Solving a proportion problem, and the Trairasika rule of three

What to assume they know

• Simplest form, and using it to test whether two ratios are proportional — reducing a ratio, and the proportion symbol
• What a ratio claims, and why it is not a difference — what a ratio claims, and why adding to both terms breaks it
• Multiplying and dividing by a fraction, and cancelling in a product of fractions
• Solving a one-step equation such as 150 × x = 21600
• Converting hours to minutes, and grams to kilograms
• Reading a floor plan with lengths marked on the edges

What they should be able to do

• Model a "three known, one unknown" situation as a proportion, putting the two quantities in the same order in both ratios
• Find the factor of change between corresponding terms and use it to get the unknown term, including when that factor is a fraction less than 1
• Derive the cross-multiplication rule from the definition of proportionality, and use it to compute the fourth term
• State Āryabhaṭa's rule of three in the chapter's four Sanskrit names and match each to its place in a : b :: c : d
• Check that both ratios of a proportion use the same units before solving
• Judge whether a situation is one the rule of three applies to, and give the reason when it is not
• Compare several mixtures given as ratios and rank them by strength
• Solve a multi-step proportion problem where the first step is a measurement read from a diagram

Where it usually goes wrong

• "Cross multiplication is a rule you memorise." It is one line of algebra away from the definition. Derive it; a class that has seen the derivation does not multiply the wrong pair.
• "Whichever way round I write the ratios, it works." Both ratios have to list the same two quantities in the same order. The chapter's own first attempt at the car problem shows what a mismatch looks like — there in the units, and the same discipline catches an order mismatch.
• "The factor has to be a whole number." The three blanks of Example 7 include a factor of three sevenths, and the rice example uses two thirds.
• "A factor less than one means I have made a mistake." Fewer students, less rice. The factor is smaller than 1 precisely when the quantity falls.
• "Any four numbers in a story can be put in a proportion." This is the chapter's own warning and the reason section 11 exists. If doubling one quantity does not double the other, the rule of three has no business there.
• "Faster means more, so the answer goes up." The single most common error in speed problems, and the chapter stages it deliberately.
• "18 more glasses means 18 glasses in total." Read the sentence aloud twice. Both readings are arithmetically fine; only one is the chapter's.
• "Bigger packs are always better value." The chapter's own shampoo table says otherwise. Let the numbers speak.
• "3.78 buses means 3 buses." A count of vehicles cannot be rounded down without leaving children behind.

Questions to check understanding

• Given three of four proportional quantities, find the fourth, by the factor method and by cross multiplication
• Given a proportion with a unit mismatch, repair it and then solve
• Derive ad = bc from the definition of proportionality
• Name the four terms of a rule-of-three problem in Āryabhaṭa's vocabulary
• Rank several mixtures by strength given as ratios with unequal totals
• Decide whether a stated situation can be solved by the rule of three, with a reason, and identify the quantity that stays fixed
• Read lengths off a floor plan, total them, and convert a per-length rate into a total quantity
• Solve a rule-of-three problem whose terms are fractions or mixed numbers
• Round a proportional answer to a whole number of indivisible objects and justify the direction of rounding

Examples worth working on the board

Values marked verified are worked out here. Where the chapter works a computation itself, that is said explicitly.

• Filter coffee, the three cups (Part I, §7.4, pp.164–165). Manjunath's regular cup takes 15 mL of coffee decoction and 35 mL of milk, so 15 : 35. The strong cup is 20 mL and 30 mL; the light cup is 10 mL and 40 mL. Two Math Talk markers ask why the second is stronger and why the third is lighter — each question carrying its own Math Talk badge in the right margin. A photograph of three glass mugs of coffee sits below both questions, directly above the strength table. Verified and worth showing: all three cups come to 50 mL, so for these three the comparison can be made by looking at the decoction alone. That is why the table that follows is a harder question.
• Filter coffee, the table (Part I, §7.4, p.165). Three columns — decoction in mL, milk in mL, and a blank column to be marked regular, strong or light. Five printed rows: 300 and 600; 150 and 500; 200 and 400; 24 and 56; 100 and 300. All five cells of the third column are blank on the page, confirmed on the page image. Verified as an error-check as a check: the five rows have five different totals, so none of them can be judged the way the three cups were; against the regular ratio two come out stronger, two lighter, and exactly one is the regular mixture again at quite different amounts. That last row is the point of the exercise, and the class should be the one to find it.
• Kesang's lemonade (Part I, §7.4, Example 2, pp.162–163). Six glasses of lemonade were made with 10 spoons of sugar. Her father asks for 18 more glasses. The page models this as 6 : 10 :: 18 : ?, divides 18 by 6 to get the factor 3, applies 3 to the second term and states the answer as 30 spoons. Two cautions: the chapter reads "18 more glasses" as a fresh batch of 18, not as a new total of 24 — say so out loud, because a student who reads it the other way gets 40 spoons and is not being careless; and this is one of the few places where the chapter prints its own answer.
• Three blanks against 14 : 21 (Part I, §7.4, Example 7, p.164). The printed blanks are __ : 42, 6 : __ and 2 : __. The page works all three: 42 is twice 21 so the first is 28 : 42; for the second it sets up 14y = 6, gets the factor three sevenths, applies it to 21 and reaches 6 : 9; for the third it divides both terms of 14 : 21 by 7 to get 2 : 3. A small marginal diagram beside the text shows 14 : 21 above, a blank paired with 42 below, and a doubling arrow on each side. This is the chapter's demonstration that the factor need not be a whole number, and it is worth the whole of section 4.
• Rice on a rainy day (Part I, §7.4, Example 8, p.167). A school of 120 students normally needs 15 kg of rice for the mid-day meal; on a wet day only 80 come. The page models 120 : 15 :: 80 : ?, computes the factor as 80 over 120, that is two thirds, applies it to 15 and prints the answer 10 kg. The framing question is about not wasting food.
• The derivation (Part I, §7.4, p.168, worked on the page). From a : b :: c : d, the third term is the first times some factor f and the fourth is the second times the same f: c = fa and d = fb. Hence f is both c/a and d/b, so c/a = d/b; multiplying both sides by ab clears the fractions and gives bc = ad. The page then states d = bc/a and encloses the rule in a tinted box. Everything in section 6 and 7 is on this one page.
• Āryabhaṭa's statement (Part I, §7.4, p.168). The chapter names the three knowns and the unknown as pramāṇa, phala, ichchhā and ichchhāphala, glossing them as measure, fruit, requisition and yield, and gives the recipe: multiply the fruit by the requisition and divide by the measure. It then writes the proportion and the cross-multiplied equation in those four words, and on Part I p.169 the resulting formula. The chapter dates Āryabhaṭa to 199 CE. See Notes — report the date as the chapter gives it, and do not embellish it.
• The car (Part I, §7.4, Example 9, p.169). It covers 90 km in 150 minutes; the question asks the distance in 4 hours at the same speed. The page first writes 150 : 90 :: 4 : ?, asks whether that is right, answers no because one time is in minutes and the other in hours, converts 4 hours to 240 minutes, and only then solves: 150 × x = 240 × 90, so x = 144, and the distance is 144 km. The printed working shows the cancelling. A Note to the Teacher on Part I p.169 asks that students be allowed several strategies rather than one method — worth honouring by showing the factor method and the cross multiplication side by side, since 240 is 1.6 times 150.
• **The saffron problem from the *Lilavati*** (Part I, pp.176–177, exercise item 5). Weights are in palas and money in niskas. Two and a half palas of saffron cost three sevenths of a niska; the question asks how much saffron nine niskas buys. Extraction trap: the printed three-sevenths is a stacked fraction, and pdftotext shows it as the two digits 3 and 7 side by side, so the item reads as "37 niskas" in the text layer. It is not 37. An explanation built on the extracted text would pose a different and much duller problem.
• Puneeth's father (Part I, p.171, a Math Talk item). He rode from Lucknow to Kanpur in 2 hours at 50 km/h. The question asks how long the same trip takes at 75 km/h, offers the model 50 : 2 :: 75 : __, asks whether the time should go up or down, and then states plainly that this problem cannot be handled by the rule of three because the time falls as the speed rises. It does not solve it. A thinking-face illustration sits beside the passage. Verified, for an added argument only: the quantity that cannot change here is the journey, 50 × 2, and it is a product rather than a quotient — which is exactly why a rule built on a shared factor has nothing to grip.
• The mason's house (Part I, pp.170–171, exercise item 2). Ten feet of wall takes about 1450 bricks; all walls are the same height and thickness. The plan on Part I p.171 is a rectangle divided by one vertical line, with a smaller rectangle hanging below it. Marked lengths, read off the drawing: the left part of the top rectangle is 9 ft wide, the right part 15 ft wide, the whole rectangle 12 ft tall, and the box below is 6 ft wide and 9 ft tall, its left edge continuing the line of the internal wall. Verified by measuring the drawing on the printed page: it is to scale, about 13 px to the foot in every direction, so the reading above is not a guess. Verified arithmetic: the drawn line work totals 108 ft — 72 ft around the top rectangle, the 12 ft internal wall, and 9 + 6 + 9 ft of new edge for the box below — and at 145 bricks a foot that is about fifteen and a half thousand bricks. See Notes: the question says "the inner wall" in the singular and the drawing has more than one internal segment, so the total depends on a reading the class should be asked to state before computing.
• The Earth's orbit (Part I, p.170, exercise item 1). About 940 million km in a year; the item asks for a week. The modelling choice is exposed here and should be made out loud: 52 weeks to the year, or 365 days divided by 7. The two give slightly different answers, and neither is wrong.
• The shampoo shelf (Part I, Activity 2, pp.171–172). A sample table with four rows: sachet, 6 mL, ₹2; small bottle, 180 mL, ₹154; medium bottle, 340 mL, ₹276; large bottle, 1000 mL, ₹540. The page compares the sachet with the small bottle, 6 : 180 against 2 : 154, and asks why the prices are not proportional to the volumes, then asks for a discussion of packaging and ecological footprint. Verified, and counter to what a teacher will assume: on these printed numbers the sachet is the cheapest per millilitre and the small bottle the dearest, so "buying big saves money" is not what this table says. Check it before scripting.
• Buses for the school trip (Part I, p.176, exercise item 2). Last year 3 buses carried 162 students and teachers, all full; this year there are 204 students. The item asks how many buses are needed and whether they will all be full. Two wrinkles to keep: the answer is not a whole number of buses, and last year's count included teachers while this year's does not.
• Oxen and a tractor (Part I, p.177, exercise item 11). Two oxen working together need 6 hours for one acre; the field is 20 acres; a tractor covers the same ground four times faster. This is the same structure as Puneeth's father and belongs next to it in the explanation's final section.
• Activity 1 (Part I, p.170). Scale a family recipe up to 15 guests: list the ingredients and their quantities, then find the new quantities. A data-collection task, so the explanation can only set it up.

Figures to have open

• The three coffee cups as stacked bars of equal total. An added figure; the chapter has a photograph, which does not carry the numbers.
• The five-row strength table, with an empty third column that fills in as the explanation works. The chapter's own figure (Part I p.165).
• The derivation on Part I p.168, redrawn as annotated lines rather than a reproduced page. Essential.
• A four-position diagram carrying the Sanskrit names. Standard schematic.
• The mason's floor plan of Part I p.171, redrawn to scale with all five marked lengths and with the internal segment distinguished from the outer boundary. Essential, and must be redrawn rather than reproduced — the printed plan does not distinguish outer from inner line work, which is the ambiguity.
• A fixed-area rectangle for section 12, one side labelled speed and the other time. An added figure and the one that makes the failure of the rule intelligible.
• No photograph is needed anywhere.

Where this sits in the book

• NCERT Ganita Prakash Class 8, Part I, printed Chapter 7, "Proportional Reasoning-1", §7.4 "Problem Solving with Proportional Reasoning", Part I pp.162–172. Within §7.4 the chapter prints two unnumbered subheadings that can be named but not cited by number: "Filter Coffee!" (Part I p.164) and "Trairasika — The Rule of Three" (Part I p.167).
• Examples used here: 2 (Part I pp.162–163), 7 (Part I p.164), 8 (Part I p.167), 9 (Part I p.169, wholly on that page; Part I p.170 carries Example 10's solution). Examples 3 and 6 are carried in What a ratio claims, and why it is not a difference; Example 1 in Simplest form, and using it to test whether two ratios are proportional; Example 5 in Why some resized images look right and others look stretched; Example 10 in Unit conversion is a proportion in disguise.
• The derivation and the Āryabhaṭa passage: Part I p.168, continuing to the formula on Part I p.169.
• Notes to the Teacher at Part I p.169 (several strategies, not one method) and Part I p.172 (the shop-price project).
• Exercises: Part I p.170 item 1, Part I pp.170–171 item 2, Part I p.176 item 2, Part I pp.176–177 item 5, Part I p.177 item 11. Activity 1 (Part I p.170) and Activity 2 (Part I pp.171–172).
• The chapter's SUMMARY (Part I p.177) states the proportionality test in the cross-multiplied form, which is this topic's result.
• Forward pointer: the relation the chapter refuses to name on Part I p.171 is taken up in "Proportional Reasoning-2", Part II, printed Chapter 3 (Proportional Reasoning-2).