Direct variation as a ratio
Express direct variation as a ratio: in y=kx the ratio y/x equals constant k for all points, so any two ordered pairs satisfy y1/x1=y2/x2 to confirm a direct variation relationship.
Key Concepts
The statement $A$ varies directly as $B$ also implies the proportion: $$\frac{A 1}{A 2} = \frac{B 1}{B 2}$$.
1. If 12 items cost 78 dollars, how much for 42 items? $\frac{78}{C 2} = \frac{12}{42} \rightarrow 78 \cdot 42 = 12 \cdot C 2 \rightarrow C 2 = 273$ dollars. 2. If 15 ties are for 30 feet of track, how many for 50 feet? $\frac{15}{T 2} = \frac{30}{50} \rightarrow 15 \cdot 50 = 30 \cdot T 2 \rightarrow T 2 = 25$ ties.
This is a fantastic shortcut for solving problems when you don't need to find the constant 'k'. Just set up the two pairs of values as a proportion, cross multiply, and you've found your answer!
Common Questions
What is direct variation and how is it expressed as a ratio?
Direct variation is the relationship y=kx where k is the constant of variation. Because y/x=k for every point on the line, any two valid data points satisfy the proportion y1/x1=y2/x2. This ratio property is useful for solving problems without first finding k explicitly.
How do you determine the constant of variation k from a table?
Calculate y/x for each row of the table. If every ratio is identical, the relationship is direct variation and k equals that common ratio. If the ratios differ, the relationship is not direct variation and a different model is needed.
How does direct variation differ from a general linear equation?
Direct variation y=kx always passes through the origin (0,0). A general linear equation y=mx+b has a y-intercept b that is not zero. All direct variations are linear, but not all linear equations represent direct variation.