Why is partitioning a directed line segment into a ratio of $1: 3$ not the same as finding $\frac{1}{3}$ 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 $3-1=2$.
C. The ratio given is part to part. The total number of parts in the whole is $1+3=4$.
D. The ratio given is part to whole, but the associated fraction is $\frac{3}{1}$.
Answer (1)
Partitioning a line segment in the ratio 1 : 3 means dividing it into two parts such that one part is one-third the length of the other.
The total number of parts is 1 + 3 = 4 , so the first part is 4 1 of the whole segment.
Finding 3 1 of the length of the segment means taking one-third of the total length.
Since 4 1 of the segment is not equal to 3 1 of the segment, the two operations are different. They are not the same .
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 a ratio represents and how it relates to fractions of the whole segment.
Partitioning in the ratio 1:3 Consider a directed line segment A B . When we partition it in the ratio 1 : 3 , we are dividing it into two parts, A P and PB , such that the length of A P is one part and the length of PB is three parts. The total number of parts is 1 + 3 = 4 . Therefore, A P is 4 1 of the total length A B . That is, A P = 4 1 A B .
Finding 1/3 of the length Now, let's consider finding 3 1 of the length of the directed line segment A B . This means we are looking for a point Q on the segment such that A Q = 3 1 A B .
Comparison and Conclusion Comparing the two scenarios, we have A P = 4 1 A B and A Q = 3 1 A B . Since 4 1 A B is not equal to 3 1 A B , partitioning the segment in the ratio 1 : 3 is not the same as finding 3 1 of its length. The ratio 1 : 3 divides the segment into 4 parts, with one part being A P , while the fraction 3 1 refers to one-third of the total length.
Examples
Imagine you're baking a cake and the recipe says to divide the batter in a ratio of 1:3 to make two layers of different thickness. This means you'll have 4 total 'parts,' and the smaller layer will use 1 part of the batter while the larger layer uses 3 parts. If you instead tried to use 1/3 of the batter for the smaller layer, you wouldn't be following the recipe's intended proportions, and the cake layers wouldn't have the desired relative sizes. Ratios help maintain proportions within a whole, while fractions represent a portion of the entire quantity.
