When the Rule of Three does not apply

A method is only as good as knowing when not to use it. Here's a problem that looks exactly like the others and isn't one of them.

Puneeth's father rides from Lucknow to Kanpur in 2 hours at 50 km/h. If he rides at 75 km/h, how long will it take?

It's tempting to write 50 : 2 :: 75 : ?. Do that and the answer comes out as 3 hours: longer than before, from riding faster. Obviously wrong. (The road is 100 km, so at 75 km/h it actually takes 1 hour 20 minutes.)

The trouble is the direction of the relationship. In every earlier problem, when one quantity went up, the other went up too: more students, more rice; more glasses, more sugar.

Here, going faster means arriving sooner: time decreases as speed increases. A proportion of the form a : b :: c : d assumes both terms move the same way, so it simply doesn't describe this situation. (You'll meet the right tool for it in Proportional Reasoning-2. For now, the important skill is spotting that the Rule of Three doesn't fit.)

A second case where proportionality fails, for a different reason: shampoo.

Compare the sachet with the small bottle. Volumes are 6 : 180, so 1 : 30. Prices are 2 : 154, so 1 : 77. Not proportional at all: the big bottle is far better value per millilitre. Why? Packaging, transport and the convenience of a small size all cost something. Worth thinking about for the company, for the customer, and for the waste all those sachets leave behind.