Direct proportion restated: the quotient that stays constant

What to assume they know

• What a ratio claims about two quantities, and that it is a statement about their relative size rather than their difference — What a ratio claims, and why it is not a difference
• Reducing a ratio to simplest form and using it to test two ratios against each other — Simplest form, and using it to test whether two ratios are proportional
• The rule of three as a way of getting a fourth quantity out of three known ones — Solving a proportion problem, and the Trairasika rule of three
• Multiplying and dividing whole numbers and simple decimals, and cancelling a common factor in a fraction
• That a fraction with the same units above and below is a plain number with no unit left on it

What they should be able to do

• Write a recipe or mixture stated in amounts as a ratio, and say what the ratio is claiming that the amounts alone do not
• Compute the within-pair quotient for each of two ratios and check whether they agree
• Compute the between-pair factor that carries one ratio onto the other, term by term, and check that a single factor works for both terms
• Show that the equal-quotients form and the equal-factors form both rearrange to the same product equality, and use the product form as the test
• Apply the product test to decide whether two stated mixtures behave alike
• Construct a mixture that fails the test, and say which quantity would have to change and by how much to make it pass
• Set up a proportion with one unknown term from a worded situation and solve for the unknown, naming which two quantities were compared against which
• State what proportionality between two ingredients does not settle about the finished product

Where it usually goes wrong

• "Proportional means equal." 6 : 3 and 4 : 2 are not the same amounts and will not feed the same number of people. They are the same recipe. Keep the two words apart from the first minute.
• "Cross-multiplication is a rule about fractions I learnt somewhere." It is the statement that one pair is a scaled copy of the other, with the division cleared away. Show the clearing.
• "You can keep a ratio by adding the same amount to both terms." Going from 6 : 3 to 7 : 4 adds one cup to each and changes the recipe. Test the two against each other: 6 × 4 = 24 and 3 × 7 = 21, which disagree. Ratios are preserved by multiplying, not by adding.
• "Same difference, same ratio." 6 : 3 and 8 : 5 both have a gap of three. One is two parts to one; the other is not. This is the misconception the earlier chapter attacked, and it comes straight back the moment the numbers get bigger.
• "If the ratio checks out, the two dishes are identical." The chapter refuses to say that. Everything not named in the ratio is unconstrained.
• "The fraction form must be first-over-second." The chapter prints first-over-first across the two ratios; the SUMMARY prints first-over-second within each pair. Both are correct and neither is the definition. The product equality is what they share.
• "You can always divide, so use the fraction form." You cannot — a zero term leaves the fraction form with nothing to divide by, while the product form still computes. The two forms agree only provided no term is zero, and once a term is zero the product form stops being a test at all. Set 0 : 0 against 3 : 5 and both products are nought, so they match and the product test passes a pair that is no proportion. Keep the practical point — nothing to divide by zero — and state the condition next to it. One line each.

Questions to check understanding

• Given two ratios, decide whether they are proportional, showing the two products
• Given two mixtures that are not proportional, state the smallest change to one quantity that makes them proportional
• Given three terms of a proportion and one blank, find the missing term
• Given a worded rate situation, write the statement of proportionality yourself before solving — the step boards most often mark separately
• Explain why adding the same number to both terms of a ratio does not preserve it, with a counter-instance
• Justify, in a sentence, whether two scaled recipes can be guaranteed to taste alike — the competency-style item this section invites

Examples worth working on the board

Values marked printed are worked out on the page; values marked not in the book are worked out here on the chapter's inputs and must not be presented as something the chapter states.

• The batter proportion (Part II §3.1, p.55). One regional mixing rule for idli batter is given as 2 cups of rice to 1 cup of urad dal, written 2 : 1. The chapter says openly that the proportion varies by region, so the 2 : 1 is one choice and not a law.
• The two cooks. Viswanath's batch takes rice 6 cups, dal 3 cups; Puneet's takes rice 4 cups, dal 2 cups. The question posed is whether the two batches would taste alike if everything else were done the same way. Printed: the two mixtures are written 6 : 3 and 4 : 2, both cross products come to 12, and the chapter concludes the ratios are proportional.
• The two readings of that one fact. Not in the book: within each pair, 6 ÷ 3 = 2 and 4 ÷ 2 = 2 — both cooks are at two parts rice per part dal. Between the pairs, 6 ÷ 4 = 1.5 and 3 ÷ 2 = 1.5 — Viswanath's batch is one and a half times Puneet's, in both ingredients. Section 5 has to show that 6 × 2 = 3 × 4 is what both of those sentences reduce to.
• The general statement (Part II §3.1, p.55). Printed: for two ratios written a : b and c : d, the chapter gives the product equality a × d = b × c, and beside it the fraction form that sets a over c equal to b over d. Note the fraction form pairs first term with first term and second with second across the two ratios — it is the between-pair reading, not the within-pair one.
• The SUMMARY's form of the same fact (Part II p.69, the third bullet). Printed: for two directly proportional quantities with corresponding value lists, the SUMMARY sets the first x over the first y equal to the second x over the second y and so on, all equal to a constant. That is the within-pair reading. A student who meets only one of them will think the other is a different rule.
• A mixture that fails (not in the book — the chapter sets no failing case here). If Puneet's batch had taken rice 4 cups, dal 3 cups, the cross products are 6 × 3 = 18 against 3 × 4 = 12. They disagree, so the batches are not proportional; Puneet's is dal-heavy. To fix it he must drop to 2 cups of dal, or raise the rice to 6 cups. Two repairs, one test.
• The bricks proportion (Part II §3.6, p.63, Example 1). Inputs: 5 workers shift 4500 bricks in a day; the target is 18000 bricks in a day. Printed: the chapter lays it out as 4500 : 18000 :: 5 : x and answers 20 workers. Two Note: the proportion compares bricks against bricks and workers against workers, so both ratios are unit-free; and the between-pair factor here is 4, because 18000 is four times 4500.
• The taste caveat (Part II §3.1, p.55). The chapter's own conclusion is hedged: the batches would probably taste alike provided the other ingredients are in proportion too. Two ingredients matching is a claim about two ingredients.

Figures to have open

• The chapter's cross-multiplication diagram (Part II §3.1, p.55): the two ratios 6 : 3 and 4 : 2 set side by side, with one arc arching over the top from the first term of the left ratio to the second term of the right (the 6 to the 2), and a second arc dipping underneath from the second term of the left to the first term of the right (the 3 to the 4). Each arc carries a small red multiplication sign just outside its apex — above the upper arc, below the lower one — and both arcs are drawn in teal. The two arcs are nested, not crossed: one encloses the other and they never intersect, so the printed figure is not the conventional X. Redraw as a schematic — the whole of section 7 depends on the arcs being labelled with which pairing they represent, which the printed version does not do. If the redraw switches to crossing arcs, say that the book draws the pairing as two nested arcs, or a student holding the explanation against the page will think one of the two is wrong.
• A two-column strip showing the same four numbers read twice: down the columns (within-pair) and across the rows (between-pair). Standard schematic, and the key visual of the whole topic.
• Two measuring cups per cook, drawn to scale, so that 6 : 3 and 4 : 2 look visibly different in amount and visibly the same in shape. Standard schematic; no textbook art needed.
• No photograph or map is needed for this topic.

Where this sits in the book

• NCERT Ganita Prakash Class 8, Part II, printed Chapter 3, "Proportional Reasoning-2", §3.1 "Proportionality — A Quick Recap", Part II p.55. The section occupies that page alone.
• Part II p.69, the chapter SUMMARY, third bullet, for the within-pair form of direct proportion. The SUMMARY box has that page to itself.
• Part II p.63, the head of §3.6 "Inverse Proportions", for the rule of three and for Example 1 (the bricks). That page opens §3.6 with a direct-proportion recap before the inverse material starts; this topic owns the recap and Example 1, and When one quantity rises and the other falls by the inverse factor picks the page up from the question immediately after it.
• Backward pointer: Part I printed Chapter 7, "Proportional Reasoning-1", §7.2 and §7.3 for ratio and simplest form, and §7.4 including "Trairasika — The Rule of Three".