Is each expression less than, equal to, or greater than $\frac{1}{2}$?
Options:
To determine whether each given expression is less than, equal to, or greater than \(\frac{1}{2}\), we must compute the exact value of each expression by finding a common denominator for the fractions involved and then comparing the result to \(\frac{1}{2}\). To simplify comparison, we will represent \(\frac{1}{2}\) using the same common denominator as the expression being evaluated.
For the first expression, \(\frac{5}{6} - \frac{1}{4}\), the least common denominator for 6 and 4 is 12. We can rewrite the fractions as follows:
\[\frac{5}{6} = \frac{5 \times 2}{6 \times 2} = \frac{10}{12}\]
\[\frac{1}{4} = \frac{1 \times 3}{4 \times 3} = \frac{3}{12}\]
Subtracting these gives:
\[\frac{10}{12} - \frac{3}{12} = \frac{7}{12}\]
Now, compare \(\frac{7}{12}\) to \(\frac{1}{2}\). Since \(\frac{1}{2} = \frac{6}{12}\), and \(\frac{7}{12} > \frac{6}{12}\), we conclude that \(\frac{5}{6} - \frac{1}{4}\) is greater than \(\frac{1}{2}\).
For the second expression, \(\frac{3}{4} - \frac{2}{5}\), the least common denominator for 4 and 5 is 20. We rewrite the fractions:
\[\frac{3}{4} = \frac{3 \times 5}{4 \times 5} = \frac{15}{20}\]
\[\frac{2}{5} = \frac{2 \times 4}{5 \times 4} = \frac{8}{20}\]
Subtracting these gives:
\[\frac{15}{20} - \frac{8}{20} = \frac{7}{20}\]
Now, compare \(\frac{7}{20}\) to \(\frac{1}{2}\). Expressing \(\frac{1}{2}\) with the denominator 20, we have \(\frac{1}{2} = \frac{10}{20}\). Since \(\frac{7}{20} < \frac{10}{20}\), we conclude that \(\frac{3}{4} - \frac{2}{5}\) is less than \(\frac{1}{2}\).
For the third expression, \(\frac{2}{3} - \frac{1}{6}\), the least common denominator for 3 and 6 is 6. We rewrite the fractions:
\[\frac{2}{3} = \frac{2 \times 2}{3 \times 2} = \frac{4}{6}\]
Subtracting these gives:
\[\frac{4}{6} - \frac{1}{6} = \frac{3}{6}\]
Simplifying \(\frac{3}{6}\), we get:
\[\frac{3}{6} = \frac{1}{2}\]
Therefore, \(\frac{2}{3} - \frac{1}{6}\) is equal to \(\frac{1}{2}\).
In cooking, you often need to compare quantities to see if you have enough ingredients. Suppose a recipe requires exactly 1/2 cup of flour. If you have various partial bags of flour measuring 5/6, 3/4, 2/3, 1/4, 2/5, and 1/6 of a cup, you can use subtraction to determine how much you actually have available after portioning them out. By comparing these resulting differences to 1/2, you can quickly identify whether you have enough, too little, or exactly the right amount of ingredients to proceed with your recipe. This type of mental fraction comparison is an essential skill for efficient kitchen management.