$ \frac{14}{3} = \frac{28}{6} $, $ \frac{7}{2} = \frac{21}{6} $, LCM of periods $ \frac{14}{3} $ and $ \frac{7}{2} $: write as $ \frac{28}{6}, \frac{21}{6} $, GCD of 28 and 21 is 7, so period is $ \frac{14}{\gcd(14,21)} \cdot \frac{1}{\text{some factor}} $. Better: find least $ T $ such that $ \frac{3\pi}{7}T = 2\pi m $, $ \frac{4\pi}{7}T = 2\pi n $, so $ T = \frac{14m}{3} = \frac{7n}{2} \Rightarrow \frac{14m}{3} = \frac{7n}{2} \Rightarrow 28m = 21n \Rightarrow 4m = 3n $. Small

["Understanding Equivalent Fractions and Meeting Periods: Solving $ \frac{14}{3} = \frac{28}{6} $ and $ \frac{7}{2} = \frac{21}{6} $", "When working with fractions, equivalent representations are invaluable for simplifying calculations and comparisons. Two classic examples demonstrate this clearly: proving $ \frac{14}{3} = \frac{28}{6} $ and $ \frac{7}{2} = \frac{21}{6} $, and finding the least common period of two oscillating systems.", "---", "### Equivalent Fractions: Why $ \frac{14}{3} = \frac{28}{6} $?", "At first glance, $ \frac{14}{3} $ and $ \frac{28}{6} $ look different, but they represent the same value. To confirm:\n$$\n\frac{14}{3} = \frac{14 \ imes 2}{3 \ imes 2} = \frac{28}{6}\n$$\nSimilarly,\n$$\n\frac{7}{2} = \frac{7 \ imes 3}{2 \ imes 3} = \frac{21}{6}\n$$\nThese fractions are equivalent because multiplying numerator and denominator by the same factor — in this case, 2 — preserves the fraction’s value.", "---", "### Finding the Least Common Period: $ T $ for $ \frac{3\pi}{7}T = 2\pi m $ and $ \frac{4\pi}{7}T = 2\pi n $", "Now suppose we want to find the smallest $ T $ such that two periodic motions align:\n$$\n\frac{3\pi}{7}T = 2\pi m \quad \ ext{and} \quad \frac{4\pi}{7}T = 2\pi n\n$$\nfor integers $ m $ and $ n $. These equations describe angular motion — one repeating every $ \frac{2\pi}{3\pi/7} = \frac{14}{3} $, the other every $ \frac{2\pi}{4\pi/7} = \frac{7}{2} $.", "To find the least $ T $ where both motions complete full cycles simultaneously, we compute the least common multiple (LCM) of their periods.", "Start by simplifying:\nWe solve for $ T $ in both equations:\nFrom $ \frac{3\pi}{7}T = 2\pi m $:\n$$\n\frac{3T}{7} = 2m \Rightarrow T = \frac{14m}{3}\n$$\nFrom $ \frac{4\pi}{7}T = 2\pi n $:\n$$\n\frac{4T}{7} = 2n \Rightarrow T = \frac{7n}{2}\n$$", "Set the two expressions equal:\n$$\n\frac{14m}{3} = \frac{7n}{2}\n$$\nMultiply both sides by 6 to eliminate denominators:\n$$\n28m = 21n \Rightarrow 4m = 3n\n$$\nThis Diophantine equation has smallest solution when $ m = 3 $, $ n = 4 $ (since 4 and 3 are coprime). Substitute back:\n$$\nT = \frac{14 \cdot 3}{3} = 14 \quad \ ext{or} \quad T = \frac{7 \cdot 4}{2} = 14\n$$", "So the smallest $ T $ for both motions to align is $ \boxed{14} $.", "---", "### GCD and Period Simplification", "Interestingly, the period can also be expressed via GCD:\nSince the LCM of $ \frac{14}{3} $ and $ \frac{7}{2} $ in time corresponds to reducing the ratio of periods, we consider:\nThe ratio of periods $ \frac{14/3}{7/2} = \frac{14}{3} \cdot \frac{2}{7} = \frac{28}{21} = \frac{4}{3} $.\nBut the least common period $ T $ satisfies:\n$$\nT = \frac{\ ext{LCM of numerators}}{\ ext{GCD of denominators}} \ imes \frac{1}{\ ext{GCD scaling}}\n$$\nMore precisely, the fundamental period is:\n$$\nT = \frac{\ ext{lcm}(14, 21)}{7} \cdot \frac{1}{\ ext{factor}} \quad \ ext{(conceptual)}\n$$\nBut our earlier direct computation via LCM confirms $ T = 14 $ is correct.", "---", "### Final Takeaways", "- Equivalent fractions like $ \frac{14}{3} = \frac{28}{6} $ show how scaling preserves value.\n- The least common period of $ \frac{3\pi}{7}T = 2\pi m $ and $ \frac{4\pi}{7}T = 2\pi n $ is found by solving $ \frac{14m}{3} = \frac{7n}{2} $, leading to $ \boxed{T = 14} $.\n- This has applications in physics, engineering, and any periodic system analysis.", "Understanding fractions and periodic alignment empowers deeper insight into recurring phenomena — whether in timekeeping, wave motion, or harmonic oscillators.", "---", "Keywords: equivalent fractions $ \frac{14}{3} = \frac{28}{6} $, equivalent fractions $ \frac{7}{2} = \frac{21}{6} $, least common multiple, periodic motion, fundamental period, rational numbers, sync period."]