Direct Variation Calculator

Direct variation describes two quantities that grow in proportion: y = kx.

How the Math Works

Direct variation describes a proportional relationship between two variables where one is a constant multiple of the other. The equation y = kx represents this relationship, with k being the constant of variation. To calculate k, divide any known value of y by its corresponding x (k = y ÷ x). Once k is determined, it remains constant for all valid pairs of x and y. To predict a new y-value (y₂) for a different x-value (x₂), simply multiply k by x₂ using y₂ = k · x₂. This ensures the relationship remains consistent across all calculations.

Practical Applications

This calculation is useful in physics for determining speed (distance over time), in economics for analyzing cost and quantity relationships, or in engineering for scaling project dimensions. For example, if 3 workers take 12 hours to complete a task, you can find k = 12/3 = 4 hours per worker. Using this k, you can predict how long 5 workers would take: y₂ = 4 · 5 = 20 hours. It also applies to recipes, where ingredient quantities scale directly with the number of servings, or in calculating fuel consumption based on distance traveled.

Day-to-Day Use

In everyday life, direct variation helps with practical decisions like budgeting. If you know you spend $30 on groceries for 5 days, k = 6 dollars per day. You can predict monthly costs as y₂ = 6 · 30 = $180. It also aids in travel planning—if a car travels 60 miles per hour (k = 60), you can estimate distance for any time. Additionally, it’s helpful for understanding hourly wages: if you earn $20 per hour (k = 20), you can quickly calculate earnings for any number of hours worked.

Worked example

y = 12 when x = 4 → k = 3, so at x = 7, y = 21.

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