Arrange the fractions \(\dfrac{2}{3},\ \dfrac{3}{4},\ \dfrac{1}{2},\ \dfrac{5}{6}\) in ascending order.
\(\dfrac{1}{2} < \dfrac{2}{3} < \dfrac{3}{4} < \dfrac{5}{6}\)
Idea: Compare the fractions by changing them to the same denominator.
Write the denominators: 3, 4, 2, 6.
The lowest common multiple (LCM) of 3, 4, 2, 6 is \(12\).
Change each fraction to twelfths (denominator \(12\)):
\(\dfrac{2}{3} = \dfrac{2\times 4}{3\times 4} = \dfrac{8}{12}\)
\(\dfrac{3}{4} = \dfrac{3\times 3}{4\times 3} = \dfrac{9}{12}\)
\(\dfrac{1}{2} = \dfrac{1\times 6}{2\times 6} = \dfrac{6}{12}\)
\(\dfrac{5}{6} = \dfrac{5\times 2}{6\times 2} = \dfrac{10}{12}\)
Now compare the numerators (because all have denominator \(12\)):
\(6 < 8 < 9 < 10\)
So, \(\dfrac{6}{12} < \dfrac{8}{12} < \dfrac{9}{12} < \dfrac{10}{12}\).
Write the answer in the original fractions:
\(\dfrac{1}{2} < \dfrac{2}{3} < \dfrac{3}{4} < \dfrac{5}{6}\)
Quick check (optional): decimals are \(0.5,\ 0.666\ldots,\ 0.75,\ 0.833\ldots\), which match the same order.