The question hiding inside a circle
Two school surveys display their findings as equally large circles. One shows a 90-degree slice for visitors who chose a science trail; the other shows a 120-degree slice. Can we tell which event had more visitors choosing the trail? Not yet. The slices describe proportions, but the total numbers of visitors may differ. A larger angle in one survey can represent fewer people than a smaller angle in another. This guide teaches you to move fluently between sector angles, fractions, percentages, frequencies and the total sample. It also teaches you when the display does not contain enough evidence for a claim.
What this volume owns
The earlier sampling and denominator guide, Vol 0214 establishes what a surveyed group represents. Vol 0186 on relative and expected frequency explains count-versus-proportion distinctions, while Vol 0210 on misleading graphs audits presentation. Our new task is different: reverse-read a pie chart to reconstruct unknown frequencies, recover totals and test comparisons between unequal groups. We build especially on Vol 0378, where a graph about inherited colour cannot establish a natural-selection mechanism without biological evidence. In this Mathematics chapter, first establish what the numbers actually say.
The correct official examination route
The SEAB 2027 G3 school-candidate directory identifies G3 Mathematics as K310 (the earlier code 4052). The 2027 K310 Mathematics syllabus includes pie charts and other statistical diagrams, interpretation of data and real-world mathematical problems. It specifies Paper 1 and Paper 2 at 90 marks each, each lasting 2 hours 15 minutes, each contributing 50% of the subject assessment. An approved scientific calculator is permitted in both papers, but essential working still matters. Exact wording and requirements must come from the current syllabus for your own examination year rather than from a recycled revision sheet.
A circle is 360 degrees, not 100 people
A full pie chart always represents 360 degrees and 100% of the group being displayed. The total group might contain 20, 120, 1,000 or another number. The angle is a share of that full circle. It is not a direct count until you know the whole. This is the first check whenever a question says ‘how many’. A 90-degree sector is one quarter of whatever total has been specified. If the group contains 48 respondents, that sector represents 12 people. If the group contains 400, the same sector represents 100.
Three representations of the same share
For a sector of 72 degrees, the fraction of the circle is 72/360 = 1/5. Its percentage is 20%. If the survey total is 150, its frequency is 150 × 1/5 = 30. You should be able to move in both directions without memorising a separate formula for each task. The controlling idea is always part divided by whole. Write the total beside the circle before calculating. That small habit protects your answer from the common mistake of treating a sector marked 72° as 72 students.
Forward calculation: frequency to angle
Suppose a class of 40 pupils chooses among four revision methods. Twelve prefer worked examples. Their sector angle is (12/40) × 360° = 108°. The numerator counts those choosing worked examples; the denominator counts all pupils in that class. Check reasonableness: twelve out of forty is 30%, so the slice should be a little smaller than a third of the circle. Because one third of 360° is 120°, 108° makes sense. The check is conceptual; it catches a wrong denominator without redoing all arithmetic.
Reverse calculation: angle to frequency
A fictional garden-survey chart has 180 respondents and a 60-degree sector for birds. Its frequency is (60/360) × 180 = 30. Work backwards to check: 30/180 = 1/6, and one sixth of a circle is 60°. If a calculation gives 300 birds from 180 respondents, stop before copying it into the answer. That result violates the total. Checking the possible range is especially important in examinations because an incorrect operation can yield a neat-looking whole number.
Recover the total from one labelled slice
A poster states that 24 respondents selected a particular route and that their sector covers 54 degrees. The full sample size is 24 × 360/54 = 160. This is an inverse-proportion problem. The 24 people make up 54/360 of the whole, so divide 24 by that fraction. Write the relationship first: 24/N = 54/360. Solving gives N = 160. When an exam question supplies an angle and a sector frequency, do not confuse the known category count with the unknown survey total.
The missing-angle principle
A pie chart has sectors of 45°, 105°, 90° and one unlabelled sector. The missing angle is 360° − (45° + 105° + 90°) = 120°. This step can be completed before you know the total number surveyed. If the total is later stated as 60, the missing frequency is 120/360 × 60 = 20. Break an unfamiliar task into two parts: first reconstruct the missing share, then convert that share into a count. Trying to calculate everything at once invites avoidable copying errors.
A sum of angles is a quality-control check
If stated sector angles add to 355°, the remaining 5° may be a genuine unlabelled category—or the displayed numbers may have been rounded. Read the question carefully. If it states that all sectors are labelled and angles have been rounded, do not invent a new category containing exactly 5°. If it clearly asks for one missing angle in an exact diagram, subtract the listed angles from 360°. A mathematical explanation should distinguish a missing quantity from a measurement or presentation limitation. The wording determines whether an exact subtraction is justified.
Use 360 once, not repeatedly in the wrong place
A student calculates the number represented by a 144° sector out of 90 respondents as 144 × 90 / 100. That treats degrees as percentages. The correct fraction is 144/360 = 0.4, and the frequency is 36. Degrees and percentages are different scales, even though both describe parts of a whole. A useful check is to convert the angle mentally: 180° means half; 90° means a quarter; 36° means a tenth. These anchor values make many examination questions faster and safer.
A first comparison with unequal samples
Survey A contains 120 respondents, 30% of whom chose the garden. Survey B contains 200 respondents, 20% of whom chose the garden. A has the higher proportion. Its count is 36; B’s count is 40. Therefore B has more respondents choosing the garden, despite its smaller sector. If the question asks ‘which survey has a larger proportion?’, answer A. If it asks ‘which survey has more people?’, answer B. Those are different questions with different correct answers, and the total is needed to distinguish them.
Read a headline as a mathematical claim
The same two surveys are displayed with the headline ‘Garden interest dropped from 30% to 20%’. It is defensible only as a comparison of proportions under comparable sampling conditions. The headline ‘Four fewer people chose the garden’ is false: the second survey has four more garden choices in absolute numbers. The phrase ‘interest dropped’ also implies a change over time, which the datasets may not support if they represent different schools. Mathematics can identify the direction of a percentage difference; it cannot invent a common population or timeline.
Case 1 — A 90-degree sector in a group of 48
A category occupies one quarter of the circle. Its count is 48/4 = 12. If a candidate writes 48 × 90, the answer is 4,320 and is immediately impossible. This example is intentionally easy: learn to recognise benchmark angles first, because you can then use them to check more difficult fractions quickly. A quarter-circle should never represent more than the whole group. The right mathematical expression includes the fraction 90/360. State the count as 12 respondents, not merely as the unexplained number 12.
Case 2 — Three categories and one unknown
An observation chart divides visits into 126° for flowering plants, 90° for birds, 54° for insects and an unknown remainder. Add the known angles to obtain 270°. The remaining angle is 90°. If 160 visitors answered the survey, the unlabelled category represents one quarter, or 40 visitors. You can verify the other frequencies: plants 56, birds 40 and insects 24. Their total is 120, leaving 40. A reverse solution is stronger when a second calculation restores the entire survey total.
Case 3 — Finding a total that must be a whole number
In a fictional chart, 36 pupils correspond exactly to a 72° sector. Then 36/N = 72/360, so N = 180. A total of 180 is a valid whole number, and the sector is 20% of it. If your answer were 180.6 pupils, you would need to check whether the angle was rounded, whether the question says approximate, or whether the calculation went wrong. People counted in a simple survey are whole individuals. Do not simply round every impossible result to the nearest whole person without first investigating the information given.
Case 4 — Two charts with different circles
A school prints a 100-millimetre-wide pie chart for 80 responses and a 60-millimetre-wide pie chart for 500 responses. The physical diameter of each printed circle has no bearing on the respondent count. In a standard pie chart, relative sector angle represents the within-group share. The designer may have changed the drawing size to fit the page. To compare category counts, use the stated totals and sector proportions. Treating the printed area as a count would confuse page layout with mathematical encoding.
Case 5 — A category has twice the angle, not necessarily twice the count
A 120° sector in Survey A looks twice as wide as a 60° sector in Survey B. If A has 60 responses, that sector contains 20 people; if B has 240 responses, its narrower sector contains 40. The sector in B has half the share but twice the count. A ratio of angles across different pies gives a ratio of proportions only. For absolute counts, multiply by each pie’s own sample size. This distinction is the most important repeated pattern in the volume.
Case 6 — Recovering a frequency after a survey update
A school’s original survey had 100 respondents and 40 selected Science. A later survey had 150 respondents, and the Science sector measured 108°. That later share is 30%, or 45 respondents. The percentage fell from 40% to 30%, while the observed count rose from 40 to 45. Do not write only ‘Science selections declined’ unless you specify the metric. If an exam asks for the change in number, calculate 45 − 40 = 5. If it asks for percentage-point change, calculate 30 − 40 = −10 percentage points.
Case 7 — Percentage change differs from percentage-point change
Moving from 40% to 30% is a decrease of 10 percentage points. Relative to the original 40%, the proportional reduction is (40−30)/40 = 25%. These descriptions are not interchangeable. A reporter saying ‘down 10%’ may be unclear: does that mean ten percentage points or ten percent of the previous share? In a Mathematics answer, label the unit of comparison. An exact statement says ‘the share fell by 10 percentage points’ or ‘the share was 25% lower relative to its initial percentage’. Precision turns a confusing headline into a testable claim.
Case 8 — Recovering a missing sample size from two sectors
A chart shows that ‘walking’ accounts for 144° and ‘cycling’ for 72°. The survey recorded 60 walkers. Since 144/360 = 2/5, the total is 60 ÷ (2/5) = 150. Cycling therefore accounts for 72/360 × 150 = 30 respondents. This two-stage task is not two independent formulas to memorise. One sector identifies the whole; once the whole is known, every other sector can be converted to a count. Label the intermediate total clearly so it is not mistaken for a category frequency.
Case 9 — Ratio between two categories
One sector measures 120° and another 80°. The ratio of their frequencies is 120:80 = 3:2 because both categories belong to the same chart and therefore share the same denominator. If together they represent 125 people, then five ratio parts represent 125; one part represents 25. Their frequencies are 75 and 50. This approach can be faster than first calculating the survey total. State why it works: within one pie chart, category frequencies are proportional to their sector angles.
Case 10 — Do not transfer the same ratio between charts
Survey A has category angles 120° and 80°, so their frequency ratio is 3:2. Survey B has angles 135° and 90°, also 3:2. That does not mean either category has the same count in both surveys. It means each pair has the same within-chart ratio. If A surveyed 75 people and B surveyed 300, B’s category counts may be four times A’s even though the ratio is identical. Ratios summarise relationships, not raw size. Always restore the missing denominator before asserting equality of counts.
Case 11 — Using a subtraction shortcut safely
In a group of 72 pupils, the three known sector counts are 18, 24 and 12. The fourth category has 72−18−24−12 = 18 pupils. Its angle is 18/72 × 360° = 90°. You could instead calculate the known angles and subtract from 360°, but that adds arithmetic without improving the answer. Choose a route that starts from the clearest given information. The best solution is not always the longest. Use a second method as a check only when a mistake is plausible or the question specifically requests the angle.
Case 12 — The information is not enough
A question supplies only a pie chart with one 45° sector and asks how many pupils chose that option, but gives no survey total or other absolute frequency. You can state the share, 45/360 = 1/8, or 12.5%. You cannot calculate the number of pupils uniquely. A complete answer identifies the missing total rather than guessing a familiar class size. This is valuable examination judgement: a mathematically justified ‘cannot determine’ is stronger than an invented numerical answer. State what additional fact would make the calculation possible.
Case 13 — Exact people and rounded angles
A diagram labels a sector as 33° and says the chart angles were rounded to the nearest degree. Out of 200 respondents, using 33/360 × 200 gives approximately 18.33. Do not conclude that exactly 18.33 people chose the option. The true sector angle may lie within a rounding interval, and the underlying respondent count must be an integer. Check possible counts by converting each to an exact angle. For example, 18 of 200 gives 32.4°, which rounds to 32°, not 33°; 19 gives 34.2°, rounding to 34°. Here the supplied numbers may be inconsistent under the stated rule, so flag that rather than pretending false precision.
Case 14 — Distinguish drawn angle from printed label
A published chart may look like a 100° sector but be labelled 96°. Unless the task asks you to measure an unlabelled diagram, use the stated numerical information. A drawing can be not to scale, distorted on a screen or printed with an imprecise edge. When measurements are requested, use the appropriate geometrical instrument and expected tolerance. When numbers are given, treat them as the data and show your working from them. This distinction prevents visual estimates from replacing explicit mathematical facts.
Case 15 — Check a claimed percentage with an angle
A school poster labels a 135° sector as ‘30%’. Convert the angle: 135/360 × 100% = 37.5%. The percentage is incorrect if the angle is exact. Before criticising the poster, check whether the sector covers a different category or whether the image was redrawn inaccurately. If the labels genuinely describe the same exact part of the circle, both cannot be right. Write the conflicting quantities and the conversion. A precise correction is more educational than calling the entire poster false or unreliable without identifying the mismatch.
Case 16 — Compare ratios without calculating totals
Two categories in one chart have angles of 150° and 75°. Their frequency ratio is 2:1, because 150:75 simplifies to 2:1. That is enough if the question asks for the relative number of responses. You do not need the overall sample size. However, if it then asks for the actual number in the 150° group, you need an absolute count or total. Identify which information is sufficient for which task. The ability to refuse an unnecessary calculation is as important as being able to perform it.
Case 17 — Partitioning the same total into fractions
An exam club surveys 240 pupils. The Mathematics sector is 36°, the English sector 108° and Science 144°. Their counts are 24, 72 and 96 respectively. The remaining angle is 360−36−108−144 = 72°, giving 48 for the final category. Confirm that 24+72+96+48 = 240. This restores the whole and checks your work without relying on the visual shape of the chart. Write each category with its label; otherwise a correct series of numbers can be assigned to the wrong choices.
Case 18 — Reverse a half-circle and a quarter-circle
Suppose half the respondents prefer fieldwork and one quarter prefer laboratory demonstrations. If the laboratory group contains 45 students, the whole contains 180. The fieldwork group contains 90. You can see this without multiplying by 360 at every step: a quarter of the whole is 45, so the whole is four times 45. Recovering the whole mentally is often faster. Nevertheless, write the relevant fraction when presenting a formal examination solution. A correct answer without a logical path may not earn method credit in a longer question.
Case 19 — Infer a total from a percentage and an angle
A 72° sector is labelled 20%. A second 144° sector is said to contain 56 pupils. The second sector is 40%, so the survey total is 56/0.4 = 140. The first sector then contains 28. Check that 20% corresponds to 72° and 40% to 144°. Multiple forms of the same data create an opportunity for consistency checking. If an angle and percentage contradict, resolve the conflict before using either as a reliable premise. Do not silently choose the number that makes arithmetic easiest.
Case 20 — Relative frequency is not a predicted future count
A classroom survey finds 18 of 60 learners choosing Science, a relative frequency of 0.3. Predicting 30 choices among 100 similar future respondents may be reasonable under a stated assumption, but it is an estimate rather than a guarantee. A pie chart is a description of observed data. It does not make the same share inevitable in the next group. This reinforces the distinction developed in Vol 0186. Read words such as ‘estimate’, ‘expected’ and ‘exactly’; they signal different levels of certainty.
Case 21 — A changing biological population
A fictional field sample finds 120 beetles, 36 dark and 84 light. The dark share is 30%, giving a 108° pie sector. A second sample includes 200 beetles, 120 dark and 80 light. The dark share is 60%, giving 216°. Dark beetles increased in both observed count and observed share. That numerical description is well supported. It does not prove natural selection, because the survey has not measured inheritance or unequal reproduction. Refer back to Vol 0378 for the missing biological evidence. Statistics tells you the pattern; Biology tests the proposed cause.
Case 22 — Interpreting the word ‘doubling’
In the beetle data, the dark share rises from 30% to 60%, so the proportion doubles. But the dark count rises from 36 to 120, more than tripling. The survey total itself changed from 120 to 200. The word ‘doubled’ must attach to the correct quantity. A careful answer says ‘the observed proportion of dark beetles doubled’ rather than ‘the number of dark beetles doubled’. When reporting a calculation, retain the noun that identifies its measurement. Most misleading numerical claims result from removing that noun or its denominator.
Case 23 — Same count, different proportions
Survey A records 24 insects of a particular type among 80 insects, or 30%, equivalent to 108°. Survey B also records 24 of the type, but among 120 total insects, giving 20%, equivalent to 72°. The same observed count can occupy different sectors because the denominators differ. Thus ‘the category shrank’ is misleading unless it specifies that the share decreased. A sound comparison states both counts and percentages if they are central to the interpretation. The calculation itself is simple; the wording is the harder part.
Case 24 — Compare three different survey totals
A, B and C survey 60, 100 and 200 people. Their blue-sector angles are 180°, 108° and 72°. Blue counts are 30, 30 and 40. Blue proportions are 50%, 30% and 20%. C has the largest absolute blue count, A the highest blue proportion, and A and B have equal blue counts. A question can legitimately ask for any of these three findings. Read the comparative word carefully—highest share, greatest frequency or equal count—then select the correct operation rather than automatically picking the largest slice.
Case 25 — The trap of a nonrandom sample
A chart records club preferences only among pupils who attended a lunchtime display. Even if the angles and counts are computed perfectly, the results need not represent the entire school. Some pupils were absent or unable to attend. This is a sampling limitation, not an arithmetic error. A fair report distinguishes the group actually surveyed from the population someone wishes to describe. The sampling owner develops that distinction in detail. In this volume it matters because a precise sector is not a licence to make an imprecise claim about everyone.
Case 26 — Missing sector with a ratio condition
A chart has known angles of 60° and 90°. Its two missing sectors are in the ratio 3:4. The remaining angle is 210°. Seven ratio parts represent 210°, so one part is 30°. The missing angles are 90° and 120°. If the survey total is 180, these represent 45 and 60 people. This problem combines pie-chart reconstruction with ratio reasoning. First determine what remains of the circle; then divide that remainder in the stated ratio. Finally restore counts if the total is supplied. Each step has a separate purpose.
Case 27 — The total can be inferred from a difference
Two sectors have angles 144° and 90°, and the first contains 30 more respondents than the second. Their angle difference is 54°, which is 15% of the circle. Thus 15% of the survey total equals 30 people, so the total is 200. The sector counts are 80 and 50, a difference of 30. This is a useful inverse method when no single category frequency is given. Express the difference as a share of the whole, then find the whole from the absolute difference.
Case 28 — A ratio of the whole versus a ratio of the remainder
A question states that half the respondents chose an activity. It then says the other two activities divide the remaining half in the ratio 2:3. Their shares of the whole are (1/2)×(2/5)=1/5 and (1/2)×(3/5)=3/10. The sector angles are 72° and 108°. A common error is to assign 2/5 and 3/5 of the entire circle, leaving nothing for the first activity. Every ratio has a reference whole. Underline phrases such as ‘of the remainder’ before calculating.
Case 29 — Reporting an interval rather than an exact category
Suppose the sample total is 360 and a sector angle is described as ‘approximately 45°’. The corresponding count is approximately 45, not necessarily exactly 45. The label tells us the angle is an estimate. If the examiner requires an exact count, look for additional numerical data. If none exists, report an estimate and state that precision is limited. Students sometimes mistake the convenience of a 360-person total for proof of exactness. A mathematical calculation inherits the limitations of the measurements or rounded quantities on which it depends.
Case 30 — How to read a multi-part question efficiently
Part (a) may ask you to find an unknown angle, part (b) the number represented, and part (c) which school had more participants. Do not begin with part (c) simply because the diagrams look easy to compare. Record each school’s total, reconstruct missing angle shares, then convert to frequencies. Reuse earlier results, with units, rather than starting from scratch. If a later part introduces a new sample size, explicitly mark the change. Careful page organisation helps prevent a correct value from one circle being used with the wrong denominator from another.
Case 31 — Contextual units matter
If a pie chart describes 36° of a monthly $2,000 expenditure budget, the amount is $200. If a different pie describes 36° of 200 survey respondents, the count is 20 respondents. The same angular fraction can represent money, time, objects or people. The formula is unchanged, but the meaning and allowable rounding differ. You cannot have 0.6 of a surveyed person, while a money amount may have cents if the problem permits. Always write the unit at the end. It helps the reader verify that the computed quantity matches the question.
Case 32 — Do not mistake diagrams for experimental evidence
A large proportion of a phenotype observed in a habitat is not proof that the phenotype causes survival. A chart can also be affected by seasonal visibility, selection of sampling sites or differing observation effort. In a Mathematics response, your immediate duty is to describe the accurately computed proportions. In a Science explanation, additional evidence would be needed to establish heredity and differential reproduction. Learn to separate calculations from claims. This is precisely why the next English volume will examine how a caption can exaggerate what a visual source shows.
Case 33 — Check fractions before decimal arithmetic
An angle of 150° is 150/360 = 5/12 of the whole. If the total is 96, the count is 96×5/12 = 40. Reducing the fraction first makes the computation shorter and lowers the chance of a rounding error. If you immediately enter 150÷360 and use a rounded decimal, you can introduce an unnecessary approximation. For exact classroom data, keep a fraction until the final step. The same habit helps with inverse problems involving simple ratios or missing sample sizes.
Case 34 — An impossible frequency is an alarm
An exact pie chart for a survey of 100 people has one sector labelled 37°. The implied count is 37/360 × 100 ≈ 10.28, not an integer. This does not mean that 10.28 people participated. Either the sector was rounded, the number of respondents was not 100, or there is a mismatch in the information. If the question explicitly says the sector is measured to the nearest degree, 10 people give a 36° angle and 11 give about 40°; neither rounds to 37°. Under these stated conditions the data are inconsistent. Recognising such a conflict is better than silently changing one of the facts.
Case 35 — Percentage of a category versus percentage of the whole
Of 200 survey respondents, 80 prefer a garden route. Half of those garden-route supporters prefer the shaded section. That gives 40 shaded-section supporters, or 20% of all respondents. It would be wrong to report 50% of the entire survey. The sentence ‘half of those’ restricts the denominator to a subgroup. The complete pie could show 40 people, or a 72° sector, for the shaded-section preference. Nested percentages are common in real-world context questions; identify the denominator at every stage.
Case 36 — A clear worked comparison in six sentences
Survey A includes 80 people and has a 135° sector for outdoor activities; Survey B includes 150 and has a 72° sector. A’s share is 37.5%, corresponding to 30 people. B’s share is 20%, also corresponding to 30 people. Therefore the number choosing outdoor activities was equal in the two surveys, although their proportions differed. A had the larger proportion, while neither survey had a larger count. This complete answer is only six sentences, but it reports the assumptions, operations and appropriate conclusion without exaggeration.
Case 37 — What a table can reveal that a pie cannot
Suppose an exam supplies category frequencies for three years. One pie chart per year may make each year’s composition easy to see, but comparing absolute trends requires the totals or an accompanying table. A table can show counts, a line graph can show change over time, and separate pies can show the share within each year. Choose the representation that answers the question. The curriculum lists multiple forms of statistical diagrams because no single form serves every purpose. Do not insist that a beautiful pie chart is automatically the clearest one.
Case 38 — Estimation is not concealment
A question asks for the approximate count represented by a sector estimated at 118° in a sample of 300. Calculate 118/360 × 300 ≈ 98.3, so approximately 98 respondents. Make clear that the answer is approximate, since the angle itself was estimated. If the actual chart was assembled from integer counts, the true count may differ slightly. The word ‘approximately’ preserves an honest link between input and output precision. It is not a sign that you have failed to find an exact answer; it is the correct handling of estimated data.
Case 39 — Why the calculator result needs a reasonableness test
An electronic calculator may return 30.00000001 from a rearranged expression. In a problem constructed from exact integer frequencies, this tiny difference is a numerical display effect rather than an additional fractional person. But do not treat every noninteger as harmless. A result of 30.7 from exact information is a meaningful warning. Check whether the sector angle was rounded and whether the sample total fits the question. Use the correct numerical conventions from K310, and show the relationship that generated the result rather than reporting an unexplained machine output.
Case 40 — Reverse-checking one category against another
A chart has 140 responses. If the Mathematics sector is 126°, its count is 49; if English is 90°, its count is 35. These sum to 84, leaving 56 responses. The remaining 144° therefore corresponds to 56. The consistency test is 49+35+56=140 and 126°+90°+144°=360°. Both totals are restored. A reliable solution often provides this double confirmation when a question has many categories. It is especially helpful before explaining a more advanced conclusion, because a small numerical error can contaminate an entire argument.
Case 41 — From counts to a defensible caption
A poster says, ‘Of 180 respondents, 54 selected a practical workshop.’ The share is 30% and the sector angle is 108°. An accurate caption might read: ‘Thirty per cent of the 180 surveyed respondents selected a practical workshop.’ Notice that it preserves the denominator. ‘Thirty per cent of the school preferred the workshop’ would be unjustified without representative sampling of the entire school. The calculation is a Mathematics skill; choosing the words is a communication skill. They belong together when interpreting a real-world task.
Case 42 — Three ways to audit an answer
First test the arithmetic: do the category counts add to the total and do the angles add to 360°? Second test the units: is the answer an angle, a percentage, a count or an amount of money? Third test the conclusion: does it compare equivalent denominators and the same category? These checks are quick but they protect different parts of your work. A learner who completes one check only may still miss a scientific or linguistic overclaim. In an unfamiliar examination scenario, the three checks supply a dependable final pass.
Case 43 — A longer mixed worked question
A school runs two fictional wildlife surveys. In Site A, 60 of 150 recorded insects are dark. In Site B, dark insects occupy a 150° sector of a 240-insect sample. Site A’s dark share is 40%, or 144°. Site B’s share is 150/360 = 5/12, giving 100 dark insects. Site B has more dark insects in the sample (100 versus 60), and a slightly higher sample proportion (about 41.7% versus 40%). That comparison does not establish that dark colouring is increasing over time: the sites and sampling conditions may differ.
Case 44 — Make the last sentence answer the actual question
If the command is ‘calculate the number’, end with the counted quantity. If it is ‘state which group has the greater proportion’, finish with the group and the comparable fractions or percentages. If it is ‘explain why the poster may be misleading’, name the missing denominator, sampling difference or changed scale. An answer can contain correct arithmetic and still not complete the task set. Read the instruction at the start and again before the final sentence. Correct working is necessary, but clear alignment with the question completes the answer.
Four-week practice from foundation to examination judgement
Week one: convert familiar angles such as 36°, 72°, 90°, 120° and 180° into fractions and percentages. Week two: reconstruct whole survey totals from a sector count and angle, including unknown categories. Week three: compare unequal survey totals and write conclusions that distinguish frequency from proportion. Week four: tackle a mixed real-world question with incomplete or rounded information and practise explaining when a unique result cannot be found. After each session, keep one example of a denominator error, an angle error and a claim that went beyond the evidence.
Frequently asked: is this topic a separate SEC paper?
No. It is a focused G3 Mathematics learner’s guide, not a new subject or paper. The official K310 syllabus governs the examination. This guide selects one valuable statistical reasoning skill and develops it in unfamiliar contexts. The same skill may appear with transport, school activities, science surveys or spending data. Treat the fictional beetle examples as original practice rather than a prediction of an examination question. Learn the relationship between part and whole so that the context can change without destabilising your method.
Frequently asked: must the number of people always be exact?
If the problem gives exact whole-person counts and exact angles that are mathematically consistent, the recovered frequency should be a whole number. If an angle is measured from a diagram or rounded, an estimated count may be required. Always read the question’s precision instruction. A decimal number from a calculator is not automatically an exact headcount, and rounding it without explanation is not always safe. Show the fraction and state whether the answer is exact or approximate. A clear distinction between measurement and counting demonstrates understanding.
Frequently asked: can a pie chart prove adaptation?
No. A pie chart shows a distribution of recorded categories at the time and within the sampled group. To argue a natural-selection process, additional biological evidence would be required about inherited variation, environmental pressure and differences in reproduction across generations. Vol 0378 explains those biological links. The graph can motivate an investigation or describe an outcome, but the mechanism is not automatically contained in the circle. This is why careful scientific interpretation and careful numerical reading reinforce one another without becoming the same question.
A calm final checklist and related routes
Before submitting, write the relevant total; distinguish angle from count; simplify the fraction; preserve exactness when possible; verify that all shares and counts fit the whole; compare like with like; and limit the final claim to the information supplied. For extended statistical interpretation, use Vol 0210, Vol 0214 and Vol 0359. For the primary-to-secondary bridge, use the PSLE Mathematics word-problem route. The authoritative references remain SEAB’s G3 syllabus directory and Mathematics K310. The next English and integrated EMS guides develop how a chart’s numbers can be stated accurately for a particular audience.
