Solving a Variation Problem

Solving a variation problem means writing the variation equation from the words, using the given pair to find the constant k, and answering the new case with the completed formula.

Travel time varies inversely with speed, and a trip takes 4 hours at 60 miles per hour. Write the form: t = k/v. Find k from the given pair: 4 = k/60, so k = 240 — and k has meaning here: 240 is the trip's distance in miles. Answer the new case: at 80 miles per hour, t = 240/80 = 3 hours. The plain-sense audit agrees — faster should mean shorter, and 3 beats 4. Every variation problem is the same three moves: form, constant, answer, with the constant usually naming a real quantity worth pausing on.