Why is partitioning a directed line segment into a ratio of [tex]$1: 3$[/tex] not the same as finding [tex]$\frac{1}{3}$[/tex] the length of the directed line segment?
A. The ratio given is part to whole, but fractions compare part to part.
B. The ratio given is part to part. The total number of parts in the whole is [tex]$3-1=2$[/tex].
C. The ratio given is part to part. The total number of parts in the whole is [tex]$1+3=4$[/tex].
D. The ratio given is part to whole, but the associated fraction is [tex]$\frac{3}{1}$[/tex].
Answer (2)
The ratio 1 : 3 divides the line segment into two parts with a total of 1 + 3 = 4 units.
The first part is 4 1 of the total length, and the second part is 4 3 of the total length.
Finding 3 1 of the length means taking one-third of the total length.
Since 4 1 = 3 1 , partitioning in the ratio 1 : 3 is different from finding 3 1 of the length. Partitioning in the ratio 1:3 is not the same as finding 3 1 of the length.
Explanation
Problem Analysis Let's analyze the problem. We are asked to explain why partitioning a directed line segment in the ratio 1 : 3 is not the same as finding 3 1 of the length of the segment. The key is to understand what the ratio 1 : 3 means in the context of partitioning a line segment.
Understanding the Ratio The ratio 1 : 3 means that the line segment is divided into two parts, where the length of one part is one unit and the length of the other part is three units. The total number of units is 1 + 3 = 4 . Therefore, the first part represents 4 1 of the total length of the line segment, and the second part represents 4 3 of the total length of the line segment.
Comparing with 1/3 of the Length Finding 3 1 of the length of the line segment means taking one-third of the total length. This is different from dividing the line segment into parts with lengths in the ratio 1 : 3 , where the smaller part is 4 1 of the total length.
Conclusion Since 4 1 (representing the portion from the 1 : 3 partition) is not equal to 3 1 , partitioning a directed line segment into a ratio of 1 : 3 is not the same as finding 3 1 of the length of the directed line segment.
Examples
Imagine you're baking a cake and the recipe says to divide the batter in a 1:3 ratio to make two layers of different thicknesses. This means you'll have one layer that's 1/4 of the total batter and another layer that's 3/4. If you instead tried to use 1/3 of the batter for one layer, the layers wouldn't be in the correct proportions, and the cake wouldn't turn out as intended. This highlights how ratios and fractions, though related, serve different purposes in dividing quantities.
Partitioning a line segment into a ratio of 1 : 3 divides it into two parts, making the smaller part 4 1 of the total. In contrast, finding 3 1 of the length means calculating one-third of the whole segment. Therefore, these two actions represent different concepts of division and are not equal.
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