The Tiger That Wouldn't Stay in Shape Proportional Reasoning-1

Five versions of the same tiger photo -- Images A through E -- come with their width and height measured in millimetres: A is 60 by 40, B is 40 by 20, C is 30 by 20, D is 90 by 60, and E is a perfect 60 by 60 square. Laid side by side, A, C, and D genuinely look like the same photo at different sizes, while B looks stretched sideways and E looks squashed into a square. The reason has nothing to do with which image is bigger or smaller -- it is about HOW each one changed compared to A. Going from A to C, both the width (60 to 30) and the height (40 to 20) were multiplied by exactly the same factor, one half. Going from A to D, both dimensions were multiplied by exactly one and a half. But going from A to B, the width dropped by 20 mm and the height also dropped by 20 mm -- the same DIFFERENCE, not the same factor -- and a width of 40 next to a height of 20 no longer has the same shape as 60 next to 40. Changes that keep the same multiplying factor are called proportional; changes that merely keep the same difference are not, even when the numbers look deceptively similar.

Mathematics gives this same-factor relationship a name: a ratio. The ratio of width to height for image A is written 60:40, and the two numbers in it, 60 and 40, are called the terms of the ratio. To check whether two ratios truly describe the same shape, the cleanest method is to reduce each one to its simplest form, by dividing both terms by their HCF. Image A's ratio 60:40 has HCF 20, giving simplest form 3:2. Image D's ratio 90:60 has HCF 30, also giving 3:2 -- an exact match, confirming A and D are proportional. Image C's ratio 30:20 also reduces (HCF 10) to 3:2, matching again. Meanwhile image B's ratio 40:20 reduces (HCF 20) to 2:1, and image E's ratio 60:60 reduces to 1:1 -- both genuinely different simplest forms from 3:2, proving B and E are not proportional to A, exactly as the eye suspected. When two ratios share the same simplest form, they are called proportional, written with a double-colon symbol: 60:40 :: 30:20, and 60:40 :: 90:60.

Turning this idea into a working tool: Kesang made 6 glasses of lemonade using 10 spoons of sugar, and needs to scale up to 18 glasses for extra guests while keeping the same sweetness. Since 18 is 6 multiplied by 3, the sugar must also multiply by 3, giving 30 spoons -- modelled as 6:10 :: 18:30. A similar question came up between two neighbours, Nitin and Hari, building a compound wall: Nitin built a 60-foot wall using 3 bags of cement, Hari built a 40-foot wall using only 2 bags, and Nitin worried Hari's wall would be weaker. Comparing length-to-cement ratios settles it: Nitin's 60:3 reduces to 20:1, and Hari's 40:2 also reduces to 20:1 -- identical, so both walls used cement just as generously, and Nitin's worry was unfounded. A coffee-shop owner, Manjunath, uses the same reasoning to control taste: his regular filter coffee mixes 15 mL of decoction with 35 mL of milk (15:35, simplest form 3:7); a stronger cup uses 20:30 (simplest 2:3); a lighter cup uses 10:40 (simplest 1:4). Comparing any new mix's simplest form against 3:7 instantly reveals whether it will taste regular, stronger, or lighter, before a single sip is taken.

A trickier example exposes a genuine misconception. When Neelima was 3 years old, her mother was 10 times her age, 30 -- a ratio of 3:30, simplest form 1:10. Nine years later, Neelima is 12 and her mother is 39 (mothers age too), giving a ratio of 12:39, simplest form 4:13 -- clearly NOT 1:10 anymore. Adding the same number of years to both ages changed the ratio, because ratios only stay proportional when BOTH terms are multiplied (or divided) by the same factor, never when the same amount is merely added or subtracted. A related subtlety: the multiplying factor connecting two proportional ratios does not have to be a whole number. Finding ratios proportional to 14:21, when the first term is given as 6, the factor connecting 14 to 6 is 6/14, which simplifies to 3/7 -- a genuine fraction, not a whole number -- so the second term becomes 21 multiplied by 3/7, which is exactly 9, giving the ratio 6:9. Whether the connecting factor is a whole number or a fraction, the underlying rule never changes: multiply, don't add.

Problems with three known quantities and one missing turn up constantly -- a school cook usually makes 15 kg of rice for 120 students, but only 80 arrive on a rainy day, so how much rice now? Modelled as 120:15 :: 80:?, the connecting factor from 120 to 80 is 2/3, so the rice scales the same way, 15 multiplied by 2/3, giving exactly 10 kg. The general algebra behind this -- that for any proportional ratios a:b::c:d, cross-multiplying always gives a matching pair of products, a times d equals b times c -- is not a modern invention. Indian astronomer-mathematician Aryabhata, writing around 499 CE, called this exact idea Trairasika, the Rule of Three, using the terms pramana (the measure), phala (the fruit, or result), and iccha (the request), solved as icchaphala (the yield) equals phala times iccha, divided by pramana -- precisely today's cross-multiplication, more than fifteen centuries earlier. The same method solves less obvious cases too: a car covering 90 km in 150 minutes, asked for its distance in 4 hours, first needs 4 hours converted to 240 minutes (matching units matters!), then 150:90 :: 240:?, cross-multiplied to 144 km.

Not every three-known-one-missing setup is safe for the Rule of Three. Puneeth's father rides from Lucknow to Kanpur in 2 hours at 50 km/h; the tempting next step is to model his trip at 75 km/h as 50:2 :: 75:?, exactly like the rice and coffee problems above. But this hides a genuine trap: as speed goes UP, travel time goes DOWN, the opposite direction from every proportional example so far -- so the Rule of Three, built for quantities that rise and fall TOGETHER, simply cannot be applied here. The correct method instead finds the fixed distance first (50 km/h times 2 hours equals 100 km), then divides that fixed distance by the new speed (100 divided by 75, exactly 1 hour and 20 minutes). A second useful trap-check comes from comparing tea prices: a Himachal Pradesh farmer sells 200 g of tea for Rs 200 (weight:price simplest form 1:1), while a Meghalaya estate sells 1 kg (1000 g) for Rs 800 (simplest form 5:4) -- two DIFFERENT simplest forms, so the ratios are not proportional, meaning the fair way to compare is converting both to the same weight: 1 kg of Himachal tea costs Rs 1,000 by the same scaling logic, genuinely pricier than Meghalaya's Rs 800 per kg.

Ratios also answer a different kind of question: how to split a fixed total UNEVENLY, on purpose. Twelve counters split evenly between two people gives a 1:1 ratio, 6 each -- but splitting the same 12 counters in a 3:1 ratio instead means treating the total as 3+1, that is 4 equal-sized groups, each group holding 12 divided by 4, exactly 3 counters; one person collects 3 of these groups (9 counters), the other collects just 1 group (3 counters). The same reasoning scales up: 42 counters shared in a 4:3 ratio means 4+3=7 groups, each of size 42 divided by 7, exactly 6, giving 24 counters to one side and 18 to the other. Real partnerships use exactly this idea: Prashanti and Bhuvan started a food cart together, investing Rs 75,000 and Rs 25,000 (a 3:1 ratio once simplified), and agreed to split any profit the same way -- so a Rs 4,000 profit splits into 4 equal shares of Rs 1,000 each, giving Prashanti 3 shares (Rs 3,000) and Bhuvan 1 share (Rs 1,000). A 40 kg sand-cement mixture in a 3:1 ratio (30 kg sand, 10 kg cement) needing to become a 5:2 ratio, with the 30 kg of sand fixed, requires the cement to rise to exactly 12 kg -- meaning just 2 kg more cement needs adding.

Proportional reasoning frequently needs a units bridge before the maths can even start: 1 metre equals 3.281 feet, 1 hectare equals 2.471 acres, 1 litre is 1,000 mL, and temperature conversion runs on Fahrenheit equals nine-fifths of Celsius plus 32 (so 25 degrees Celsius becomes 77 degrees Fahrenheit). Armed with consistent units, some genuinely striking real problems open up. Earth travels roughly 940 million km around the Sun every year; dividing by 365 days and multiplying by 7 gives roughly 1.8 crore km covered in a single week. An centuries-old problem from Bhaskaracharya's Lilavati asks: if two-and-a-half palas of saffron cost three-sevenths of a niska, how much saffron can nine niskas buy? -- solved by the same cross-multiplication as any modern ratio problem, giving exactly 52.5 palas. Comparing Delhi (1,484 sq km, roughly 30 million people) against Mumbai (550 sq km, roughly 20 million people) by total population alone wrongly suggests Delhi is more crowded, but dividing population by area (density) reveals Mumbai actually packs roughly 36,000 people per sq km against Delhi's roughly 20,000 -- despite Mumbai's smaller population, its much smaller area makes it the genuinely denser city.