Inverse Variation xy = k

Pre-Algebra · Axiom Academy

A reciprocal relationship: as one quantity grows, the other shrinks so that their product stays fixed at k . Two quantities x and y vary inversely when their product is always the same number. As one gets bigger, the other gets smaller by just enough to keep that product fixed. If 4 workers finish a job in 6 hours , how long will 6 workers take to finish the same job? Workers and time vary inversely — more workers, less time. The total work (workers time) is the constant k . Step 1. Find the constant k from the first scenario: Step 2. Keep the product at 24 and solve for the new time t : Plot every (x, y) pair that satisfies xy = 24 and you get a smooth curve called a hyperbola . Solving for y gives , so the height drops as x climbs. gets closer and closer to both axes, but never touches them; falls steeply near the y -axis and flattens out as x grows; The axes are asymptotes : y can shrink toward 0 as , but neither x nor y can ever be 0 — a zero would make the product 0 , not 24 . Inverse variation shows up whenever two quantities multiply to a fixed total: The faster you drive, the less time the trip takes — distance is fixed. More workers finish the job in less time — total work is fixed. For a fixed area, a longer rectangle must be narrower. A rectangle has area 48 square units. If its length is 12 units, what is its width? Length width is the constant area, so :

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