60. Побудуй три дроби на одній числовій прямій. \( \begin{array}{lll}\text { a) } \frac{1}{2}, \frac{5}{6}, \frac{5}{12} ; & \text { б) } \frac{1}{2}, \frac{1}{7}, \frac{3}{14} ; & \text { в) } \frac{1}{2}, \frac{2}{3}, \frac{3}{4} \\ \text { Розкажи, який одиничний відрізок зручно взяти в кожном } \\ \text { випадку. Чому? }\end{array} \)

Alright, I need to tackle this problem about plotting three fractions on a number line. The problem presents three different sets of fractions and asks me to determine which unit interval is most convenient to use for each case. Let's break it down step by step. First, let's look at the first set of fractions: 1/2, 5/6, and 5/12. To plot these on a number line, I need to decide on a unit interval that makes it easy to represent these fractions accurately. The fractions are all less than 1, so a unit interval of 1 seems appropriate. But maybe a smaller unit interval would be better to capture the differences between the fractions more precisely. Next, the second set: 1/2, 1/7, and 3/14. Here, the fractions are also less than 1, but 1/7 and 3/14 are smaller than 1/2. A unit interval of 1 might still work, but perhaps a smaller unit would help in distinguishing between these smaller fractions. The third set is 1/2, 2/3, and 3/4. All these fractions are greater than 1/2 and less than 1, so again, a unit interval of 1 seems suitable. However, plotting 2/3 and 3/4 on a number line with a unit interval of 1 might require some precision. Now, considering the convenience of the unit interval, I should think about the denominators of the fractions. For the first set, the denominators are 2, 6, and 12. The least common multiple (LCM) of these denominators is 12, so using a unit interval of 1/12 would allow me to plot all fractions accurately without any rounding errors. For the second set, the denominators are 2, 7, and 14. The LCM here is 14, so a unit interval of 1/14 would be ideal. This would make it easier to place 1/7 and 3/14 precisely on the number line. In the third set, the denominators are 2, 3, and 4. The LCM is 12, so again, a unit interval of 1/12 would be convenient. This would allow for accurate placement of 2/3 and 3/4 without any approximation. But wait, maybe there's a more efficient way. Perhaps using a unit interval that's a common multiple of all denominators across all sets could simplify the process. The LCM of 2, 6, 12, 2, 7, 14, 2, 3, and 4 is 84. So, a unit interval of 1/84 would be the most precise, but it might be overly detailed and not necessary for plotting these fractions on a number line. Alternatively, I could consider using a unit interval of 1/12 for all three sets since 12 is a common multiple of the denominators in the first and third sets, and it's also a multiple of 7 in the second set. This would provide a balance between precision and simplicity. Another thought: maybe the choice of unit interval doesn't significantly affect the plotting process, and any unit interval that's a multiple of the denominators would suffice. However, using the smallest possible unit interval that accommodates all denominators would make the plotting more accurate and precise. Also, considering the fractions' values, some are closer to each other than others. For example, 1/2 and 5/6 are closer together than 1/2 and 5/12. So, the unit interval should be chosen to reflect these differences appropriately. In conclusion, for each set of fractions, selecting a unit interval that's a common multiple of the denominators ensures that all fractions can be plotted accurately without rounding errors. This approach provides the necessary precision for each case.