Proportional Reasoning-1 · Lesson 4 of 8
Trairasika — The Rule of Three
“Derive cross multiplication from a shared scale factor, apply the Rule of Three with consistent units, and test whether direct proportion is appropriate.”
• Derive ad = bc from equivalent ratios. • Find an unknown fourth quantity from three known quantities. • Convert corresponding quantities into matching units. • Compare prices at a common amount. • Recognise situations that do not fit direct proportion.
Imagine preparing lunch when fewer students arrive than expected. Making the usual amount of rice may waste food, but reducing it by a guessed amount may leave someone hungry. If every student receives the same portion, the student count tells you exactly how the rice quantity should change. Problems like this often give three numbers and ask for a fourth. We can turn the scale-factor reasoning you already know into a reliable general method.
The method is useful only after the relationship has been checked. A neat calculation cannot repair a proportion that pairs the wrong quantities or assumes a relationship that is not present.
Trairasika — The Rule of Three
Problem
A cook prepares 15 kg of rice for 120 students. How much is needed for 80 students with the same portion size?
Write a general proportion as a : b :: c : d, with the positive quantities in matching order. Let f be the common factor. Then c = fa and d = fb. Dividing gives c/a = f and d/b = f, so c/a = d/b.
Multiply both sides by ab. On the left, ab × c/a = bc. On the right, ab × d/b = ad. Therefore ad = bc. This explains cross multiplication: the products are equal because the two changes use the same factor, not because crossing lines is a rule to follow blindly.
If a, b and c are known and d is unknown, divide ad = bc by a. This gives d = bc/a. Read the result alongside the labels to check that you are finding the intended quantity.
For the rice example, 120 × d = 15 × 80, so d = 1200/120 = 10 kg. The formula and scale-factor method agree because one was derived from the other.
The traditional Rule of Three uses the names pramāṇa for the given measure a, phala for its corresponding result b, ichchhā for the requested measure c, and ichchhāphala for the required result d. Multiply the given result by the requested measure, then divide by the given measure.
These names describe the same structure as the modern formula. Keeping the meaning of each quantity visible helps you remember the relationship without relying only on the letters.
Problem
Find x in 8 : 14 :: 12 : x.
Corresponding quantities must use matching units. If the first time is in minutes, convert the second time to minutes too. The two different kinds of quantity in a ratio, such as minutes and kilometres, do not need the same unit as each other; time must match time and distance must match distance.
The same-speed condition is also essential. A car that stops or changes speed does not automatically keep the same time-to-distance ratio for its whole journey.
Problem
A car covers 90 km in 150 minutes. At the same speed, how far does it travel in four hours?
Problem
One seller charges ₹200 for 200 g of tea; another charges ₹800 for 1 kg. Which is more expensive per kilogram?
Figure it Out
Problem
Earth travels approximately 940 million km in a year. Estimate the distance in one week using 52 weeks per year.
Problem
The house has a 24 ft by 12 ft upper rectangle and a 6 ft by 9 ft extension. Count all outer walls, the 12 ft vertical divider and the 6 ft shared wall. At 1450 bricks per 10 ft, how many bricks are needed?
Before solving another problem, predict the direction of change. For a fixed journey, travelling faster should take less time. The proposed setup 50 : 2 :: 75 : t would make the time grow with speed, so it cannot be the correct direct proportion.
You can still solve the situation using the fixed distance. At 50 km/h for two hours, the distance is 100 km. At 75 km/h it takes 100/75 = 4/3 hours, or one hour 20 minutes. This is a check on the model, not a reason to apply the direct Rule of Three indiscriminately.
First identify the quantities and the relationship. Next match the corresponding units and predict whether the answer should increase or decrease. Only then solve the proportion and check it against that prediction.
Quiz
Why does a : b :: c : d give ad = bc?
In 120 : 15 :: 80 : x, what is x?
Which setup represents 90 km in 150 minutes and an unknown distance in four hours?
Which tea is cheaper per kilogram: ₹200 per 200 g or ₹800 per kg?
For a fixed trip, why is 50 : 2 :: 75 : t an unsuitable speed : time proportion?
Practice Problems
- Find x in 5 : 8 :: 15 : x.
- A recipe uses 750 g of rice for six equal servings. Find the amount for ten servings.
- A vehicle travels 60 km in 90 minutes at constant speed. Find its distance in 2.5 hours.
- Compare ₹150 for 250 g with ₹540 for 1 kg. Which is cheaper per kilogram?
- A rider covers a fixed trip in two hours at 50 km/h. Find the time at 75 km/h and explain why direct scaling of time by 75/50 is wrong.
1. 5x = 8 × 15 = 120. 2. x = 120/5 = 24.
1. Factor = 10/6 = 5/3. 2. Rice = 750 × 5/3 = 1250 g, or 1.25 kg.
1. Convert 2.5 hours to 150 minutes. 2. Distance = 60 × 150/90 = 100 km.
1. The first price for 1 kg is 4 × ₹150 = ₹600. 2. The second costs ₹540, which is ₹60 less per kilogram.
1. Distance = 50 × 2 = 100 km. 2. New time = 100/75 = 4/3 hours = 80 minutes. 3. A faster speed reduces the time for the fixed distance. Multiplying two hours by 75/50 would instead increase time, contradicting the situation.
Key Takeaways
• The Rule of Three finds a fourth quantity from three known quantities in a valid proportion. • The cross-product equation ad = bc follows from a common scale factor. • Corresponding measurements must use matching units and consistent order. • Compare prices for the same quantity, not merely the same number of packages. • Check the relationship before calculating: not all changing quantities are directly proportional. • State assumptions and label approximate answers honestly.