Question

Why is partitioning a directed line segment into a ratio of 1:3 not the same as finding 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 1/3.

Answer 1

The length of the directed line segment is partitioned into the ratio of $1:3$  is $\boxed{{\mathbf{not\ same}}}$  as the one third length of the directed line segment.

Further explanation:

Ratio is the quantitative relation between two different things.

Given:

It is given that a directed line segment is partitioned in the ratio of $1:3$ .

Step by step explanation:

Step 1

The directed line segment is partitioned in the ratio of $1:3$ .

First part is before the desired point and 3 parts are after the desired point.

It means that there are total 4 shares as it contains four pieces.

Step 2:

The one third length of the directed line segment means there are three shares of equal size.

Step 3:

Consider an example a cloth has length of 12cm.

Now divide the cloth into the ratio of $1:3$ .

$\begin{aligned}{\text{first part}}&=\frac{1}{4}\times12=3\hfill\\{\text{second part}}&=\frac{3}{4}\times12=9\hfill\\\end{aligned}$

Therefore, the cloth is divided in the ratio of $1:3$  as ${\text{3cm, 9cm}}$  

Now divide the cloth equally in three parts that is $\frac{1}{3}$  of the cloth.

  $\frac{1}{3}\times12=4$

Therefore, the cloth is divided into one third as $4{\text{cm, }}4{\text{cm}}$ .

Thus, it has been proved the given explanation.

Learn more:

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Answer details:

Grade: Middle school

Subject: Mathematics

Chapter: Fractions

Keywords: directed, line segment, length, one third, ratio, units, line, thing, quantitative relation, number, comparison, different things, ribbon, equally divided.

Answer 2

Answer:

The ratio given is part to part. The total number of parts in the whole is 1 + 3 = 4.

Step-by-step explanation:

I like to call the numbers in the ratio 1 : 3 "ratio units".

That is, the short length is 1 ratio unit long, and the longer length is 3 ratio units long. The total (whole) length is 1+3 = 4 ratio units long. Then the short length (1 ratio unit) is 1/4 of the whole length (4 ratio units).

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Coment on the answer wording

I think you need to be careful with terminology. The word "part" has a different meaning in the first answer sentence "... part to part" than it does in the second answer sentence "... parts in the whole ...".

In the first sentence, it refers to "a piece of the line, no matter its length." In the second sentence, it refers to "that length of the line represented by 1 in the ratio."