Rule of Three Without Memorising: Find the Right Proportion in Any Situation

The rule of three is a method for finding an unknown value when we know three related values. It is applicable whenever there is a proportional relationship between two types of quantity — amount and price, speed and time, number of workers and days of work, among many other pairs.

The method is called 'rule of three' precisely because it uses three known values to find the fourth. For example: if 5 kilos of rice cost $22, how many kilos can I buy with $44? We have three values (5 kg, $22 and $44) and we are looking for the fourth.

The rule of three appears constantly in practical situations: adjusting ingredient quantities in a recipe, converting currencies while travelling, calculating travel time at different speeds, estimating material consumption on a construction site, or measuring team productivity.

Before doing any calculation, you need to identify the type of relationship between the quantities. Ask yourself: if the first quantity increases, does the second also increase or does it decrease?

In direct proportion, the quantities move in the same direction: when one increases, the other also increases. The classic example is quantity and price: more products, more cost. Similarly, more hours worked, more production.

In inverse proportion, the quantities move in opposite directions: when one increases, the other decreases. The classic example is number of workers and time to complete a task: more workers, less time.

With the relationship identified as direct, set up a table with the two quantities in parallel columns and the known values in rows. The unknown is represented by x. Then apply cross-multiplication: the product of one diagonal equals the product of the other.

Example: if 8 apples cost $12, how much do 20 apples cost? Column 1 (apples): 8 and 20. Column 2 (dollars): 12 and x. Cross-multiply: 8 × x = 20 × 12, so 8x = 240, therefore x = 30. Twenty apples cost $30.

Common-sense check: we bought more than twice the apples (from 8 to 20), so the price should be more than twice $12. Paying $30 is coherent. Always do this sense check before considering the problem solved.

In inverse proportion, the standard cross-multiplication needs an adjustment. Multiply 'in parallel' instead: the product of values in one column remains constant.

Example: 6 workers build a wall in 4 days. How many days would 3 workers need? Since the relationship is inverse: 6 × 4 = 3 × x, so x = 24/3 = 8 days. Half the workers → twice the time. Coherent.

Another example: a car travelling at 80 km/h arrives in 3 hours. How long does it take at 120 km/h? More speed, less time — inverse relationship. So 80 × 3 = 120 × x, therefore x = 240/120 = 2 hours.

Cooking recipe: a cake recipe that serves 8 people uses 300 g of flour. To serve 12 people, how much flour is needed? More people, more flour — direct proportion. Cross-multiply: 8 × x = 12 × 300, so x = 450 g.

Fuel consumption: a car gets 12 km per litre of petrol. How many litres are needed to travel 300 km? Direct proportion. 300/12 = 25 litres.

Team production: 4 employees pack 240 boxes per hour. How many boxes would 7 employees pack in the same period? Direct proportion. 4 × x = 7 × 240, x = 420 boxes per hour.

The most serious mistake is applying a direct rule of three in an inversely proportional situation, or vice versa. The confusion produces a mathematically coherent result that is completely wrong for the real problem. Always identify the type of proportion before calculating.

Another common error is mixing units within the same column. If one row has kilometres and another has metres, the cross-multiplication will produce a meaningless result. Standardise units before setting up the table.