Comparing two proportions that have different totals

What to assume they know

• Why forcing every fraction onto a scale of 100 makes them comparable — a percentage is a fraction over 100, and a fixed denominator is what makes fractions comparable
• Computing a percentage in your head by splitting it — computing a percentage of a quantity, and recovering the quantity
• Ordering fractions with unlike denominators
• Reading a table with row and column headings, and a horizontal bar graph with a two-series legend
• Writing large numbers in standard form, and estimating a quotient of two large numbers

What they should be able to do

• Judge whether two given proportions can be compared directly, and say what makes the direct comparison invalid
• Convert two part-out-of-total figures with unequal totals into percentages and rank them
• Show that a comparison of shortfalls gives the same verdict as a comparison of scores when both are done as percentages
• Read a product label as a set of shares of a stated total weight, and complete a table of those shares
• Check that the shares of one whole total 100, and use the check to catch an error
• Explain why equal percentages of unequal totals mean unequal amounts, and give an instance
• Estimate a country's share of a world total from figures written in standard form
• Decide which of several statements a percentage bar graph does and does not support

Where it usually goes wrong

• "Fewer marks lost means better." Only against the same maximum. 8 out of 50 is a bigger fraction than 10 out of 80, which is why the percentages reverse Eesha's verdict.
• "Vishu is right — you can't compare them." Different maxima block the direct comparison, not comparison itself. Treat him as half right and say which half.
• "More grams of an additive means a larger proportion." The badam-mix labels are built to break this: 24 g and 9 g, the same 6%.
• "Percentages of two products can be compared without checking the pack size." They can be compared as shares, which is often the question you want. They cannot be compared as amounts. Say which question is on the table before converting.
• "A larger percentage means a larger quantity." Madhu at 120 g and Madhav at 95 g. The lower percentage is on the bigger pack, and the answer still goes to Madhav — but only after the arithmetic, not before.
• "The shares might not add to 100 because of rounding." In this table they add exactly, and the chapter tells the student to check. A row that misses 100 is a signal to recheck a cell, not a curiosity.
• "A bar graph in percentages tells you how many people there are." It does not. The third statement in the graph item is unanswerable from the chart, and students will answer it anyway.
• "The seniors' bars settle the statement about people aged 60 or older." Read the band labels carefully before agreeing — the chart's bands are named by decade of life, and which of them covers ages from 60 upwards has to be argued from the labels rather than assumed.

Questions to check understanding

• Two scores out of different maxima, ranked, with the reasoning shown
• Given a product label, express each ingredient as a percentage of the stated total and verify the total is 100
• Given two labels with different pack sizes, answer a share question and an amount question about the same ingredient, and say which is which
• Match large populations to approximate percentage shares of a world total, using standard form
• Percentage of a vote total, plus a bounding argument about how many candidates there must have been
• Percentage of an area from a figure on a square lattice
• Read a two-series percentage bar graph and mark a list of statements valid or invalid, including at least one that the graph cannot settle

Examples worth working on the board

Values marked arithmetic added here are added here; the chapter prints no answer key for Part II.

• Example 1, three children and two test scores (Part II, §1.3, p.14). Eesha scored 42 out of 50 in English and 70 out of 80 in Science. Three positions are printed:
  • Eesha: she dropped just 8 marks in English but 10 in Science, so she reckons English went better;
  • Reema: Eesha scored more marks in Science, so Science went better;
  • Vishu: the scores cannot be compared at all because the maximum marks differ. The page then converts: (42/50) × 100 = 84%, (70/80) × 100 = 87.5%, and concludes Science. The argument the explanation owes the student: Vishu is right that the raw figures cannot be compared and wrong that comparison is impossible — he has identified the obstacle and mistaken it for a wall. Reema is right by accident: her method (more marks wins) would have failed if the maxima had gone the other way. And Eesha's method is not wrong in kind, only in execution: arithmetic added here, the losses as percentages are 8/50 = 16% and 10/80 = 12.5%, so counting what was lost, done properly, also picks Science. Show that. A method that gives the right answer when it is fixed is far more instructive than a method that is simply banned.
• Example 2, Know Your Contents (Part II, §1.3, p.15). Madhu and Madhav are at a shop choosing between two badam drink mixes and want to know which has the larger share of badam, and which of the two has the lower share of additives. The numbers live inside the two label illustrations and do not appear in the running text at all — read off the printed page:

| | Sugar | Milk solids | Badam powder | Food chemicals | Total weight | |---|---|---|---|---|---| | DEF (a foil pouch) | 99 g | 30 g | 12 g | 9 g | 150 g | | Zacni (a carton) | 272 g | 64 g | 40 g | 24 g | 400 g |

The page works one cell for the student: DEF's sugar as a percentage of total weight = (99/150) × 100 = 66%. Then a four-column table is printed with columns Sugar, Milk Solids, Badam Powder, Food Chemicals and rows DEF and Zacni — and only the DEF/Sugar cell is filled, with 66%. Seven cells are blank. The closing instruction is to check that each product's percentages add up to 100. Two speech bubbles run beside it: one child suggests the product might as well be called a sugar drink mix, the other agrees that knowing your contents matters, and a small placard reads Do KYC! Arithmetic added here, as a check to hold and not to show early: DEF is 66%, 20%, 8%, 6%; Zacni is 68%, 16%, 10%, 6%. Both rows total exactly 100.

• The result that makes this example worth a whole section (section 7). Zacni lists 24 g of food chemicals against DEF's 9 g — nearly three times as much by weight — and the two shares are identical at 6%. So the question the page asks — which of the two has the lower share of additives — has the answer neither. An explanation that assumes one of them wins will be wrong on the page's own numbers. Meanwhile the badam question does have an answer: 40 g of 400 is 10% against 12 g of 150 at 8%, so Zacni's share is larger as well as its weight. One pair of columns goes the same way, the other pair does not. That contrast is the topic.
• Madhu and Madhav's biscuits, Example 1 of §1.2 (Part II, §1.2, p.6) — the same two children, the mirror-image question. Madhu's biscuits are 25% sugar, Madhav's are 35%; who ate more sugar. The page's answer is that comparing the percentages alone is inappropriate when they refer to different quantities: if both had eaten 100 g, Madhav's 35 g clearly beats Madhu's 25 g; but Madhu ate 120 g and Madhav 95 g. The printed working gives Madhu 30 g and Madhav 33.25 g — so Madhav still ate more, but the margin has collapsed from 10 g to about 3 g. Note: the two examples pull in opposite directions on purpose. Section 3 says convert to percentages to compare; section 9 says a percentage on its own settles nothing. Both are true, and the distinction is whether the wholes are the thing being compared or the thing being divided out. (The three computational routes on that page belong to Computing a percentage in your head by splitting it.)
• Countries against the world (Part II, §1.3 exercises, p.28, Q2). World population in 2025 taken as about 8.2 billion. Four country figures for 2025, printed in tinted boxes: Germany 83 million, India 1.46 billion, Bangladesh 175 million, USA 347 million. Nine option chips are printed below: 13%, 8%, 18%, 10%, 1%, 35%, 2%, 2%, 0.1% — read off the printed page, and yes, 2% appears twice. The printed hint is to write the numbers in standard form and estimate. Arithmetic added here: Germany ≈ 1.0%, India ≈ 17.8%, Bangladesh ≈ 2.1%, USA ≈ 4.2%. There is no chip near 4%. Either one of the two 2% chips was meant to read 4%, or the USA is intended to be matched to the nearest offered value. Flag it to the teacher; do not let an explanation assert a match it cannot justify. See Notes.
• The election item (Part II, §1.3 exercises, p.20, Q4). 1600 people voted and the winner took 500 votes; what percentage of the votes did the winner get, and — a second, harder question — what is the smallest number of candidates who could have stood. Arithmetic added here: 31.25%; and since no other candidate can have exceeded 500, the remaining 1100 votes need at least three more candidates, so at least four stood. The second half is a genuine reasoning item and the chapter's only one of its kind.
• The area item (Part II, §1.3 exercises, pp.28–29, Q6, figure printed at the top right of Part II p.29). What percentage of the area is the region marked E. Read by counting lattice columns on the printed page: a square dot lattice carries a large outer square eight units by eight, so 64 unit squares. A vertical line four units from the left splits it; the upper-left four by four square is A; on the right of the vertical, a horizontal line six units down marks off B (four wide, six tall) above C (four wide, two tall); the lower-left four by four square is cut by a diagonal from its bottom-left corner to its top-right corner, giving D above the diagonal and E below it. Arithmetic added here: the areas are A 16, B 24, C 8, D 8 and E 8, totalling 64 — the whole square — so E is 8 of 64, or 12.5%. The total of 64 is the check that the figure has been read correctly: it must equal 8 × 8. As percentages of the whole: A 25%, B 37.5%, C 12.5%, D 12.5%, E 12.5%.
• The survey graph (Part II, §1.3 exercises, p.30, Q15). A horizontal bar graph, titled for the ability to use a computer by age and gender in 2023, with a printed subtitle stating that the ability is highest among people in their twenties and among teenagers. Two series, Female and Male, legend at the top. Seven age bands down the side and a percentage axis running 0% to 40% in steps of 5%. The printed data labels, female then male:

| Age band | Female | Male | |---|---|---| | Children | 4% | no printed label | | Teenage | 24% | 29% | | Twenties | 26% | 37% | | Thirties | 14% | 25% | | Forties | 7% | 14% | | Fifties | 4% | 9% | | Seniors | 2% | 4% |

The Children male bar is drawn — it reaches to about the 5% gridline — but carries no percentage label, unlike every other bar in the chart. Read off the printed page; the text layer shows only the single 4% for that band, which is exactly the kind of hole that makes an explanation assert the bar is missing. It is not. Source line printed beneath, crediting the National Statistics Office's Comprehensive Annual Modular Survey, NSS Round 79, with a Data For India credit mark. Six statements are offered to be judged valid or not: that the twenties band tops every band for computer literacy; that women lag the men across the bands; that the twenties band holds more people than the teenage band; that over a quarter of the thirties band can use a computer; that fewer than one in ten of those aged 60 or older can; and that half the twenties band can. The reasoning: the third statement is about how many people exist in each band, and a graph of percentages carries no information about that at all. That single distinction — a share is not a count — is the thesis of this whole topic arriving in an exercise.

Figures to have open

• The two product labels of Part II p.15, redrawn as two ingredient panels with their four weights and their total weights. Chapter's own figure, and the whole of sections 5 to 8 depends on the numbers being legible. Do not reproduce the printed artwork; rebuild the panels.
• The KYC table, blank except the 66% cell.
• Two bars of unequal length for the marks comparison, then the same two rescaled to equal length. Standard schematic, and the single most useful image in the topic.
• The survey bar graph of Part II p.30, redrawn with all thirteen printed labels and the fourteenth bar drawn but unlabelled as printed. The 0–40% axis must be kept: several of the six statements are judged by reading against gridlines.
• The dot-lattice figure of Part II p.29 with regions A to E, redrawn to the read geometry (8 × 8 outer square; A and the D/E square each 4 × 4; B 4 × 6; C 4 × 2; the vertical four units from the left; the diagonal from the bottom-left corner of the lower-left square to its top-right corner). Keep the dot lattice itself in the redraw — the counting is how a student gets the areas.
• No photograph is needed.

Where this sits in the book

• NCERT Ganita Prakash Class 8, Part II, printed Chapter 1, "Fractions in Disguise", §1.3 "Using Percentages", Part II pp.14–15 — the unnumbered bold subheading "To Compare Proportions" (Part II p.14) and, second in that run, "Know Your Contents (KYC)" (Part II p.15).
• Part II p.6, Example 1 of §1.2 — the mirror-image comparison. Its computational routes belong to Computing a percentage in your head by splitting it; its argument is used here.
• Exercise items: Part II p.20 no. 4; Part II p.28 nos. 2 and 6 (the figure for no. 6 is printed at the top of Part II p.29); Part II p.30 no. 15.
• The chapter's SUMMARY, Part II p.31, records that parts of a quantity given as ratios can be converted to percentages — the KYC table's exact operation.