Why is partitioning a direct line segment of 1:3 not the same as finding 1/3 the length of the direct line segment

First 1/3

1/3 the length of something means that there are 2 pieces of (say) lumber. One of them is 1/3 the length of the other.

For example you cut a board in such a way that the shorter piece is 1 foot long and the longer piece is 3 feet long. The shorter piece is 1/3 the longer piece.

Now one to 3

If the ratio is 1:3 what you are describing is the parts to a total. The example above describes the pieces to each other.

This describes the parts to the whole

One piece is 1 foot long.

One piece is 3 feet long

the smaller piece is 1/4 when compared to the entire length of the board.

The larger piece is 3/4 when compared to the entire length of the board.

What happens when the fractions are a little bit more complicated?

Suppose you have a board that is 2 1/2 times long than a smaller board?

The fractional value is 1/2.5 or 2 / 5

Let the shorter board be 1

Let the longer one be 2.5

The ratio is 2 / (2 + 5) and 5 / (2 + 5)