Answering "could this possibly fit?" by estimating in stages

What to assume they know

• What it actually takes to make one lakh — a feel for a lakh, and comparing a large number against a familiar quantity
• Crores and millions: one number, two naming systems — reading crore-sized figures in the Indian system
• When an approximate answer is the better answer — that an approximate answer can be the right kind of answer
• Factorise and regroup instead of multiplying head-on — regrouping a product so it can be done in easy stages
• Multiplying and dividing whole numbers by 24, 60, 365 and 1000

What they should be able to do

• Turn a "could it possibly fit" question into two quantities that can be compared
• State the assumption a feasibility estimate rests on, before computing
• Compute a large product or quotient in stages rather than in one step, and say what each stage means
• Compare the two quantities and give a verdict together with the margin
• Test how far the verdict survives a change in the assumption, and find the assumption at which it flips
• Choose reasonable values for quantities not supplied — seats, hours, days — and defend the choice
• Recognise when a margin is so small that the assumptions must be sharpened before the verdict can be trusted

Where it usually goes wrong

• "A lakh buses is obviously enough for one city." It is not, at fifty a bus. Intuition about crore-sized quantities is exactly what this section distrusts.
• "The answer is a property of the question." It is a property of the question and the assumption. Fifty to a bus gives no; a hundred and twenty-five gives yes. An explanation that states the verdict without the assumption has misreported the chapter.
• "Because the buses failed, the ships will fail too." They succeed, by 57,627 people. Five thousand ships at 2,500 each hold two and a half times what a lakh of buses at fifty hold.
• "You should do the whole calculation in one multiplication." Staging is the chapter's own advice, and it is what lets a student check their work halfway. It also makes the units visible: kilometres per day, then per year, then per ten years.
• "Ten years at 100 km a day is nowhere near the Moon." It is within five per cent. Students who eyeball this get it badly wrong in both directions, and that is the reason to compute.
• "A million and a lakh are close enough for a rough answer." They differ by a factor of ten, and the babies-in-a-day question is decided by exactly that factor.
• "Reasonable assumptions means whatever numbers make it work." A defensible assumption is one you can say out loud and have a classmate accept — a bus seats fifty, a day has 86,400 seconds.

Questions to check understanding

• "Could X fit into Y?" with one quantity supplied and one to be assumed — the chapter's own form
• Compute a distance travelled over a stated period at a stated daily rate, in stages
• Decide whether a target is reachable in a lifetime, showing the per-year step
• State the assumptions used and say how the answer would change if one of them doubled
• Estimate a count from a rate — coins per second, births per minute
• Combine given number cards with any operations to land close to a target, and state the error
• Competency-style: criticise a stated conclusion whose assumption was never declared

Examples worth working on the board

Values marked verified are worked out here against the printed page. The chapter works the bus case fully and leaves the rest open.

• The buses (Part I, §1.6, pp.18–19, worked on the page). A bus is assumed to take 50 people, so a lakh of buses take 1,00,000 × 50 = 50,00,000. Mumbai's population is taken from the chapter's own earlier table as more than 1 crore 24 lakh, so the answer is no. Verified: the shortfall is about 74 lakh, and the buses hold only about two fifths of the city.
• Flipping the assumption. Verified, an added extension: at 125 to a bus a lakh of buses hold 1,25,00,000, which is more than 1,24,42,373. The break-even seating is about 125 people per bus. The chapter's verdict is correct for the chapter's assumption and for nothing else, and this is the single most important beat in the topic.
• The ships (Part I, §1.6, p.19, asked and left open, with artwork of a line of ships beside a crowded shore). The RMS Titanic is given as carrying about 2,500 passengers; can Mumbai fit into 5,000 such ships? Verified: 5,000 × 2,500 = 1,25,00,000, which exceeds 1,24,42,373 — yes, but only by 57,627, under half a per cent. Set beside the buses, this is the pair the section is built on: same method, opposite verdict, and one of the two answers is too close to trust without better figures.
• The Moon (Part I, §1.6, p.19). Travelling 100 km a day for 10 years, with the Earth–Moon separation given as 3,84,400 km. Verified in stages: 100 × 365 = 36,500 km in a year; × 10 = 3,65,000 km in ten years; short by 19,400 km, about five per cent. Counting the two or three leap days in a decade would add only 200 to 300 km and changes nothing. The chapter asks the yearly figure and the ten-yearly figure separately, which is the staging it wants shown.
• The Sun (Part I, §1.6, p.19). Travelling 1000 km a day, using the Earth–Sun figure the student uncovered earlier in the chapter as 2100 × 70,000 (Part I, §1.5, p.16). Verified: the distance is 14,70,00,000 km; 1000 km a day is 3,65,000 km a year; the journey takes about 403 years, so no.
• Three quick checks (Part I, §1.6, p.19). Verified: a lakh of sheets at 5 g each weighs 5,00,000 g, which is 500 kg — no. At 250 births a minute, a day brings 250 × 60 × 24 = 3,60,000 babies, well under a million — no. Counting one coin a second gives 60 × 60 × 24 = 86,400 in a day, under a tenth of a million — no.
• The end-of-section questions (Part I, §1.6, pp.20–21) that belong to this topic's closing beat. Coins 1 mm thick stacked to the height of the Statue of Unity: verified, 180 m is 1,80,000 mm, so 1,80,000 coins. A grey-headed albatross covering 900 to 1000 km a day on a 12,000 km trip: verified, 12 to about 13 days. A bar-tailed godwit flying 13,560 km in about 11 days from 13 October 2022: verified, about 1,233 km a day and about 51 km an hour. Bald eagles at 4500–6000 m, Everest at 8850 m and aeroplanes at 10,000–12,800 m against Somu's 40 m building: verified, about 113 to 150 times, about 221 times, and about 250 to 320 times.
• The number-card estimation game (Part I, §1.6, p.21, question 10). Cards 4000, 13000, 300, 70000, 150000, 20 and 5, each usable once, to be combined by any operations to land as close as possible to five targets. The chapter works the first: 4000 × (20 + 5) + 13000 = 1,13,000 against a target of 1,10,000. This is estimation with a deliberate error left visible, and it is the best closing exercise in the chapter.
• The two-row card puzzle (Part I, §1.6, p.20, question 9, artwork). Two rows of empty boxes — seven in the upper row and five in the lower, right aligned — to be filled from two sets of cards numbered 1 to 9, once for the largest sum and once for the smallest difference. The box counts are visible only in the drawing.

Figures to have open

• Two bars on one scale for capacity against population, redrawable at a different seating assumption. This is the topic's core picture and the chapter prints nothing like it. Standard schematic.
• A staged calculation strip — a value passing through three boxes, each labelled with its unit. Standard schematic.
• The line of ships beside the crowded shore is the chapter's own artwork (Part I, §1.6, p.19); it carries the sense of scale but no data. Redraw or replace freely.
• The two-row box puzzle (Part I, §1.6, p.20). If the explanation uses it, the seven and five box counts must be taken from the drawing, since they are not stated in words.
• No data table is required; the Mumbai figure comes from the table already drawn for When an approximate answer is the better answer.

Where this sits in the book

• NCERT Ganita Prakash Class 7, Part I, printed Chapter 1, "Large Numbers Around Us", §1.6 "Did You Ever Wonder…?", pp.18–21. The section has no subheadings; its closing "Figure it Out" runs from p.19 to p.21.
• The Mumbai population used here is the chapter's own, from the table under "Populations of Cities" (Part I, §1.4, pp.12–13).
• The Earth–Sun figure is the one the student uncovers under "Fascinating Facts about Large Numbers" (Part I, §1.5, p.16).
• The Statue of Unity height used in the coin-stacking question is given under "Getting a Feel of Large Numbers" (Part I, §1.1, p.3), as is Somu's building.
• Part I, SUMMARY, p.22 names this kind of question a thought experiment.