The meaning of ``Get together'': Part 2

A while ago, I taught mathematics to a thirteen years old boy. His family moved to another city, so we could not have more sessions, but I liked to teach him.

In the last session, we discussed about addition of fractions. If we get a half cake ($\frac{1}{2}$) and a half cake ($\frac{1}{2}$) together,
$$\begin{eqnarray*}  \frac{1}{2} + \frac{1}{2} = \frac{1+1}{2} = \frac{2}{2} = 1, \end{eqnarray*}$$
it becomes 1. If we add fractions with like denominators, we only add numerators. We discussed why we did in this way and what this meant.

Then I told him a story. The boy's favorite soccer player tried to shoot the goal twice in a game, and he succeeded once. His goal ratio of this game is $\frac{1}{2}$. (Two shoots, one goal) The next game, again he tried to shoot the goal twice in the game, and again he succeeded once. His goal ratio of the game is again $\frac{1}{2}$. Let's think about his goal ratio to get two games together. For these two games, he tried four goals in total, but succeeded twice in total. Therefore, his total goal ratio is:
$$\begin{eqnarray*}  \frac{1}{2} + \frac{1}{2} = \frac{1+1}{2+2} = \frac{2}{4} = \frac{1}{2}. \end{eqnarray*}$$
The goal success ratio is correct. However, how is $\frac{1}{2}$ + $\frac{1}{2}$ equal to $\frac{1}{2}$. That must be 1. This also violated the rule of fraction addition. If we got two $\frac{1}{2}$ cakes together,
$$\begin{eqnarray*}  \frac{1}{2} + \frac{1}{2} = 1, \end{eqnarray*}$$
but, when we got two games goal ratio together,
$$\begin{eqnarray*}  \frac{1}{2} + \frac{1}{2} = \frac{1}{2}. \end{eqnarray*}$$
I asked him why this happened. It is quite strange that one addition expression has two different answers. 1+1 is only equal to 2. There should not be two different answers of one addition, like 1 + 1 = 2 and 1 + 2 = 1 at the same time. Therefore, at least one should be wrong.

We discussed this problem. He first told me the total goal ratio must be 1 as the cake's addition. But I added one more game, the player tried two goals and one success, then in total three games,
$$\begin{eqnarray*}  \frac{1}{2} + \frac{1}{2} + \frac{1}{2} = \frac{3}{2}. \end{eqnarray*}$$
Now the goal success ratio is $\frac{3}{2}$. This means this player tried two kicks and he got three points. How it could be possible? At that time, his father just passed by us, and made a joke, ``He is the best player, he could get three points with two shoots.'' But we all know that is not possible, so something wrong in this calculation. When we want to get all the three games together, his goal ratio should be
$$\begin{eqnarray*}  \frac{1}{2} + \frac{1}{2} + \frac{1}{2} = \frac{1+1+1}{2+2+2} =   \frac{3}{6} = \frac{1}{2}. \end{eqnarray*}$$
Since he always success one goal per two shoots. In other words, he tried six shoots and succeeded three goals, this means his goal success ratio is $\frac{3}{6}$ which is $\frac{1}{2}$.

Our discussion continued more than 20 minutes, but at the end, he found what is the problem. Why this strange thing happens. I was impressed. It was a great moment for me and hopefully also for him.

The following article will be the conclusion of this story.