If $x = 6$ and $y = 10$, find $\frac{|x|-|y|}{|xy|}$.
To calculate the value of the expression \(\frac{|x|-|y|}{|xy|}\) for \(x=6\) and \(y=10\), we first substitute these values into the expression. The expression becomes \(\frac{|6|-|10|}{|6 \times 10|}\). Recall that the absolute value function \(|z|\) returns the non-negative magnitude of a number; therefore, \(|6| = 6\) and \(|10| = 10\). Also, \(|6 \times 10| = |60| = 60\). Substituting these values back, we get \(\frac{6 - 10}{60}\). Performing the subtraction in the numerator yields \(-4\). Thus, the expression simplifies to \(\frac{-4}{60}\). We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is \(4\). This gives \(\frac{-1}{15}\).
In finance, absolute differences are often used to determine the variance between two values, such as comparing a target budget and actual spending. For example, if you planned to spend \(\$60\) on an item but the actual cost was \(\$100\), you might look at the absolute difference relative to the total scale to understand the impact of the deviation. This simple arithmetic helps in tracking budgets and analyzing performance metrics.