Solve for $q$.
$\frac{5}{6} q-\frac{1}{3} q=\frac{2}{5}$
$q=$
The given equation is \(\frac{5}{6} q - \frac{1}{3} q = \frac{2}{5}\). Our goal is to isolate the variable \(q\) by simplifying the expression on the left-hand side and solving for \(q\).
To combine the terms on the left-hand side, we need a common denominator for the fractions \(\frac{5}{6}\) and \(\frac{1}{3}\). The common denominator is 6. We can write \(\frac{1}{3}\) as \(\frac{2}{6}\). Now the equation becomes:
\[\left(\frac{5}{6} - \frac{2}{6}\right) q = \frac{2}{5}\]
\[\left(\frac{5-2}{6}\right) q = \frac{2}{5}\]
\[\frac{3}{6} q = \frac{2}{5}\]
Simplifying \(\frac{3}{6}\) gives \(\frac{1}{2}\), so the equation is:
\[\frac{1}{2} q = \frac{2}{5}\]
To isolate \(q\), we multiply both sides of the equation by 2:
\[q = \frac{2}{5} \times 2\]
\[q = \frac{4}{5}\]
Thus, the value of \(q\) is \(\frac{4}{5}\).
This type of linear equation appears in many practical situations, such as calculating the amount of an ingredient needed for a recipe if you have a fraction of the original amount left, or adjusting a budget when you are dealing with portions of a total. For example, if you have a certain fraction of a stock investment and you want to calculate the total value based on a known current share, you solve for the total amount \(q\) using simple algebraic manipulation.