Question

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?

• the ratio given is part to whole, but fractions compare part to part.
• the ratio given is part to part. the total number of parts in the whole is 3 - 1 = 2.
• the ratio given is part to part. the total number of parts in the whole is 1 + 3 = 4.
• the ratio given is part to whole, but the associated fraction is \\(\frac{3}{1}\\).

Explanation:

Analyze the partition ratio definition

$$ \text{Ratio } a:b = 1:3 \implies \text{Part}_1 = 1, \text{Part}_2 = 3 $$

Determine the total parts of the whole segment

$$ \text{Total Parts} = a + b = 1 + 3 = 4 $$

Compare with the fraction of the whole length

$$ \text{Fraction of whole} = \frac{a}{a+b} = \frac{1}{4} eq \frac{1}{3} $$

Answer:

• The ratio given is part to whole, but fractions compare part to part.
• The ratio given is part to part. The total number of parts in the whole is \(3 - 1 = 2\).
• The ratio given is part to part. The total number of parts in the whole is \(1 + 3 = 4\) (Correct answer)
• The ratio given is part to whole, but the associated fraction is \(\frac{3}{1}\).