To address the question of why partitioning a directed line segment into a ratio of [tex]\( 1:3 \)[/tex] is not the same as finding [tex]\( \frac{1}{3} \)[/tex] the length of the directed line segment, let's closely examine the concept of ratios and how they apply to partitioning a segment.
### Understanding Ratios
1. Ratios Explained:
- Part-to-Part Ratio: The ratio [tex]\( 1:3 \)[/tex] means that for every one part, there are three parts of something else. If this represents a directed line segment, it means the line is divided into two segments where one segment is one part and the other segment is three parts.
- Part-to-Whole Ratio: Conversely, if the line segment was described in terms of fractions of the whole, [tex]\( \frac{1}{3} \)[/tex] would mean the line segment is divided such that one segment is [tex]\( \frac{1}{3} \)[/tex] of the total length.
### Partition of the Line Segment
2. Total Number of Parts:
- When given a ratio [tex]\( 1:3 \)[/tex], this refers to part-to-part comparison. To find the total number of parts, we add the parts together: [tex]\( 1 + 3 = 4 \)[/tex].
- Therefore, the directed line segment is divided into four parts where one part is [tex]\( \frac{1}{4} \)[/tex] of the total length, and the remaining three parts together constitute [tex]\( \frac{3}{4} \)[/tex] of the total length.
### Comparing Ratios and Fractions
3. Difference Between [tex]\( \frac{1}{3} \)[/tex] and [tex]\( 1:3 \)[/tex]:
- [tex]\( \frac{1}{3} \)[/tex] indicates a segment that is one-third of the total length of the line.
- [tex]\( 1:3 \)[/tex] indicates that the line segment is divided into four parts, with one part being [tex]\( \frac{1}{4} \)[/tex] of the total length.
### Conclusion
The given partition ratio of [tex]\( 1:3 \)[/tex] means the segment is divided into four parts, with one part measuring [tex]\( \frac{1}{4} \)[/tex] of the total length. This is fundamentally different from simply taking [tex]\( \frac{1}{3} \)[/tex] of the length of the segment.
Thus, the total number of parts in the whole is summed as [tex]\( 1 + 3 = 4 \)[/tex]. This answer correctly explains why the partition [tex]\( 1:3 \)[/tex] is not the same as [tex]\( \frac{1}{3} \)[/tex] the length of the segment. The key understanding is that the ratio [tex]\( 1:3 \)[/tex] represents a total of four parts, not a simple one-third fraction.
