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 divides it into two parts with lengths in that ratio.
This means the segment is divided into 1 + 3 = 4 parts, and the first part is 4 1 of the total length.
Finding 3 1 of the length means taking a segment that is 3 1 of the total length.
Since 4 1 = 3 1 , the two operations are different. The ratio given is part to part. The total number of parts in the whole is 1 + 3 = 4 .
Explanation
Problem Analysis Let's analyze the difference between partitioning a directed line segment in a ratio of 1 : 3 and finding 3 1 of its length.
Partitioning in the ratio 1:3 Consider a directed line segment A B . Partitioning it in the ratio 1 : 3 means dividing it into two parts, A P and PB , such that the ratio of their lengths is A P : PB = 1 : 3 . This implies that A P is 4 1 of the total length A B , since the total number of parts is 1 + 3 = 4 .
Finding 1/3 of the length Finding 3 1 of the length of the directed line segment A B means finding a point Q on A B such that the length A Q is 3 1 of the total length A B .
Comparison Since 4 1 A B = 3 1 A B , the point P (partitioning in the ratio 1 : 3 ) and the point Q (finding 3 1 of the length) are different.
Conclusion Therefore, partitioning a directed line segment into a ratio of 1 : 3 is not the same as finding 3 1 of the length of the segment because the ratio 1 : 3 implies dividing the segment into 1 + 3 = 4 parts, while finding 3 1 of the length refers to a fraction of the total length. The correct answer is: The ratio given is part to part. The total number of parts in the whole is 1 + 3 = 4 .
Examples
Imagine you're baking a cake and the recipe says to divide the batter in a ratio of 1:3 to create two layers of different thickness. This means you'll have one layer that's 1/4 of the total batter and another layer that's 3/4. Now, if you were to simply take 1/3 of the batter, you wouldn't achieve the same layer thicknesses. The 1:3 ratio ensures a specific proportion, while 1/3 represents a fraction of the whole, leading to different results.
