When one quantity rises and the other falls by the inverse factor

What to assume they know

• Direct proportion and the rule of three — Direct proportion restated: the quotient that stays constant, Solving a proportion problem, and the Trairasika rule of three
• Reading a two-row table of paired values
• Multiplying and dividing by decimals such as 1.5, and by simple fractions
• Adding fractions with unlike denominators, and taking the reciprocal of a fraction
• Recognising a repeating decimal as an exact fraction written approximately
• Speed as distance divided by time

What they should be able to do

• Read a table of paired values and decide, by multiplying each pair, whether the pairing is inverse
• State the defining relation as a constant product, and identify what the constant physically is in a given situation
• Derive the two-pair form of the relation from the constant-product form, and the sideways form in which one ratio equals the other's reverse
• Use the constant-product relation to find a missing value in either quantity
• Complete a partly filled inverse-proportion table
• Given a described pair of quantities, classify the relation as direct, inverse or neither, by naming what the situation holds fixed
• Show that a fall in one quantity as the other rises is not sufficient for inverse proportion, and that agreement in the first columns of a table settles nothing — testing every column of the chapter's tables (i) and (ii), which share their x row and their first two y values
• Solve a combined-rate problem by adding rates rather than times, and explain why times cannot be added
• State the modelling assumptions an inverse-proportion answer depends on

Where it usually goes wrong

• "Inverse proportion means one goes up while the other goes down." Necessary, nowhere near sufficient. Table (ii) on Part II p.65 is the chapter's own failing table — but note what it fails at: not opposite movement (it does not move oppositely throughout) but the constant product, and only in its last two columns. Use it to attack the weaker habit of testing one column and stopping. No table in this chapter really does fall the whole way and still fail, but one is easy to make: x = 1, 2, 3, 4 against y = 12, 6, 3, 1 falls throughout, yet the products are 12, 12, 9 and 4. Show a failure before the definition, not after.
• "Every problem about more workers is inverse." Item 2 of the second exercise set is about buying more pencils and is direct; item 7 is about filling more tanks and is direct; item 10 is a rate addition. Sorting them is the skill the set is testing, and a student who has learnt "workers means inverse" will get three wrong.
• "You can add the times." Ram at 1 hour and Shyam at 1.5 hours do not make 2.5 hours, or 1.25. Rates add; times do not. And the sanity check is free — together must be faster than the faster of them alone.
• "k is just a letter." k is 90 km. It is the road between two cities. Once students can name the constant in a problem they stop guessing which kind of proportion it is.
• "The chapter's n and the SUMMARY's n are the same n." They are not, and a teacher reading both aloud will contradict himself inside a minute. Use k for the constant product throughout and say once that the SUMMARY writes it n.
• "0.083333… only nearly equals 0.083333…" Both are one twelfth exactly. The decimal is a truncated way of writing a clean fraction, and the check is exact.
• "Doubling the workforce always halves the time." Only if every worker is interchangeable, nobody gets in anyone's way and the job divides freely. The chapter asks for the assumptions twice, at Part II p.67 item 3 and Part II p.68 item 6, and both are Math-Talk-style questions with no printed answer. Take them seriously rather than treating them as decoration.
• "Inverse proportion is a different rule to memorise." It is the same proportional reasoning with a product held fixed instead of a quotient. That is the whole chapter in one sentence.

Questions to check understanding

• Test a table of paired values for inverse proportion by computing products
• Complete a partly filled inverse-proportion table
• Find a missing value from a stated inverse relation using constant products
• Classify each of several described pairings as direct, inverse or neither, with a reason naming what stays fixed
• Solve a workers-and-days, pumps-and-hours or provisions-and-days problem
• Solve a combined-rate problem by adding rates
• Convert a speed change into a journey-time change, and the reverse
• State the assumptions an answer rests on — the item type this chapter asks for explicitly, twice
• Explain, with a counter-instance, why "one falls as the other rises" is not enough

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. The chapter prints answers for all six of its Examples and none for any Figure it Out item; Part II carries no answer appendix.

• The direct recap and its worked case (Part II §3.6, p.63). Printed: for a proportion between four quantities the chapter restates the rule of three, naming this kind of proportion a direct proportion, and works Example 1: 5 workers shift 4500 bricks in a day, and 20 workers are needed for 18000 bricks in a day. This brief treats that as the setup; the topic Direct proportion restated: the quotient that stays constant owns the direct-proportion argument itself.
• Example 2, the pivot question (Part II p.63, marked Math Talk in the margin). Printed with a question the page answers only through the table that follows: may the motorcycle-and-car situation be written as the statement 30 : 60 :: 3 : x, and does travel time grow or shrink as speed grows? Inputs: the motorcycle journey from Lucknow to Kanpur takes 3 hours at 30 km/h, and the same trip is to be driven at 60 km/h. Not in the book: the naive rule of three gives x = 6 hours, which is worse than useless — it doubles a journey the driver has just made twice as fast. Showing that wrong answer and letting it be absurd is the strongest opening this topic has.
• The transport table (Part II p.63, a pink-ruled three-row table). Printed: four columns headed Walk, Bicycle, Motorcycle, Car; the speed row reads 5, 15, 30 and 60 km/h; the time row reads 18, 6, 3 and 1.5 hours. Beside it a small illustration of a walker, a cyclist, a scooter rider and a red car on a winding road.
• The matched factors (Part II p.64). Printed: a bicycle at 15 km/h is three times the walking speed of 5, and the 18-hour walk becomes a 6-hour ride, which is three times shorter; the chapter then asks the reader to check the other columns, and states that the two quantities move by one common factor but in opposite directions, which is what earns this kind of proportion its name.
• The factor diagram (Part II p.64, artwork wrapped around the table). Read off the printed page, because the brackets are artwork and their spans are the whole content: above the table, ×3 spans the 5 and the 15, ×2 spans the 15 and the 30, and ×4 spans the 15 and the 60 on an outer bracket; below the table, ×⅓ spans the 18 and the 6, ×½ spans the 6 and the 3, and ×¼ spans the 6 and the 1.5, nested the same way. Each multiplier above sits directly over its own reciprocal below. A teacher must preserve that vertical pairing — it is the figure's entire argument, and a redraw that merely lists the factors loses it.
• The constant product (Part II p.64). Printed: every column's speed times time comes to the same value, given as 90 km; the relation is stated as xy = k with k a constant; and the chapter says outright that here k is the distance from Lucknow to Kanpur, which does not change however you travel. Not in the book: check all four — 5 × 18, 15 × 6, 30 × 3 and 60 × 1.5 each give 90, and the unit works out as km/h times h, which is km. The units are what prove k is a distance and not a coincidence, and the chapter does not do the unit check.
• The two-pair and sideways forms (Part II p.64). Printed: for two value pairs the relation is written as the first product equal to the second product equal to k, and from it the chapter derives the form in which the ratio of the two x-values equals the reverse ratio of the two y-values. It then checks that form on the walk and the car: 5 ÷ 60 gives 0.083333…, and 1.5 ÷ 18 gives 0.083333… too. Not in the book: both are exactly one twelfth; the repeating decimal is the same fraction written approximately, and saying so avoids a student thinking the check only nearly worked.
• Figure it Out, Part II p.65, two items. Inputs, no printed answers:
  • Decide which of three tables is inverse. Table (i): x is 40, 80, 25, 16 and y is 20, 10, 32, 50. Table (ii): x is 40, 80, 25, 16 and y is 20, 10, 12.5, 8. Table (iii): x is 30, 90, 150, 10 and y is 15, 5, 3, 45.
  • Fill the gaps in a table that is stated to be inverse: the x row reads 16, 12, blank, 36; the y row reads 9, blank, 48, blank. Not in the book: in (i) all four products are 800, so it is inverse; in (iii) all four are 450, so it is inverse; in (ii) the products run 800, 800, 312.5 and 128 — the first two agree and the last two do not, so it is not inverse. Table (ii) is the most important object in the section, and the reason is its relationship to table (i). Table (ii) copies table (i)'s entire x row and its first two y values, then diverges: 12.5 and 8 where (i) had 32 and 50. So a student who tests only the leftmost column, or only the leftmost two, cannot tell (i) from (ii) at all — the two tables are built to be indistinguishable exactly as far as a lazy check reaches. That is section 8: check every column, not the first one or two. Two things. It does not fall throughout as x rises: read in ascending x the pairs are (16, 8), (25, 12.5), (40, 20), (80, 10), so y rises across the first three and only then falls. And those first three all satisfy y = x/2, which makes table (ii) closer to a direct table with one broken column than to an opposite-moving one. If section 8 wants a table that genuinely falls throughout and still fails the constant-product test, no table in this chapter is one. For item 2, k is 16 × 9 = 144, so the gaps are 12, 3 and 4.
• Example 3, the road gang (Part II p.65). Inputs: 20 workers finish laying a road in 4 days; 10 workers are to lay the same road. Printed: the quantities are inverse, so 20 × 4 = 10 × y and the answer is 8 days; the chapter then observes that halving the workforce doubled the days, and states the general form — one quantity times a factor, the other times that factor's inverse.
• Example 4, the pumps (Part II p.66). Inputs: 2 pumps fill a tank in 18 hours; 2 more of the same kind are added. Printed: 4 pumps, 2 × 18 = 4 × x, and the answer is 9 hours.
• Example 5, the provisions (Part II p.66). Inputs: food for 80 students for 15 days; 20 more students arrive. Printed: 80 × 15 = 100 × x, and the answer is 12 days.
• Example 6, Ram and Shyam (Part II pp.66–67). Inputs: Ram cuts a given quantity of vegetables in 1 hour, Shyam takes 1.5 hours for the same quantity, and they work together. Printed: the job is taken as one unit of work; Ram does 1 unit an hour, Shyam does 1 ÷ 1.5, which the chapter writes as two thirds; together they do five thirds of the job an hour. The chapter then asks, in a Math Talk marker, whether quantity of work and time are directly or inversely proportional, answers directly, writes the statement five thirds : 1 :: 1 : x, and solves it to three fifths of an hour.
• The three conserved quantities named (not in the book, section 9). Example 3 holds a fixed amount of road, so workers times days is constant at 80 worker-days. Example 4 holds one tankful, so pumps times hours is constant at 36 pump-hours. Example 5 holds one store of food, so students times days is constant at 1200 student-days. Naming the unit of the constant in each case is what turns three look-alike problems into one idea, and the chapter names the constant only for the Lucknow–Kanpur case.
• Why times cannot be added (not in the book, section 10). Ram's hour and Shyam's hour and a half do not combine to two and a half hours, or to anything else you can reach by adding times — two people working together finish faster than either alone, so the answer must be under 1 hour, and three fifths is. Rates add because work done in an hour is additive; times do not, because they are not measuring an amount of anything that piles up. The chapter performs the rate addition and never says why the naive alternative is wrong.
• Figure it Out, Part II pp.67–68, twelve items. Inputs, no printed answers:
  • Which of six described pairings is inverse: taps filling a tank against filling time; painters hired against days to paint a wall of fixed size; how far a car can go against the petrol in its tank; a cyclist's speed against time over a fixed route; length of cloth bought against price at a fixed rate per metre; number of pages against reading time at a fixed reading speed.
  • 24 pencils cost ₹120; find the cost of 20.
  • A building's tank supplies 20 families for 6 days; 10 more families move in. How long does the water last, and what has to be assumed? Marked Math Talk.
  • The eight sleep rings — handled by Building a pie chart: turning counts into angles, see Notes.
  • The transport pie chart and its four questions — also handled by Building a pie chart: turning counts into angles.
  • Three workers paint a fence in 4 days; a fourth joins. How many days, and what has to be assumed?
  • One pump fills 2 same-sized tanks in 6 hours; how long for 5 tanks?
  • A set of chairs stands as 25 rows of 12; rearranged at 20 to a row, how many rows? (The item gives only the two row figures — the total of 300 is not printed and has to be computed; see Notes.)
  • A school day of 8 periods of 45 minutes is reorganised into 9 periods with the same total teaching time; how long is a period?
  • A small pump fills a tank in 3 hours, a large one in 2 hours; both together?
  • 42 machines make a batch of toys in 63 days; how many machines to make the same batch in 54 days?
  • A car covers a route in 2 hours at 60 km/h; how long at 80 km/h? Two of these items are illustrated, and only two: the twin-tap tank beside item 10 and the conveyor line of toys beside item 11. Item 12 has no artwork — do not brief a car illustration for it.
• Worked answers to those (not in the book, none printed). Item 1: inverse are (i), (ii) and (iv), because each holds something fixed as a product — one tankful, one wall, one route. The others are direct: petrol and distance share a fixed consumption rate, cloth and price a fixed price per metre, pages and time a fixed reading speed. Item 2 is direct and the answer is ₹100 — the trap of the set. Item 3: 20 × 6 = 120 family-days, so 30 families get 4 days. Item 6: 3 × 4 = 12 worker-days, so 4 workers take 3 days. Item 7 is direct — 3 hours a tank, so 5 tanks take 15 hours. Item 8: 25 × 12 = 300 chairs, so 20 to a row gives 15 rows. Item 9: 8 × 45 = 360 minutes, so 9 periods run 40 minutes each. Item 10 is a rate-addition problem like Example 6 — one third plus one half is five sixths of a tank an hour, so the fill takes six fifths of an hour, that is 1 hour 12 minutes. Item 11: 42 × 63 = 2646 machine-days, and 2646 ÷ 54 = 49 machines. Item 12: 60 × 2 = 120 km, so at 80 km/h it takes 1.5 hours.
• The SUMMARY (Part II p.69, fourth bullet). Printed: the inverse rule is stated in the factor form — one quantity times a factor, the other times its inverse — and then in the constant-product form across a list of value pairs. Note as a check: that bullet names the constant product n, while Part II p.64 names it k and Part II p.65 uses n for the factor. Same mathematics, two incompatible uses of one letter within six pages. See Notes.

Figures to have open

• The chapter's transport table with its factor diagram (Part II pp.63–64), redrawn so that the vertical pairing of each factor with its reciprocal is unmistakable. This is the one figure in the chapter that carries an argument rather than a result, and it is worth showing as a movement.
• A constant-product strip: a rectangle of fixed area whose width and height are dragged against each other while the area label stays put. Standard schematic, and it is the best single picture of inverse proportion available. The chapter does not have it.
• Tables (i) and (ii) from Part II p.65 set one above the other, with the shared x row and the two shared y values marked, the column where they diverge marked, and each column's product computed underneath both — four agreeing under (i), two agreeing and two not under (ii). Standard schematic; section 8 needs the products visible and needs the two tables together, since the argument is about how far along they stay identical.
• A separate, clearly-labelled not in the book table that does fall throughout and still fails — x = 2, 4, 8 against y = 12, 6, 5, products 24, 24, 40. Standard schematic. It must not be presented as the book's, since no printed table does this.
• Three panels for section 9 — road, tank, food store — each labelled with its conserved product and the unit of that product. Standard schematic.
• A rate bar for Example 6: Ram's hourly share and Shyam's hourly share stacked to overshoot one whole job, with the overshoot showing why the answer is under an hour. Standard schematic.
• A two-branch decision diagram for section 12. Standard schematic.
• The chapter's small illustrations of the four travellers, the twin-tap tank and the toy conveyor are decorative and can be replaced by icons.

Where this sits in the book

• NCERT Ganita Prakash Class 8, Part II, printed Chapter 3, "Proportional Reasoning-2", §3.6 "Inverse Proportions", Part II pp.63–68. The direct recap and Example 1 and Example 2 (the pivot question) and the transport table are on Part II p.63; the matched factors, the factor diagram, the constant product and the two derived forms are on Part II p.64; the first Figure it Out and Example 3 are on Part II p.65; Examples 4, 5 and 6 are on Part II p.66; Example 6 finishes and the twelve-item Figure it Out begins on Part II p.67 and runs to the foot of Part II p.68.
• Part II p.69, the chapter SUMMARY, fourth bullet, for the inverse rule in both its forms — and for the change of letter noted above.
• Handover: within that twelve-item set, item 4 (Part II p.67, the sleep rings) and item 5 (Part II p.68, the transport pie chart) are chart-reading work and are covered by Building a pie chart: turning counts into angles, not here.
• Backward pointers: Direct proportion restated: the quotient that stays constant for direct proportion and for Example 1 of this section; Part I printed Chapter 7, §7.4 including "Trairasika — The Rule of Three".