Andy owns a car with 20 km/L mileage and Becky has one with 8. They both purchased new cars to tackle the rising fuel price. Andy’s new car is now 30 km/L, and Becky’s is 10. If they both drive similar kilometres, who will save more money?
A quick glance at the problem tells you a 50% improvement for Andy (20 to 30), whereas only 25% for Becky (8 to 10). So Andy, right? Not so fast. Because you are dealing with a compound unit (kilometre per litre) with the actual quantity, we are after (litre), which is in the denominator. So our intuition based on simple numbers goes for a toss.
Imagine they both drive 2000 kilometres this year. Andy consumed 2000 (km) / 20 (km/L) = 100 L in the past, but will consume 2000/30 = 66.7 L this year. On the other hand, Becky’s consumption will reduce from 2000/8 = 250 to 2000/10 = 200. So Becky saves 17 L more than Andy.
Speed paradox
A similar problem of averaging denominators exists in the famous challenge of average speeds: Cathy travelled from A to B at 30 km/h and returned (B to A) at 60 km/h. What was Cathy’s average speed? Needless to say, the intuitive answer (30+60)/2 = 45 is wrong. It is easy to solve if you assume a fixed distance (magnitude doesn’t matter), say, 100 km. For A to B, she took 100 (km)/30 (km/h) = 3.33 h and for the return 100 (km)/60 (km/h) = 1.67 h. So she travelled 200 km in 5 hours. The average speed is 200 (km) / 5 (h) = 40 km/h.
