- Identify the constant terms \(\frac{11}{12}\) and \(-\frac{1}{3}\) and the variable terms \(-\frac{1}{6}q\) and \(\frac{5}{6}q\).
- Combine the constant terms: \(\frac{11}{12} - \frac{4}{12} = \frac{7}{12}\).
- Combine the variable terms: \(-\frac{1}{6}q + \frac{5}{6}q = \frac{2}{3}q\).
- State the final combined expression as \(\boxed{\frac{2}{3}q + \frac{7}{12}}\).
Explanation
- Identifying and Grouping Like Terms
To simplify the expression \(\frac{11}{12} - \frac{1}{6}q + \frac{5}{6}q - \frac{1}{3}\), we need to group the terms that are alike. The terms can be classified into two types: those that are constant (numbers) and those that are variable (terms involving \(q\)).
The constant terms in the expression are \(\frac{11}{12}\) and \(-\frac{1}{3}\).
The variable terms in the expression are \(-\frac{1}{6}q\) and \(\frac{5}{6}q\).
- Combining the Constant Terms
To combine the constant terms \(\frac{11}{12}\) and \(-\frac{1}{3}\), we must find a common denominator. The least common multiple of 12 and 3 is 12. Therefore, we can rewrite \(-\frac{1}{3}\) as \(-\frac{4}{12}\).
Now, we perform the subtraction:
\[\frac{11}{12} - \frac{4}{12} = \frac{11 - 4}{12} = \frac{7}{12}\]
- Combining the Variable Terms
Next, we combine the variable terms \(-\frac{1}{6}q\) and \(\frac{5}{6}q\). These terms already have a common denominator of 6. We sum the coefficients as follows:
\[\left(-\frac{1}{6} + \frac{5}{6}\right)q = \left(\frac{-1 + 5}{6}\right)q = \frac{4}{6}q\]
We simplify the fraction \(\frac{4}{6}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 2:
\[\frac{4 \div 2}{6 \div 2}q = \frac{2}{3}q\]
- Final Expression
By combining the results from the previous steps, we get the simplified expression:
\[\frac{2}{3}q + \frac{7}{12}\]
Thus, the equivalent expression is \(\boxed{\frac{2}{3}q + \frac{7}{12}}\).
Examples
This process of simplifying expressions is very useful in everyday life, for instance, when calculating budgets or managing inventory. Imagine you are tracking a grocery list where you have a fixed budget for items (constants) and variable costs based on weight or quantity of specific goods (variable terms). By combining like terms, you can quickly determine how much of your total budget is spent on fixed items versus variable items, making it much easier to adjust your spending as needed.