Dividing line segments with ratios vs. fractions

Question

I know that $2:3$ is actually $\frac {2}{3}$. So when you split a line segment by a ratio, you would add $2$ and $3$ to get a fraction of $\frac {2}{5}$ that will be used to solve the problem.

I can't really seem to find much information on these kind of problems, though. Therefore, I am confused if I am given a fraction instead, such as $\frac {2}{5}$ should I add the numerator and the denominator to get $\frac {2}{7}$?

Here's a problem to make this question more clear.

Please note that the answer is wrong since I discovered the actual solution.

Answer 1

If you are given a ratio for a line, then the line is split into segments, whose lengths are in the given ratio. For example, if you had the ratio $2:3$, you are right in thinking that the line can scale to $2$ segments, each of length $2$ and $3$. Obviously the total is $5$ so they each amount to $\frac{2}{5}$ and $\frac{3}{5}$ of the total length.

But if you are given a fraction, then that should represent how much of the $total$ length is taken up. For example, $\frac{2}{5}$ is a length of $2$ out of a total length of $5$ and $not$ a total of $7$.

In my experience, ratios aren't given as often as fractions and I much prefer fractions.

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Also, jj172 is right in saying that $2:3$ is not $\frac{2}{3}$.

Answer 2

I'm not sure if I grasp exactly what you are asking, but I hope the following information can help.

Your initial assumption that the ratio 2:3 is equal to $2/3$ is false. In fact, the ratio 2:3 means that, out of a total of $2+3=5$ objects, that there are 2 of one and 3 of another.

Thus, given a ratio, you add the two numbers to get the total value, and then take each part of the ratio (for example 2 and 3 in the ratio 2:3) and use that number over the total to get the equivalent fraction.

For example:

The ratio 5:3.

There are $5+3= 8$ (or a multiple of 8) total objects. Then, $5/8$ of the total number of objects is the first kind, and $3/8$ of the total number of objects is the second kind.

Answer 3

If you have given a relation of two fraction like 2/5 to 3/5 it is the same as $\frac{\frac{2}{5}}{\frac{3}{5}}=\frac{2}{5} \cdot \frac{5}{3}=2/3 $

The 5 in the numerator shows you only, that you need at least 5 (or the multiple of 5) parts, to get whole numbers for the two parts.

Edit:

To solve the exercise you have to consruct a vector of the two given points:

${2 \choose -6} + {-2 \choose 11} \cdot r$

Now set r=1/5 and you get the point Y