded 72 submitted records, 60 valid classifications and 30 positive outcomes. Before using a calculator, identify the population described by the question. The numbers concern records, not the nutrient content of any particular item. The derived classified negative count is 30 and unresolved count is 12. Check the table explicitly: 30 + 30 + 12 = 72. Every entry has a meaning and the categories are mutually exclusive for the particular test in this exercise.
Core worked answer: The positive fraction of valid classifications is 30/60, giving 50.0% when expressed as a percentage to one decimal place. The share of all submitted records that already have a confirmed positive outcome is 30/72, giving 41.7%. Both use the same numerator, but the populations differ. When writing a result, add the words among valid records or among all submissions; otherwise the calculated figure may be misleading despite correct arithmetic.
Extension 13: Calculate the possible range of final positive shares if every unresolved result eventually becomes positive or negative. The minimum is 30/72 = 41.7%; the maximum is (30+12)/72 = 58.3%. The interval is a logical bound under the stated assumption, not a prediction. There is not enough information to name one exact final rate. The important checking habit is to retain the fraction and its definitions until the final sentence, rather than making an unqualified statement about the entire set of real-world samples.
Practice 14: Fictional snack samples
The fictional teaching group recorded 64 submitted records, 56 valid classifications and 21 positive outcomes. Before using a calculator, identify the population described by the question. The numbers concern records, not the nutrient content of any particular item. The derived classified negative count is 35 and unresolved count is 8. Check the table explicitly: 21 + 35 + 8 = 64. Every entry has a meaning and the categories are mutually exclusive for the particular test in this exercise.
Core worked answer: The positive fraction of valid classifications is 21/56, giving 37.5% when expressed as a percentage to one decimal place. The share of all submitted records that already have a confirmed positive outcome is 21/64, giving 32.8%. Both use the same numerator, but the populations differ. When writing a result, add the words among valid records or among all submissions; otherwise the calculated figure may be misleading despite correct arithmetic.
Extension 14: Suppose a valid record is chosen uniformly at random. The probability it is positive is 21/56 = 37.5%. If instead a submission is chosen at random from all records, the probability that it already has a confirmed positive result is 21/64 = 32.8%. These are two different events and sample spaces. Name the sample space in the answer. The important checking habit is to retain the fraction and its definitions until the final sentence, rather than making an unqualified statement about the entire set of real-world samples.
Practice 15: Garden club records
The fictional teaching group recorded 50 submitted records, 45 valid classifications and 9 positive outcomes. Before using a calculator, identify the population described by the question. The numbers concern records, not the nutrient content of any particular item. The derived classified negative count is 36 and unresolved count is 5. Check the table explicitly: 9 + 36 + 5 = 50. Every entry has a meaning and the categories are mutually exclusive for the particular test in this exercise.
Core worked answer: The positive fraction of valid classifications is 9/45, giving 20.0% when expressed as a percentage to one decimal place. The share of all submitted records that already have a confirmed positive outcome is 9/50, giving 18.0%. Both use the same numerator, but the populations differ. When writing a result, add the words among valid records or among all submissions; otherwise the calculated figure may be misleading despite correct arithmetic.
Extension 15: Use the table to compare count and rate. The absolute positive count is 9, but the valid positive rate is 9/45 = 20.0%. A group with a larger positive count may have a smaller rate if its valid denominator is sufficiently large. A fair comparison with a second group requires that second group’s positive count and relevant valid total. Do not invent them. The important checking habit is to retain the fraction and its definitions until the final sentence, rather than making an unqualified statement about the entire set of real-world samples.
Practice 16: Booklet examples
The fictional teaching group recorded 24 submitted records, 20 valid classifications and 8 positive outcomes. Before using a calculator, identify the population described by the question. The numbers concern records, not the nutrient content of any particular item. The derived classified negative count is 12 and unresolved count is 4. Check the table explicitly: 8 + 12 + 4 = 24. Every entry has a meaning and the categories are mutually exclusive for the particular test in this exercise.
Core worked answer: The positive fraction of valid classifications is 8/20, giving 40.0% when expressed as a percentage to one decimal place. The share of all submitted records that already have a confirmed positive outcome is 8/24, giving 33.3%. Both use the same numerator, but the populations differ. When writing a result, add the words among valid records or among all submissions; otherwise the calculated figure may be misleading despite correct arithmetic.
Extension 16: Calculate the negative share among valid results: 12/20 = 60.0%. This counts classified negatives and uses only the valid-result population. An unresolved record cannot silently become negative. To check, the valid positive share 40.0% and valid negative share 60.0% should total about one hundred percent, allowing for rounding. Explain why the population of interest excludes the 4 unresolved submissions. The important checking habit is to retain the fraction and its definitions until the final sentence, rather than making an unqualified statement about the entire set of real-world samples.
Practice 17: Training cards
The fictional teaching group recorded 84 submitted records, 70 valid classifications and 35 positive outcomes. Before using a calculator, identify the population described by the question. The numbers concern records, not the nutrient content of any particular item. The derived classified negative count is 35 and unresolved count is 14. Check the table explicitly: 35 + 35 + 14 = 84. Every entry has a meaning and the categories are mutually exclusive for the particular test in this exercise.
Core worked answer: The positive fraction of valid classifications is 35/70, giving 50.0% when expressed as a percentage to one decimal place. The share of all submitted records that already have a confirmed positive outcome is 35/84, giving 41.7%. Both use the same numerator, but the populations differ. When writing a result, add the words among valid records or among all submissions; otherwise the calculated figure may be misleading despite correct arithmetic.
Extension 17: Calculate the unresolved share of all submissions: 14/84 = 16.7%. The denominator is the full set of submissions because unresolved items still belong to that set. The percentage is not a rate of negative tests, nor a rate of any food substance. It describes only how much of the data remains unclassified, an important limitation on stronger conclusions. The important checking habit is to retain the fraction and its definitions until the final sentence, rather than making an unqualified statement about the entire set of real-world samples.
Practice 18: Reagent simulation
The fictional teaching group recorded 40 submitted records, 32 valid classifications and 24 positive outcomes. Before using a calculator, identify the population described by the question. The numbers concern records, not the nutrient content of any particular item. The derived classified negative count is 8 and unresolved count is 8. Check the table explicitly: 24 + 8 + 8 = 40. Every entry has a meaning and the categories are mutually exclusive for the particular test in this exercise.
Core worked answer: The positive fraction of valid classifications is 24/32, giving 75.0% when expressed as a percentage to one decimal place. The share of all submitted records that already have a confirmed positive outcome is 24/40, giving 60.0%. Both use the same numerator, but the populations differ. When writing a result, add the words among valid records or among all submissions; otherwise the calculated figure may be misleading despite correct arithmetic.
Extension 18: Calculate the possible range of final positive shares if every unresolved result eventually becomes positive or negative. The minimum is 24/40 = 60.0%; the maximum is (24+8)/40 = 80.0%. The interval is a logical bound under the stated assumption, not a prediction. There is not enough information to name one exact final rate. The important checking habit is to retain the fraction and its definitions until the final sentence, rather than making an unqualified statement about the entire set of real-world samples.
Practice 19: Supply cartons
The fictional teaching group recorded 120 submitted records, 100 valid classifications and 55 positive outcomes. Before using a calculator, identify the population described by the question. The numbers concern records, not the nutrient content of any particular item. The derived classified negative count is 45 and unresolved count is 20. Check the table explicitly: 55 + 45 + 20 = 120. Every entry has a meaning and the categories are mutually exclusive for the particular test in this exercise.
Core worked answer: The positive fraction of valid classifications is 55/100, giving 55.0% when expressed as a percentage to one decimal place. The share of all submitted records that already have a confirmed positive outcome is 55/120, giving 45.8%. Both use the same numerator, but the populations differ. When writing a result, add the words among valid records or among all submissions; otherwise the calculated figure may be misleading despite correct arithmetic.
Extension 19: Suppose a valid record is chosen uniformly at random. The probability it is positive is 55/100 = 55.0%. If instead a submission is chosen at random from all records, the probability that it already has a confirmed positive result is 55/120 = 45.8%. These are two different events and sample spaces. Name the sample space in the answer. The important checking habit is to retain the fraction and its definitions until the final sentence, rather than making an unqualified statement about the entire set of real-world samples.
Practice 20: Worksheet packs
The fictional teaching group recorded 96 submitted records, 80 valid classifications and 32 positive outcomes. Before using a calculator, identify the population described by the question. The numbers concern records, not the nutrient content of any particular item. The derived classified negative count is 48 and unresolved count is 16. Check the table explicitly: 32 + 48 + 16 = 96. Every entry has a meaning and the categories are mutually exclusive for the particular test in this exercise.
Core worked answer: The positive fraction of valid classifications is 32/80, giving 40.0% when expressed as a percentage to one decimal place. The share of all submitted records that already have a confirmed positive outcome is 32/96, giving 33.3%. Both use the same numerator, but the populations differ. When writing a result, add the words among valid records or among all submissions; otherwise the calculated figure may be misleading despite correct arithmetic.
Extension 20: Use the table to compare count and rate. The absolute positive count is 32, but the valid positive rate is 32/80 = 40.0%. A group with a larger positive count may have a smaller rate if its valid denominator is sufficiently large. A fair comparison with a second group requires that second group’s positive count and relevant valid total. Do not invent them. The important checking habit is to retain the fraction and its definitions until the final sentence, rather than making an unqualified statement about the entire set of real-world samples.
Practice 21: Teaching posters
The fictional teaching group recorded 30 submitted records, 27 valid classifications and 9 positive outcomes. Before using a calculator, identify the population described by the question. The numbers concern records, not the nutrient content of any particular item. The derived classified negative count is 18 and unresolved count is 3. Check the table explicitly: 9 + 18 + 3 = 30. Every entry has a meaning and the categories are mutually exclusive for the particular test in this exercise.
Core worked answer: The positive fraction of valid classifications is 9/27, giving 33.3% when expressed as a percentage to one decimal place. The share of all submitted records that already have a confirmed positive outcome is 9/30, giving 30.0%. Both use the same numerator, but the populations differ. When writing a result, add the words among valid records or among all submissions; otherwise the calculated figure may be misleading despite correct arithmetic.
Extension 21: Calculate the negative share among valid results: 18/27 = 66.7%. This counts classified negatives and uses only the valid-result population. An unresolved record cannot silently become negative. To check, the valid positive share 33.3% and valid negative share 66.7% should total about one hundred percent, allowing for rounding. Explain why the population of interest excludes the 3 unresolved submissions. The important checking habit is to retain the fraction and its definitions until the final sentence, rather than making an unqualified statement about the entire set of real-world samples.
Practice 22: Library lab stories
The fictional teaching group recorded 66 submitted records, 55 valid classifications and 22 positive outcomes. Before using a calculator, identify the population described by the question. The numbers concern records, not the nutrient content of any particular item. The derived classified negative count is 33 and unresolved count is 11. Check the table explicitly: 22 + 33 + 11 = 66. Every entry has a meaning and the categories are mutually exclusive for the particular test in this exercise.
Core worked answer: The positive fraction of valid classifications is 22/55, giving 40.0% when expressed as a percentage to one decimal place. The share of all submitted records that already have a confirmed positive outcome is 22/66, giving 33.3%. Both use the same numerator, but the populations differ. When writing a result, add the words among valid records or among all submissions; otherwise the calculated figure may be misleading despite correct arithmetic.
Extension 22: Calculate the unresolved share of all submissions: 11/66 = 16.7%. The denominator is the full set of submissions because unresolved items still belong to that set. The percentage is not a rate of negative tests, nor a rate of any food substance. It describes only how much of the data remains unclassified, an important limitation on stronger conclusions. The important checking habit is to retain the fraction and its definitions until the final sentence, rather than making an unqualified statement about the entire set of real-world samples.
Practice 23: Classroom stations
The fictional teaching group recorded 42 submitted records, 35 valid classifications and 14 positive outcomes. Before using a calculator, identify the population described by the question. The numbers concern records, not the nutrient content of any particular item. The derived classified negative count is 21 and unresolved count is 7. Check the table explicitly: 14 + 21 + 7 = 42. Every entry has a meaning and the categories are mutually exclusive for the particular test in this exercise.
Core worked answer: The positive fraction of valid classifications is 14/35, giving 40.0% when expressed as a percentage to one decimal place. The share of all submitted records that already have a confirmed positive outcome is 14/42, giving 33.3%. Both use the same numerator, but the populations differ. When writing a result, add the words among valid records or among all submissions; otherwise the calculated figure may be misleading despite correct arithmetic.
Extension 23: Calculate the possible range of final positive shares if every unresolved result eventually becomes positive or negative. The minimum is 14/42 = 33.3%; the maximum is (14+7)/42 = 50.0%. The interval is a logical bound under the stated assumption, not a prediction. There is not enough information to name one exact final rate. The important checking habit is to retain the fraction and its definitions until the final sentence, rather than making an unqualified statement about the entire set of real-world samples.
Practice 24: Reading circles
The fictional teaching group recorded 54 submitted records, 45 valid classifications and 18 positive outcomes. Before using a calculator, identify the population described by the question. The numbers concern records, not the nutrient content of any particular item. The derived classified negative count is 27 and unresolved count is 9. Check the table explicitly: 18 + 27 + 9 = 54. Every entry has a meaning and the categories are mutually exclusive for the particular test in this exercise.
Core worked answer: The positive fraction of valid classifications is 18/45, giving 40.0% when expressed as a percentage to one decimal place. The share of all submitted records that already have a confirmed positive outcome is 18/54, giving 33.3%. Both use the same numerator, but the populations differ. When writing a result, add the words among valid records or among all submissions; otherwise the calculated figure may be misleading despite correct arithmetic.
Extension 24: Suppose a valid record is chosen uniformly at random. The probability it is positive is 18/45 = 40.0%. If instead a submission is chosen at random from all records, the probability that it already has a confirmed positive result is 18/54 = 33.3%. These are two different events and sample spaces. Name the sample space in the answer. The important checking habit is to retain the fraction and its definitions until the final sentence, rather than making an unqualified statement about the entire set of real-world samples.
Practice 25: Study groups
The fictional teaching group recorded 70 submitted records, 56 valid classifications and 28 positive outcomes. Before using a calculator, identify the population described by the question. The numbers concern records, not the nutrient content of any particular item. The derived classified negative count is 28 and unresolved count is 14. Check the table explicitly: 28 + 28 + 14 = 70. Every entry has a meaning and the categories are mutually exclusive for the particular test in this exercise.
Core worked answer: The positive fraction of valid classifications is 28/56, giving 50.0% when expressed as a percentage to one decimal place. The share of all submitted records that already have a confirmed positive outcome is 28/70, giving 40.0%. Both use the same numerator, but the populations differ. When writing a result, add the words among valid records or among all submissions; otherwise the calculated figure may be misleading despite correct arithmetic.
Extension 25: Use the table to compare count and rate. The absolute positive count is 28, but the valid positive rate is 28/56 = 50.0%. A group with a larger positive count may have a smaller rate if its valid denominator is sufficiently large. A fair comparison with a second group requires that second group’s positive count and relevant valid total. Do not invent them. The important checking habit is to retain the fraction and its definitions until the final sentence, rather than making an unqualified statement about the entire set of real-world samples.
Practice 26: Demonstration logs
The fictional teaching group recorded 88 submitted records, 80 valid classifications and 52 positive outcomes. Before using a calculator, identify the population described by the question. The numbers concern records, not the nutrient content of any particular item. The derived classified negative count is 28 and unresolved count is 8. Check the table explicitly: 52 + 28 + 8 = 88. Every entry has a meaning and the categories are mutually exclusive for the particular test in this exercise.
Core worked answer: The positive fraction of valid classifications is 52/80, giving 65.0% when expressed as a percentage to one decimal place. The share of all submitted records that already have a confirmed positive outcome is 52/88, giving 59.1%. Both use the same numerator, but the populations differ. When writing a result, add the words among valid records or among all submissions; otherwise the calculated figure may be misleading despite correct arithmetic.
Extension 26: Calculate the negative share among valid results: 28/80 = 35.0%. This counts classified negatives and uses only the valid-result population. An unresolved record cannot silently become negative. To check, the valid positive share 65.0% and valid negative share 35.0% should total about one hundred percent, allowing for rounding. Explain why the population of interest excludes the 8 unresolved submissions. The important checking habit is to retain the fraction and its definitions until the final sentence, rather than making an unqualified statement about the entire set of real-world samples.
Practice 27: Science lesson batches
The fictional teaching group recorded 25 submitted records, 20 valid classifications and 15 positive outcomes. Before using a calculator, identify the population described by the question. The numbers concern records, not the nutrient content of any particular item. The derived classified negative count is 5 and unresolved count is 5. Check the table explicitly: 15 + 5 + 5 = 25. Every entry has a meaning and the categories are mutually exclusive for the particular test in this exercise.
Core worked answer: The positive fraction of valid classifications is 15/20, giving 75.0% when expressed as a percentage to one decimal place. The share of all submitted records that already have a confirmed positive outcome is 15/25, giving 60.0%. Both use the same numerator, but the populations differ. When writing a result, add the words among valid records or among all submissions; otherwise the calculated figure may be misleading despite correct arithmetic.
Extension 27: Calculate the unresolved share of all submissions: 5/25 = 20.0%. The denominator is the full set of submissions because unresolved items still belong to that set. The percentage is not a rate of negative tests, nor a rate of any food substance. It describes only how much of the data remains unclassified, an important limitation on stronger conclusions. The important checking habit is to retain the fraction and its definitions until the final sentence, rather than making an unqualified statement about the entire set of real-world samples.
Practice 28: Week-one records
The fictional teaching group recorded 40 submitted records, 35 valid classifications and 7 positive outcomes. Before using a calculator, identify the population described by the question. The numbers concern records, not the nutrient content of any particular item. The derived classified negative count is 28 and unresolved count is 5. Check the table explicitly: 7 + 28 + 5 = 40. Every entry has a meaning and the categories are mutually exclusive for the particular test in this exercise.
Core worked answer: The positive fraction of valid classifications is 7/35, giving 20.0% when expressed as a percentage to one decimal place. The share of all submitted records that already have a confirmed positive outcome is 7/40, giving 17.5%. Both use the same numerator, but the populations differ. When writing a result, add the words among valid records or among all submissions; otherwise the calculated figure may be misleading despite correct arithmetic.
Extension 28: Calculate the possible range of final positive shares if every unresolved result eventually becomes positive or negative. The minimum is 7/40 = 17.5%; the maximum is (7+5)/40 = 30.0%. The interval is a logical bound under the stated assumption, not a prediction. There is not enough information to name one exact final rate. The important checking habit is to retain the fraction and its definitions until the final sentence, rather than making an unqualified statement about the entire set of real-world samples.
Practice 29: Week-two records
The fictional teaching group recorded 48 submitted records, 40 valid classifications and 16 positive outcomes. Before using a calculator, identify the population described by the question. The numbers concern records, not the nutrient content of any particular item. The derived classified negative count is 24 and unresolved count is 8. Check the table explicitly: 16 + 24 + 8 = 48. Every entry has a meaning and the categories are mutually exclusive for the particular test in this exercise.
Core worked answer: The positive fraction of valid classifications is 16/40, giving 40.0% when expressed as a percentage to one decimal place. The share of all submitted records that already have a confirmed positive outcome is 16/48, giving 33.3%. Both use the same numerator, but the populations differ. When writing a result, add the words among valid records or among all submissions; otherwise the calculated figure may be misleading despite correct arithmetic.
Extension 29: Suppose a valid record is chosen uniformly at random. The probability it is positive is 16/40 = 40.0%. If instead a submission is chosen at random from all records, the probability that it already has a confirmed positive result is 16/48 = 33.3%. These are two different events and sample spaces. Name the sample space in the answer. The important checking habit is to retain the fraction and its definitions until the final sentence, rather than making an unqualified statement about the entire set of real-world samples.
Practice 30: Pilot exhibition
The fictional teaching group recorded 100 submitted records, 75 valid classifications and 30 positive outcomes. Before using a calculator, identify the population described by the question. The numbers concern records, not the nutrient content of any particular item. The derived classified negative count is 45 and unresolved count is 25. Check the table explicitly: 30 + 45 + 25 = 100. Every entry has a meaning and the categories are mutually exclusive for the particular test in this exercise.
Core worked answer: The positive fraction of valid classifications is 30/75, giving 40.0% when expressed as a percentage to one decimal place. The share of all submitted records that already have a confirmed positive outcome is 30/100, giving 30.0%. Both use the same numerator, but the populations differ. When writing a result, add the words among valid records or among all submissions; otherwise the calculated figure may be misleading despite correct arithmetic.
Extension 30: Use the table to compare count and rate. The absolute positive count is 30, but the valid positive rate is 30/75 = 40.0%. A group with a larger positive count may have a smaller rate if its valid denominator is sufficiently large. A fair comparison with a second group requires that second group’s positive count and relevant valid total. Do not invent them. The important checking habit is to retain the fraction and its definitions until the final sentence, rather than making an unqualified statement about the entire set of real-world samples.
Practice 31: Student learning stations
The fictional teaching group recorded 35 submitted records, 28 valid classifications and 14 positive outcomes. Before using a calculator, identify the population described by the question. The numbers concern records, not the nutrient content of any particular item. The derived classified negative count is 14 and unresolved count is 7. Check the table explicitly: 14 + 14 + 7 = 35. Every entry has a meaning and the categories are mutually exclusive for the particular test in this exercise.
Core worked answer: The positive fraction of valid classifications is 14/28, giving 50.0% when expressed as a percentage to one decimal place. The share of all submitted records that already have a confirmed positive outcome is 14/35, giving 40.0%. Both use the same numerator, but the populations differ. When writing a result, add the words among valid records or among all submissions; otherwise the calculated figure may be misleading despite correct arithmetic.
Extension 31: Calculate the negative share among valid results: 14/28 = 50.0%. This counts classified negatives and uses only the valid-result population. An unresolved record cannot silently become negative. To check, the valid positive share 50.0% and valid negative share 50.0% should total about one hundred percent, allowing for rounding. Explain why the population of interest excludes the 7 unresolved submissions. The important checking habit is to retain the fraction and its definitions until the final sentence, rather than making an unqualified statement about the entire set of real-world samples.
Practice 32: Afternoon practicals
The fictional teaching group recorded 63 submitted records, 54 valid classifications and 27 positive outcomes. Before using a calculator, identify the population described by the question. The numbers concern records, not the nutrient content of any particular item. The derived classified negative count is 27 and unresolved count is 9. Check the table explicitly: 27 + 27 + 9 = 63. Every entry has a meaning and the categories are mutually exclusive for the particular test in this exercise.
Core worked answer: The positive fraction of valid classifications is 27/54, giving 50.0% when expressed as a percentage to one decimal place. The share of all submitted records that already have a confirmed positive outcome is 27/63, giving 42.9%. Both use the same numerator, but the populations differ. When writing a result, add the words among valid records or among all submissions; otherwise the calculated figure may be misleading despite correct arithmetic.
Extension 32: Calculate the unresolved share of all submissions: 9/63 = 14.3%. The denominator is the full set of submissions because unresolved items still belong to that set. The percentage is not a rate of negative tests, nor a rate of any food substance. It describes only how much of the data remains unclassified, an important limitation on stronger conclusions. The important checking habit is to retain the fraction and its definitions until the final sentence, rather than making an unqualified statement about the entire set of real-world samples.
Practice 33: Morning practicals
The fictional teaching group recorded 56 submitted records, 49 valid classifications and 21 positive outcomes. Before using a calculator, identify the population described by the question. The numbers concern records, not the nutrient content of any particular item. The derived classified negative count is 28 and unresolved count is 7. Check the table explicitly: 21 + 28 + 7 = 56. Every entry has a meaning and the categories are mutually exclusive for the particular test in this exercise.
Core worked answer: The positive fraction of valid classifications is 21/49, giving 42.9% when expressed as a percentage to one decimal place. The share of all submitted records that already have a confirmed positive outcome is 21/56, giving 37.5%. Both use the same numerator, but the populations differ. When writing a result, add the words among valid records or among all submissions; otherwise the calculated figure may be misleading despite correct arithmetic.
Extension 33: Calculate the possible range of final positive shares if every unresolved result eventually becomes positive or negative. The minimum is 21/56 = 37.5%; the maximum is (21+7)/56 = 50.0%. The interval is a logical bound under the stated assumption, not a prediction. There is not enough information to name one exact final rate. The important checking habit is to retain the fraction and its definitions until the final sentence, rather than making an unqualified statement about the entire set of real-world samples.
Practice 34: Review exercise sheets
The fictional teaching group recorded 52 submitted records, 40 valid classifications and 20 positive outcomes. Before using a calculator, identify the population described by the question. The numbers concern records, not the nutrient content of any particular item. The derived classified negative count is 20 and unresolved count is 12. Check the table explicitly: 20 + 20 + 12 = 52. Every entry has a meaning and the categories are mutually exclusive for the particular test in this exercise.
Core worked answer: The positive fraction of valid classifications is 20/40, giving 50.0% when expressed as a percentage to one decimal place. The share of all submitted records that already have a confirmed positive outcome is 20/52, giving 38.5%. Both use the same numerator, but the populations differ. When writing a result, add the words among valid records or among all submissions; otherwise the calculated figure may be misleading despite correct arithmetic.
Extension 34: Suppose a valid record is chosen uniformly at random. The probability it is positive is 20/40 = 50.0%. If instead a submission is chosen at random from all records, the probability that it already has a confirmed positive result is 20/52 = 38.5%. These are two different events and sample spaces. Name the sample space in the answer. The important checking habit is to retain the fraction and its definitions until the final sentence, rather than making an unqualified statement about the entire set of real-world samples.
Practice 35: School festival
The fictional teaching group recorded 90 submitted records, 72 valid classifications and 36 positive outcomes. Before using a calculator, identify the population described by the question. The numbers concern records, not the nutrient content of any particular item. The derived classified negative count is 36 and unresolved count is 18. Check the table explicitly: 36 + 36 + 18 = 90. Every entry has a meaning and the categories are mutually exclusive for the particular test in this exercise.
Core worked answer: The positive fraction of valid classifications is 36/72, giving 50.0% when expressed as a percentage to one decimal place. The share of all submitted records that already have a confirmed positive outcome is 36/90, giving 40.0%. Both use the same numerator, but the populations differ. When writing a result, add the words among valid records or among all submissions; otherwise the calculated figure may be misleading despite correct arithmetic.
Extension 35: Use the table to compare count and rate. The absolute positive count is 36, but the valid positive rate is 36/72 = 50.0%. A group with a larger positive count may have a smaller rate if its valid denominator is sufficiently large. A fair comparison with a second group requires that second group’s positive count and relevant valid total. Do not invent them. The important checking habit is to retain the fraction and its definitions until the final sentence, rather than making an unqualified statement about the entire set of real-world samples.
Practice 36: Revision sample bank
The fictional teaching group recorded 48 submitted records, 32 valid classifications and 8 positive outcomes. Before using a calculator, identify the population described by the question. The numbers concern records, not the nutrient content of any particular item. The derived classified negative count is 24 and unresolved count is 16. Check the table explicitly: 8 + 24 + 16 = 48. Every entry has a meaning and the categories are mutually exclusive for the particular test in this exercise.
Core worked answer: The positive fraction of valid classifications is 8/32, giving 25.0% when expressed as a percentage to one decimal place. The share of all submitted records that already have a confirmed positive outcome is 8/48, giving 16.7%. Both use the same numerator, but the populations differ. When writing a result, add the words among valid records or among all submissions; otherwise the calculated figure may be misleading despite correct arithmetic.
Extension 36: Calculate the negative share among valid results: 24/32 = 75.0%. This counts classified negatives and uses only the valid-result population. An unresolved record cannot silently become negative. To check, the valid positive share 25.0% and valid negative share 75.0% should total about one hundred percent, allowing for rounding. Explain why the population of interest excludes the 16 unresolved submissions. The important checking habit is to retain the fraction and its definitions until the final sentence, rather than making an unqualified statement about the entire set of real-world samples.
Practice 37: Practice board A
The fictional teaching group recorded 36 submitted records, 24 valid classifications and 6 positive outcomes. Before using a calculator, identify the population described by the question. The numbers concern records, not the nutrient content of any particular item. The derived classified negative count is 18 and unresolved count is 12. Check the table explicitly: 6 + 18 + 12 = 36. Every entry has a meaning and the categories are mutually exclusive for the particular test in this exercise.
Core worked answer: The positive fraction of valid classifications is 6/24, giving 25.0% when expressed as a percentage to one decimal place. The share of all submitted records that already have a confirmed positive outcome is 6/36, giving 16.7%. Both use the same numerator, but the populations differ. When writing a result, add the words among valid records or among all submissions; otherwise the calculated figure may be misleading despite correct arithmetic.
Extension 37: Calculate the unresolved share of all submissions: 12/36 = 33.3%. The denominator is the full set of submissions because unresolved items still belong to that set. The percentage is not a rate of negative tests, nor a rate of any food substance. It describes only how much of the data remains unclassified, an important limitation on stronger conclusions. The important checking habit is to retain the fraction and its definitions until the final sentence, rather than making an unqualified statement about the entire set of real-world samples.
Practice 38: Practice board B
The fictional teaching group recorded 60 submitted records, 45 valid classifications and 27 positive outcomes. Before using a calculator, identify the population described by the question. The numbers concern records, not the nutrient content of any particular item. The derived classified negative count is 18 and unresolved count is 15. Check the table explicitly: 27 + 18 + 15 = 60. Every entry has a meaning and the categories are mutually exclusive for the particular test in this exercise.
Core worked answer: The positive fraction of valid classifications is 27/45, giving 60.0% when expressed as a percentage to one decimal place. The share of all submitted records that already have a confirmed positive outcome is 27/60, giving 45.0%. Both use the same numerator, but the populations differ. When writing a result, add the words among valid records or among all submissions; otherwise the calculated figure may be misleading despite correct arithmetic.
Extension 38: Calculate the possible range of final positive shares if every unresolved result eventually becomes positive or negative. The minimum is 27/60 = 45.0%; the maximum is (27+15)/60 = 70.0%. The interval is a logical bound under the stated assumption, not a prediction. There is not enough information to name one exact final rate. The important checking habit is to retain the fraction and its definitions until the final sentence, rather than making an unqualified statement about the entire set of real-world samples.
Practice 39: Campus display C
The fictional teaching group recorded 80 submitted records, 64 valid classifications and 48 positive outcomes. Before using a calculator, identify the population described by the question. The numbers concern records, not the nutrient content of any particular item. The derived classified negative count is 16 and unresolved count is 16. Check the table explicitly: 48 + 16 + 16 = 80. Every entry has a meaning and the categories are mutually exclusive for the particular test in this exercise.
Core worked answer: The positive fraction of valid classifications is 48/64, giving 75.0% when expressed as a percentage to one decimal place. The share of all submitted records that already have a confirmed positive outcome is 48/80, giving 60.0%. Both use the same numerator, but the populations differ. When writing a result, add the words among valid records or among all submissions; otherwise the calculated figure may be misleading despite correct arithmetic.
Extension 39: Suppose a valid record is chosen uniformly at random. The probability it is positive is 48/64 = 75.0%. If instead a submission is chosen at random from all records, the probability that it already has a confirmed positive result is 48/80 = 60.0%. These are two different events and sample spaces. Name the sample space in the answer. The important checking habit is to retain the fraction and its definitions until the final sentence, rather than making an unqualified statement about the entire set of real-world samples.
Practice 40: Final dataset D
The fictional teaching group recorded 120 submitted records, 96 valid classifications and 36 positive outcomes. Before using a calculator, identify the population described by the question. The numbers concern records, not the nutrient content of any particular item. The derived classified negative count is 60 and unresolved count is 24. Check the table explicitly: 36 + 60 + 24 = 120. Every entry has a meaning and the categories are mutually exclusive for the particular test in this exercise.
Core worked answer: The positive fraction of valid classifications is 36/96, giving 37.5% when expressed as a percentage to one decimal place. The share of all submitted records that already have a confirmed positive outcome is 36/120, giving 30.0%. Both use the same numerator, but the populations differ. When writing a result, add the words among valid records or among all submissions; otherwise the calculated figure may be misleading despite correct arithmetic.
Extension 40: Use the table to compare count and rate. The absolute positive count is 36, but the valid positive rate is 36/96 = 37.5%. A group with a larger positive count may have a smaller rate if its valid denominator is sufficiently large. A fair comparison with a second group requires that second group’s positive count and relevant valid total. Do not invent them. The important checking habit is to retain the fraction and its definitions until the final sentence, rather than making an unqualified statement about the entire set of real-world samples.
Transfer test and self-review
Make two fictional datasets with the same number of positive observations and different numbers of valid tests. Calculate both valid-result rates. Then reverse the challenge by constructing two datasets with different positive counts but identical proportions. Compare the results in complete sentences that name the denominators. This exercises the mathematical method rather than one memorised answer.
Prepare a third dataset with unresolved observations. Compute a possible lower and upper bound for the final positive share of all records. Explain why an exact final percentage cannot be deduced without classifying the unresolved items. Next, design a bar chart of the counts, label its axis correctly, and check that the displayed scale does not exaggerate differences.
Review Vol 0559: Biological food-test observations, Vol 0556: serving-label Mathematics, and the PSLE decision-first route. A test-result frequency is not the same as a mass concentration. These exercises train reliable mathematical interpretation while respecting that scientific difference.
