Splitting a segment with a ratio

Question

I came across the homework question that I attempted to do. After looking at the answers, and getting it wrong I didn't understand why.

I'm specifically lost at why we would get a fraction of 2/5 at the first two sentences. Could someone explain how we got this fraction, and what does it mean by 5 pieces?

Answer 1

If we share sweets in a ratio of 2-to-3 then for every 2 sweets that I get, you get three sweets.

Depending on the number of sweets, they could be shared out as:

Two to me and three to you, four to me and six to you, six to me and nine to you, etc.

Let's look at the first case: I get two sweets and you get three. There are five sweets altogether, so I get 2-out-of-5 and you get 3-out-of-5. As fractions these are $\frac{2}{5}$ of the sweets and $\frac{3}{5}$ of the sweets.

Let's look at the second case: I get four sweets and you get six. There are 10 sweets altogether, so I get 4-out-of-10 and you get 6-out-of-10. As fractions these are $\frac{4}{10}$ of the sweets and $\frac{6}{10}$ of the sweets. But look: $\frac{4}{10} = \frac{2}{5}$ and $\frac{6}{10} = \frac{3}{5}$.

Let's look at the third case: I get six sweets and you get nine. There are 15 sweets altogether, so I get 6-out-of-15 and you get 9-out-of-15. As fractions these are $\frac{6}{15}$ of the sweets and $\frac{9}{15}$ of the sweets. But look: $\frac{6}{15} = \frac{2}{5}$ and $\frac{9}{15} = \frac{3}{5}$.

Dude, you must know about ratio.

Answer 2

The segment is broken into ratio of 2:3. Say the point which splits the segment into 2:3 is Y(a,b). Now if you imagine segment XZ to be made up of 5 unit pieces then total length of the segment is 5 units. NB: 1 Unit here is 1 unit of the distance between X and Z i.e. $\frac{\sqrt{4^2 + 8^2}}{5} = \frac{\sqrt{80}}{5}$

Splitting this into 2:3 would mean XY is of length 2 units and YZ is of length 3 units.

So to find Y(a,b) you have $a = \frac{3*(-2) + 2*(6)}{5} = \frac{6}{5} = 1.2 $ which is same as $a = -2 + \frac{2}{5}*8 = \frac{6}{5} = 1.2 $. Here 8 is the projection of XZ on x-axis.

Similarly you can compute $b = 2 + \frac{2}{5}*4 = \frac{18}{5} = 3.6$

You can verify the result by computing distance between X and Y which should be equal to $2* \frac{ \sqrt{80}}{5}$

cheers!