7.5 Trairāśika: The Rule of Three
Many proportion problems share the same shape: four quantities are linked, three are known, and we must find the fourth. Ancient Indian mathematicians gave this pattern a name and a method.
Worked Example 6: School Canteen
A canteen normally rolls out 200 rotis for 160 children. On a stormy day only 100 children show up. How many rotis avoid waste?
Rotis should stay proportional to the number of children: 160 : 200 :: 100 : ? The first term changed by a factor \tfrac{100}{160} = \tfrac{5}{8}. Applying the same factor to the rotis, 200 \times \tfrac{5}{8} = 125. The canteen should roll out 125 rotis.
How does this always work? Write the two proportional ratios as a : b :: c : d. Being proportional means each term of the second ratio is the same multiple f of the matching term of the first: c = f a \quad\text{and}\quad d = f b. Then f = \dfrac{c}{a} = \dfrac{d}{b}, so \dfrac{c}{a} = \dfrac{d}{b}. Multiplying both sides by ab: \boxed{\,ad = bc\,}
Cross Multiplication Two ratios are proportional exactly when their cross products are equal: a : b :: c : d \iff ad = bc. This lets us recover any missing term. From ad = bc we get, for instance, \;d = \dfrac{bc}{a}.
Did You Know: Āryabhaṭa’s Rule of Three In classical India, Āryabhaṭa (5th century CE) and his successors named these Trairāśika, Rule of Three, problems, since three numbers are supplied and a fourth is wanted. The three carried names: the pramāṇa (measure, our a), the phala (fruit, our b), and the icchā (requirement, our c). To obtain the icchāphala (result, our d), Āryabhaṭa’s instruction reads:
“Multiply the phala by the icchā, then divide the product by the pramāṇa.”
That is precisely \;\text{icchāphala} = \dfrac{\text{phala} \times \text{icchā}}{\text{pramāṇa}} = \dfrac{bc}{a}\;, cross multiplication, spelled out in words over fifteen centuries ago.
Worked Example 7: Watch the Units
A bus covers 120 km in 180 minutes at a steady speed. How far does it go in 5 hours?
Tempting but wrong: 180 : 120 :: 5 : ?, because 180 is in minutes and 5 is in hours. Convert first: 5 hours = 300 minutes. Now 180 : 120 :: 300 : x. Cross-multiplying, 180x = 300 \times 120, so x = \frac{300 \times 120}{180} = \frac{36000}{180} = 200. The bus covers 200 km in 5 hours.
Exam Tip Before you set up any proportion, check that matching quantities are in the same units. A quick worked check: 2 m : 50 cm is not 2 : 50; convert first to 200 cm : 50 cm = 4 : 1. Converting before computing, not after, is where most easy marks are won or lost.
Worked Example 8: Which Honey Is Dearer?
A beekeeper in Kerala sells a 250 g jar for ₹300. An apiary in Uttarakhand sells a 1 kg jar for ₹900. Are the weight-to-price ratios proportional, and which honey is dearer?
Use the same unit of weight (grams). Kerala: 250 : 300 = 5 : 6. Uttarakhand: 1000 : 900 = 10 : 9. Since 5:6 \neq 10:9, the ratios are not proportional.
To compare price, find the cost of the same weight, say 1 kg. Uttarakhand: ₹900 per kg. Kerala: 250 g costs ₹300, and 1 kg is 4 \times 250 g, so 1 kg costs $4 = $ ₹1200. The Kerala honey is more expensive (₹1200/kg versus ₹900/kg).
Worked Example 9: Shadows and Heights (extra)
At the same moment, a 6 m flagpole casts a 9 m shadow, while a nearby neem tree casts a 24 m shadow. Assuming the sun’s angle is identical for both, how tall is the tree?
Because the sunlight strikes both objects at the same angle, height and shadow stay proportional. Using height : shadow, 6 : 9 :: h : 24. Cross-multiplying, 9h = 6 \times 24 = 144, so h = \dfrac{144}{9} = 16. The neem tree is 16 m tall, measured without ever climbing it.
Worked Example 10: Scaling a Recipe (extra)
A street-food vendor knows that 3 cups of batter yield 8 uttapams. He has an order for 20 uttapams. How many cups of batter must he prepare?
Batter and uttapams are proportional, so using batter : uttapams, 3 : 8 :: x : 20. Cross-multiplying, 8x = 3 \times 20 = 60, giving x = \dfrac{60}{8} = 7.5. He needs 7\tfrac{1}{2} cups of batter.
Use the Rule-of-Three solver to work any “a : b :: c : x” problem and watch the cross multiplication unfold.
Common Pitfall: Line Up the Terms When you write a : b :: c : x, the same kind of quantity must sit in the same position in both ratios. If the first ratio is minutes : kilometres, the second must also be minutes : kilometres, never kilometres : minutes. A frequent slip is to mix up which number is the “measure” and which is the “fruit”; if your answer comes out wildly too big or too small, check that the matching quantities are aligned and in the same units before doing any arithmetic.
Figure it Out: Rule of Three
Practice
- The Earth travels about 940 million km around the Sun in a year. How far does it travel in a week?
- To build a wall 12 ft long, a mason needs about 1740 bricks. For a house whose walls add up to a certain total length, how many bricks are needed? (All walls share the same height and thickness, so bricks are proportional to total wall length.)
- Math Talk (a trap!) Imran’s uncle rode at 60 km/h and took 3 hours from Bhopal to Indore. At 90 km/h, how long will the trip take? Can this be modelled as 60 : 3 :: 90 : \underline{\ \ }?
- There are 52 weeks in a year, so distance per week = \dfrac{940}{52} \approx \mathbf{18.08\text{ million km}} (about 18 million km).
- Bricks are proportional to wall length: 12 ft needs 1740 bricks, so each foot needs 1740 \div 12 = 145 bricks. Find the total length of all the walls (outer walls plus any inner dividing wall) by adding the given side-lengths of the house, then multiply by 145. For example, if the walls total L feet, the number of bricks is 145 \times L. (Add the labelled sides of the plan to get L, then apply the rule.)
- No, this is a trap! Here time and speed pull in opposite directions: a faster speed means less time, so it is not a direct proportion and cannot be written 60 : 3 :: 90 : \underline{\ \ }. Imran’s uncle will take less time at 90 km/h. (The distance is 60 \times 3 = 180 km, so the time is 180 \div 90 = 2 hours, an inverse relationship we will study in full in Proportional Reasoning-2.)