Two camera crops can retain the same percentage of a photograph and still show different evidence. Mathematics can tell us how many pixels remain, whether an aspect ratio is possible and how a display has been scaled. It cannot tell us what was outside a cropped frame. This G3 SEC guide teaches that boundary between calculation and judgement.
Begin with width, height and rectangular area, then move to ratios, similar figures, target-screen constraints and realistic rounding. All numerical datasets in this lesson are invented for practice. They are not records of a particular school exhibition or examination question.
Quick answer: find the original dimensions; identify the operation; simplify the required ratio; solve for the allowable crop dimensions; calculate retained area as a fraction of original area; and check feasibility, units, placement and any rounding rule before making a recommendation.
Contents
- What a crop changes
- Quick calculation route
- The official 2027 SEC Mathematics reference
- Why aspect ratio is a fraction
- Three operations that must not be mixed
- A full-width conversion to 16:9
- A square crop and its position
- Linear scale and area scale
- Retained area and removed area
- Pixel position is not physical size
- The danger of rounding at a boundary
- The missing-corner problem
- Compare two proposed poster layouts
- A measurement and checking ledger
- From PSLE fractions to advanced G3 judgement
- Thirty worked crop-decision cases
- Practice plan
- FAQ
- Official sources
What a crop changes
Cropping removes part of an image. It can change what the reader sees, but it does not change what was originally in the full camera frame. The Mathematics begins by identifying the rectangle before the crop and the rectangle that remains. If an original frame is 4000 pixels wide and 3000 pixels high, it contains 12,000,000 pixel positions. A crop 3000 pixels wide by 3000 pixels high contains 9,000,000. Under these stated dimensions, 75% of the original pixel area is retained and 25% has been removed. This is a numerical statement about dimensions, not a judgement about which photograph is more truthful.
The difference matters because a small-looking visual change can remove a large area, and a picture displayed at a larger size can still contain fewer original pixels. This volume extends the physics camera-focus evidence in Vol 0791 into calculations about the displayed frame. The camera focuses light; the crop determines which recorded portion is shown. Treat the two stages separately. A correct image formation explanation does not, by itself, justify a misleading selection of the final picture.
Quick calculation route
Start with the original width and height, including their units. Reduce the width-to-height ratio to its simplest form. Record the crop width and height. Find its area as width multiplied by height, and divide that area by the original area when the problem asks for the proportion retained. Convert the resulting fraction to a percentage if required. For a resized copy of the same crop, use a common scale factor for both dimensions. If the scales differ, the result has been stretched and no longer preserves similarity.
The most common error is to subtract percentages of width and height rather than comparing areas. If a crop keeps 80% of the original width and 80% of the original height, it retains 0.8 times 0.8, or 64%, of the original area, not 80%. A second error is to mistake a change of display size for new information captured by the camera. An image stretched to twice the displayed width and height occupies four times the screen area, but the original pixel record has not automatically gained four times as much detail. Follow the quantity named in the question.
The official 2027 SEC Mathematics reference
The new Singapore–Cambridge Secondary Education Certificate begins in 2027. In the school-candidate G3 Mathematics syllabus, the code is K310. The assessment has two written papers, each 2 hours 15 minutes, 90 marks and 50% of the examination result. Geometry, measurement, ratios, similarity and problem-solving with contextual information all support the kind of reasoning used in this constructed camera-crop context. This is not a claim that an image-editing practical or a separate digital photography paper is prescribed by SEAB.
Examination questions supply their own diagrams and numerical conditions. Read those conditions carefully. If a problem describes a rectangle in centimetres, answer in square centimetres when calculating area. If it describes pixel dimensions, it is natural to count pixel positions or report a ratio without pretending that the result is a physical paper size. The SEAB G3 syllabus index for 2027 is the definitive reference for subject codes and format. Earlier O-Level codes should not be casually substituted for SEC codes.
Why aspect ratio is a fraction
An aspect ratio compares width with height. In the 4000-by-3000 teaching frame, width:height is 4000:3000, or 4:3 after dividing both terms by 1000. A square image has ratio 1:1. A widescreen image may have ratio 16:9. Ratios describe shape, not the total number of pixels. Frames 4000 by 3000 and 800 by 600 share the ratio 4:3, although the second contains far fewer pixel positions. A candidate should not conclude that two same-ratio frames have equal resolution.
Simplification matters because it lets you compare shapes quickly. Ratio 3840:2160 reduces to 16:9; ratio 1920:1080 also reduces to 16:9. When deciding whether one picture can be uniformly resized to another without cropping or stretching, compare these simplified ratios. A matching ratio supports a uniform scale transformation. A non-matching ratio means at least one further operation—cropping, padding or distortion—is needed to fill the target exactly. The word ‘fit’ in an examination question is not enough: determine which operation is permitted.
Three operations that must not be mixed
Cropping removes part of the original visible field. Uniform resizing changes displayed dimensions by the same factor in each direction. Stretching changes width and height by different factors and alters shapes. They can be combined in a real editing workflow, but the question should say which is done. Imagine a 4000 by 3000 image being resized to 2000 by 1500: each dimension is halved and the 4:3 shape is preserved. Imagine instead a 4000 by 3000 image being changed to 2000 by 1000 without cropping: the horizontal scale is one-half and the vertical scale is one-third, so circles become distorted.
A common wrong answer says ‘the photograph lost 50% of its subject’ because the display width was halved. Uniform resizing does not remove half the subject; it changes display dimensions. Cropping the left half does remove part of the original scene. The best answer describes exactly which stage could alter visual evidence, then calculates the relevant change. If the problem says a digital photograph is ‘compressed’, do not immediately interpret that as geometric shrinking or cropping. Digital compression may reduce file storage without removing the same geometric region.
A full-width conversion to 16:9
Suppose an original 4000 by 3000 image must become a 16:9 rectangle while keeping all 4000 pixels of its width and cropping only from the top and bottom. Let the new height be h. The equation 4000:h = 16:9 gives h = 4000 × 9 ÷ 16 = 2250 pixels. The crop retains 4000 × 2250 = 9,000,000 pixel positions, or 75% of the original 12,000,000. If the crop is centred vertically, 750 pixels are removed in total: 375 from the top and 375 from the bottom.
Now test the opposite direction. Suppose we insist on keeping the full original height of 3000 pixels and still demand a 16:9 shape. The required width would be 3000 × 16 ÷ 9, about 5333.33 pixels, wider than the original 4000-pixel frame. That cannot be accomplished by cropping alone, because cropping cannot add width. The only possibilities under different rules are padding, extending the canvas, inventing new content or distorting the image. When a question sets a constraint, check feasibility before calculating a supposed crop that exceeds its original.
A square crop and its position
For a 4000 by 3000 original, the largest square crop has side length 3000 pixels if it stays inside the frame. The area is 9,000,000 pixels squared in a mathematical area comparison, 75% retained. But the position of the square is not determined by the dimensions alone. There are 1000 excess pixels in the horizontal direction. A centred crop removes 500 pixels from each side; a left-anchored crop removes all 1000 from the right. Both have the same numerical area, but they show different visual evidence.
This illustrates a limit of mathematical proof. The area computation establishes the proportion retained, not which person or object was removed. For a claim about whether a photo depicts an entire group, the crop boundary and original scene matter. A school exhibit that shows only the central person cannot be called a photo of everyone present unless evidence outside the crop supports the claim. In the later English volume, readers will examine how a caption can overstate what a crop establishes.
Linear scale and area scale
When a rectangle is uniformly scaled by factor k, each length becomes k times the original, and its area becomes k squared times the original. For example, a 12 cm by 9 cm print has area 108 square centimetres. Enlarging both dimensions by a factor of 1.5 gives 18 cm by 13.5 cm with area 243 square centimetres. The area factor is 2.25, which equals 1.5 squared. It would be wrong to say that the area increased by only 50% because each side grew by 50%.
The reverse problem is equally important. If a similar enlarged rectangle has four times the area, its linear scale factor is the square root of 4, or 2. If it has 2.25 times the area, the linear factor is 1.5. These principles connect to K310 similarity questions beyond photography. A camera print is simply a familiar rectangle for practising them. Always check that similarity is justified; the square-law does not apply to arbitrarily stretched shapes when the horizontal and vertical scale factors differ.
Retained area and removed area
Let A be the original rectangular area and B the crop area. The retained fraction is B/A, provided the crop lies completely inside the original and the dimensions use compatible units. The removed fraction is (A − B)/A, which also equals 1 − B/A. A crop from 4000 by 3000 to 3200 by 1800 has area 5,760,000 compared with 12,000,000 originally. The retained fraction is 0.48 or 48%, and the removed proportion is 52%. These are not two unrelated percentages; together they must total 100%.
The percentage of original width retained is 3200/4000, or 80%, while the height retained is 1800/3000, or 60%. Multiplying 0.8 by 0.6 gives 0.48, the correct area proportion. This is a useful arithmetic check whenever the crop is an axis-aligned rectangle. If someone reports 70% because they averaged 80% and 60%, challenge the reasoning: area is two-dimensional. If an examination asks how much of the photograph remains, check whether it means geometric area, width, subject count or some other quantity before selecting the formula.
Pixel position is not physical size
A digital image can be 4000 by 3000 pixels but printed at different physical sizes. Its aspect ratio stays 4:3 if the print is not distorted or cropped. To calculate physical print dimensions, extra information is needed, such as printing resolution or one physical dimension. You cannot conclude that 4000 pixels equals 40 cm without a specified conversion. The same is true for a graph image shown in a textbook: a centimetre on the page may represent many pixels or an unrelated scale.
If a question states that a 4000-pixel-wide image is printed 20 cm wide, the linear conversion is 200 pixels per centimetre under that print arrangement. Its 3000-pixel height would then occupy 15 cm if the image remains similar. Its printed area is 300 square centimetres. That numerical conversion does not mean every camera with 4000 horizontal pixels is physically 20 cm wide. Keep the conversion attached to the supplied setup. Distinguish ‘pixel’ as a count dimension from ‘cm’ as a physical length.
The danger of rounding at a boundary
Many crop calculations produce non-integer dimensions. For example, keeping a width of 3500 pixels with 16:9 ratio requires a height of 3500 × 9 ÷ 16 = 1968.75 pixels. A practical image grid cannot have a fractional pixel row. The question might request rounding to a whole pixel or to a value that does not exceed a maximum. The instruction matters. Choosing 1969 gives a ratio close to, but not exactly, 16:9. Choosing 1968 also approximates it. Neither should be called mathematically exact.
When a mathematical problem asks for the largest whole-number crop that fits a constraint, apply the inequality before rounding. If the maximum permitted height is 1968.75 pixels, 1969 exceeds it. Thus 1968 is the largest integer not exceeding the bound. If instead a printing service tolerates a tiny approximate ratio, the rule may be different. State the condition first. This is an examination habit that transfers to budgets, packing and capacity problems: solving the equation is not always the final decision.
The missing-corner problem
A student claims that two crops with equal area contain identical evidence. But consider a 3000-by-3000 square anchored at the left and another square anchored at the right of the same 4000-by-3000 frame. Each contains 9,000,000 original pixels. Their overlap is only 2000 pixels wide by 3000 pixels high, or 6,000,000 pixels. Equal area therefore does not mean equal content. The location of the crop rectangle is additional information that the percentage alone cannot supply.
One way to represent the left crop is horizontal coordinates from 0 to 3000; the right crop spans 1000 to 4000. Their common horizontal interval is 1000 to 3000, width 2000. The vertical interval is the same for both, height 3000. This elementary coordinate method reveals why visual arguments should preserve the original frame. In a real classroom question you might not need formal coordinate notation, but tracing the left and right boundaries prevents the false conclusion that all 75%-retained crops are interchangeable.
Compare two proposed poster layouts
Suppose a school must choose between layout A, a 3000-by-3000 square, and layout B, a 3600-by-2400 rectangle cropped from the same 4000-by-3000 original. Layout A retains 9,000,000 of 12,000,000 pixels, or 75%. Layout B retains 8,640,000, or 72%. Layout A therefore preserves more original pixel area by 360,000 positions, or 3 percentage points. But neither number proves that A tells a more complete story; its crop could omit crucial material near a side, while B could be positioned to keep the relevant event.
A decision requires both mathematics and stated purpose. If the poster must include a warning sign at the far left, ask whether that sign falls within each proposed crop. If the design has a required 3:2 shape, B meets that shape while A does not. A candidate who simply chooses the larger retained percentage may miss a feasibility constraint. The strongest examination answer checks area, target ratio and required contents before selecting an option. Clear reasoning is more important than allegiance to one visual style.
A measurement and checking ledger
Use a simple ledger: original width, original height, target ratio, permitted operation, crop width, crop height, retained area, retained percentage and the condition governing placement. Label units for dimensions and area. For a 4000-by-3000 original, 3600-by-2400 crop: 3600 × 2400 = 8,640,000, and 8,640,000 ÷ 12,000,000 = 0.72. Cross-check using independent fractions: 3600/4000 × 2400/3000 = 0.9 × 0.8 = 0.72. Two approaches yielding the same result reduce the chance of a place-value mistake.
Do not let a calculator conceal nonsense. The retained proportion of a crop fully inside the original cannot exceed 100%. Its width and height must each be no larger than the corresponding available original dimension, unless the task explicitly involves rotation or other transformations. The sum of removed and retained proportions should equal 1. A successful answer includes this reasonableness check even if the examination does not request a full paragraph about it.
From PSLE fractions to advanced G3 judgement
The same underlying skills begin with fractions, percentages, rectangles and ratio in the PSLE learner route and Mathematics learning hub. G3 problems then combine them with similarity, conditions, constraints and decisions. A learner who is weak at converting 0.72 to 72% should repair that skill first. There is no value in memorising a complex ‘camera crop formula’ when ordinary ratio and area reasoning already explain the calculation.
For a complete learning sequence, begin with the original frame and a drawn crop rectangle; next calculate dimensions and area; then test whether a proposed output ratio is achievable; finally decide whether the numerical comparison justifies the recommendation. This last stage is essential. Mathematics can establish exactly how much of the original rectangular area is retained. It cannot establish what an unseen person was doing outside the crop. That is an evidence question which the following English and EMS volumes will take up.
Thirty worked crop-decision cases
The following constructed cases progress from one-step arithmetic to composite constraints. Hide the solutions, attempt each calculation, explain which operation the scenario uses, and test the answer for reasonableness. Do not merely hunt for the nearest familiar formula: a question about crop position may require geometry and visual context rather than a new percentage.
Case 1: The original rectangle
Problem. The school camera records 4000 by 3000 pixels. Solution. The aspect ratio is 4:3 and the total pixel positions are 12,000,000. Transfer check. Do not confuse either result with a physical print size.
Case 2: A central square
Problem. A 3000 by 3000 square is taken from the 4000 by 3000 original. Solution. It retains 9,000,000/12,000,000 = 75% of the original area. Transfer check. Centred placement removes 500 pixels at each side, but location changes the subjects shown.
Case 3: Widescreen at full width
Problem. Keep 4000 pixels of width in a 16:9 crop. Solution. The height must be 2250 pixels; 75% of original area remains. Transfer check. A symmetrical crop removes 375 pixels from top and bottom.
Case 4: Widescreen at full height
Problem. Keep all 3000 pixels of height yet demand 16:9 by cropping alone. Solution. The required width is about 5333.33 pixels, larger than the original width. Transfer check. This crop is impossible under the rule because cropping does not add pixels.
Case 5: A three-to-two poster
Problem. Crop a 4000 by 3000 original to 3600 by 2400. Solution. The ratio is 3:2; retained area 8,640,000/12,000,000 = 72%. Transfer check. A claim that only 10% is removed has confused width reduction with area loss.
Case 6: A small but substantial crop
Problem. Use a 3200 by 1800 crop of a 4000 by 3000 image. Solution. Retained proportion is 0.8 × 0.6 = 0.48 or 48%. Transfer check. Averaging 80% and 60% would give a false 70% result.
Case 7: Uniform reduction
Problem. Resize a 4000 by 3000 image to 2000 by 1500 without cropping. Solution. Each dimension has scale factor one-half and the aspect ratio remains 4:3. Transfer check. The displayed area is one quarter; this does not mean three quarters of the original scene have been cropped away.
Case 8: Stretching, not similarity
Problem. Change an image from 4000 by 3000 to 2000 by 1000 with no cropping. Solution. Horizontal scale is one-half but vertical scale one-third. Transfer check. The resulting image is distorted; do not use similar-figure reasoning for the transformation.
Case 9: Square enlargement
Problem. Print a 12 cm by 12 cm square at 18 cm by 18 cm. Solution. The linear factor is 1.5 and the area factor is 2.25. Transfer check. The percentage increase in area is 125%, not 50%.
Case 10: An area quadruples
Problem. A similar rectangular print becomes four times the original area. Solution. Its linear dimensions are twice the original under uniform scaling. Transfer check. The square root of the area factor gives the length factor.
Case 11: The 16:9 check
Problem. A 3840 by 2160 image is compared with a 1920 by 1080 display. Solution. Both have aspect ratio 16:9; a uniform scale factor of one-half connects them. Transfer check. Their pixel-position totals are not equal; the smaller has one quarter as many.
Case 12: Area loss by subtraction
Problem. Original area is 12,000,000 and crop area is 5,760,000. Solution. Removed area is 6,240,000, which is 52% of the original. Transfer check. Check that 52% removed plus 48% retained equals 100%.
Case 13: Printing conversion
Problem. A 4000-pixel-wide 4:3 image is printed 20 cm wide. Solution. At uniform scale, its height is 15 cm and its area is 300 square centimetres. Transfer check. The conversion applies to this print, not to every camera recording 4000 pixels wide.
Case 14: Missing conversion
Problem. Only pixel dimensions of a photograph are provided, with no print information. Solution. Physical width in centimetres cannot be determined uniquely. Transfer check. Keep the answer in pixels or dimensionless ratio instead of inventing a print resolution.
Case 15: Non-integer bound
Problem. Keep 3500 pixels of width in an exact 16:9 target. Solution. Required height is 1968.75 pixels. Transfer check. If only whole-pixel heights that do not exceed the bound are allowed, the maximum is 1968.
Case 16: Equal area, different region
Problem. Two 3000-by-3000 crops are anchored to opposite sides of a 4000-by-3000 original. Solution. Both retain 75%, but they do not contain identical content. Transfer check. Their overlap is 2000 by 3000, or 6,000,000 pixel positions.
Case 17: Which has more retained area?
Problem. A is 3000 by 3000 and B is 3600 by 2400. Solution. A retains 9,000,000, B 8,640,000; A has 360,000 more positions. Transfer check. Area alone cannot prove that the layout contains the necessary evidence.
Case 18: A ratio is not a count
Problem. A 600 by 450 and a 4000 by 3000 image are compared. Solution. Both are 4:3 despite having different numbers of pixels. Transfer check. Ask whether the question wants shape or total captured positions.
Case 19: A 90% width crop
Problem. Keep 90% of width and 100% of height in an axis-aligned rectangular crop. Solution. Retained area is 90% of original because the height factor is 1. Transfer check. Do not assume the loss occurred equally on left and right.
Case 20: Two independent losses
Problem. Keep 70% of the width and 80% of the height. Solution. Retained fraction is 0.70 × 0.80 = 0.56, or 56%. Transfer check. The removed proportion is 44%, not the sum of 30% and 20%.
Case 21: Total 75% removed?
Problem. A student says a half-width and half-height crop retains 50%. Solution. A half-width and half-height rectangle retains 0.5 × 0.5 = 25%. Transfer check. Therefore 75% of original area is removed.
Case 22: Width from area
Problem. A crop has 1800-pixel height and 5,760,000-pixel area. Solution. Its width is 5,760,000 ÷ 1800 = 3200 pixels. Transfer check. Check that both dimensions fit the original before calling it a crop.
Case 23: Height from ratio
Problem. A 3:2 poster crop has width 3600 pixels. Solution. The height is 3600 × 2 ÷ 3 = 2400 pixels. Transfer check. An apparently precise ratio means the exact supplied dimensions must be consistent.
Case 24: Placement of a sign
Problem. A required sign lies near the right edge, outside a proposed left-anchored square. Solution. The square may retain 75% of area while omitting the key sign. Transfer check. Recommendation must consider the stated content constraint, not percentage alone.
Case 25: A padded display
Problem. A 4:3 image is placed inside a 16:9 window with blank margins. Solution. The source frame need not be cropped at all if padding is allowed. Transfer check. State that the displayed window and the content rectangle have different ratios.
Case 26: A percentage-point decision
Problem. Layout A keeps 75% of the original area and B keeps 72%. Solution. A retains 3 percentage points more of the original. Transfer check. Do not call the difference 3% of B without computing the relative percentage separately.
Case 27: Compare crop dimensions
Problem. Crop A is 3600 by 2400; crop B is 3200 by 1800. Solution. A area is 8,640,000 and B area is 5,760,000. Transfer check. A has 1.5 times B’s pixel area, though both can still be positioned differently.
Case 28: A wrong rounding answer
Problem. A required width is 3500 and the maximum exact-ratio height is 1968.75. Solution. For the largest whole-pixel crop not exceeding that height, choose 1968. Transfer check. Rounding to 1969 violates the stated maximum.
Case 29: Unclear units
Problem. A classroom note says ‘size 20 by 15’ and asks for area. Solution. Numerically area is 300 square units, but units need clarification. Transfer check. Do not label it 300 square centimetres unless the measurement unit was given.
Case 30: A final composite problem
Problem. A 4000-by-3000 original must fit a 16:9 display using a full-width crop; the final display is 1920 by 1080. Solution. First crop to 4000 by 2250, retaining 75%; then resize uniformly by factor 0.48 to 1920 by 1080. Transfer check. The second step changes display scale, not which region of the original was retained.
Four-week route from basic arithmetic to examination decisions
Week one is for width, height, area, simple fractions and converting decimals into percentages without a camera context. Draw rectangles and label the missing side of each. In week two, calculate 4:3, 3:2, 1:1 and 16:9 crops under stated constraints, showing each step and checking that the result fits the source rectangle. Week three introduces uniform enlargement, similar figures and the difference between the scale of lengths and the scale of areas. In week four, solve combined problems with word limits and a short justification of why the recommended crop is or is not suitable.
For each wrong answer, diagnose its source: an arithmetic slip, wrong ratio order, a mistaken unit, an unrecognised impossibility, or an unsupported inference. Repair that specific link before repeating the whole worksheet. Complete at least five mixed problems using both direct area multiplication and retained-fraction multiplication so you can cross-check under time pressure. In the examination, a clear labelled computation followed by one proportionate conclusion is more persuasive than a paragraph that repeats the question.
Frequently asked questions
How do I find a ratio?
Write width:height, divide both by their greatest common factor, and keep order consistent.
Does resizing mean cropping?
No. Uniform resizing changes scale but retains the full field; cropping removes a selected region.
Why multiply two retained fractions?
A rectangle has two independent dimensions. Its area scales by width factor times height factor.
Can a crop contain more than 100% of the original?
Not if it stays inside the original without adding content; an answer over 100% signals a mistake or a different operation.
What if the target ratio cannot fit?
Test constraints. The requested dimensions may require padding, stretching or a different crop rule.
Are pixels a physical length?
No. Pixel dimensions become centimetres only after a specific scale or print condition is supplied.
Can I say the photograph is more honest because it retains more pixels?
No. Retained area is relevant but does not prove that the crucial event or person was included.
What does a similarity question require?
Use the same linear factor in both directions and square that factor for areas.
Is this a new examination paper?
No. It is a constructed context for K310 Mathematics, not an additional assessed subject.
How should I revise?
First practise fractions and percentages, then area and ratio, then similarity, constraints and evidence-safe conclusions.
Official source and linked learning route
Check the official 2027 G3 Mathematics K310 syllabus and SEAB’s G3 syllabus index for current papers and scope. Revisit the earlier mathematics data-evidence volume 0788 when distinguishing a calculated result from a claim about people. Continue into the following English visual-reading guide to test whether the chosen crop and caption are meaningfully aligned.
