Increasing or Decreasing? Variation 2

Task

Consider the expression $$\frac{R_1 + R_2}{R_1R_2}$$ where $R_1$ and $R_2$ are positive.

Suppose we increase the value of $R_1$ while keeping $R_2$ constant. Find an equivalent expression whose structure makes clear whether the value of the expression increases, decreases, or stays the same.

IM Commentary

The purpose of this task is to help students see manipulation of expressions as an activity undertaken for a purpose.

Variation 1 of this task presents a related more complex expression already in the correct form to answer the question.

The expression arises in physics as the reciprocal of the combined resistance of two resistors in parallel. However, the context is not explicitly considered here.

Solution

We rewrite $$\frac{R_1+R_2}{R_1R_2}$$ in the form $$\frac{R_1}{R_1R_2} + \frac{R_2}{R_1R_2} = \frac{1}{R_2} + \frac{1}{R_1}.$$ Now the bigger a (positive) number is, the smaller its reciprocal is, and so the value of this expression decreases as $R_1$ increases.