1. Inverse Variation
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Study the ratio of two quantities that vary directly and the product of two quantities that vary inversely.
See solution.
Suppose that we are in a store where any item is worth $ 0.50. This means that if we buy x items the total amount of money y that we will have to pay is 0.5x dollars. Total &= Price per item * Nº of items y &= 0.5* x dollars Let's make a table with some values of x and y.
| x | y=0.5x |
|---|---|
| 1 | 0.5 |
| 2 | 1 |
| 3 | 1.5 |
| 4 | 2 |
| 5 | 2.5 |
Let's add one more column where we find the ratio between y and x.
| x | y=0.5x | y/x |
|---|---|---|
| 1 | 0.5 | 0.5/1 = 0.5 |
| 2 | 1 | 1/2 = 0.5 |
| 3 | 1.5 | 1.5/3 = 0.5 |
| 4 | 2 | 2/4 = 0.5 |
| 5 | 2.5 | 2.5/5 = 0.5 |
Notice that the ratio between y and x is constant. When this happens we say that y varies directly with x.
|
Two quantities (variables) vary directly when their ratio is constant. y/x=k |
On the other hand, let's consider a rectangle with an area of 36 cm^2. Below, we show a list of possible dimensions for this rectangle.
| Length (l) | Width (w) |
|---|---|
| 36 | 1 |
| 18 | 2 |
| 12 | 3 |
| 6 | 6 |
Notice that the product between l and w is always equal to 36. When this situation happens, we say that l and w show inverse variation.
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Two quantities (variables) vary inversely when their product is constant. xy=k |