Adding drawnfractions (Very simple question)

Question

I have been surprised not being able to solve this:

+---------------------------+
| XXXXXXXXXXXX|             |
|   XXXXXXXXXX|             |
|     XXXXXXXX|             |
|       XXXXXX|             |
|         XXXX|             |
|            X|             |
+------+--------------------+
|      |      |             |
|      |      |             |
|      |      |             |
+-------------+             |
|      |XXXXXX|             |
|      |XXXXXX|             |
|      |XXXXXX|             |
+---------------------------+

So what I looked here is, the first one is the half, so 1/2. The other is 0/1, the 3rd is 1/4, and the 4th is 0/1.

When I make the count (sum 1/2 + 0/1 + 1/4 + 0/1), I get 3/4, while just looking at the graphic, I can see there is 3/16 of the whole figure painted. What am I doing wrong?

Answer 1

If you consider the entire square to be $1$, then the triangular piece is not a half. It is one half of a quarter, that is $\frac12\cdot\frac14 = \frac18.$ The small square piece is one quarter of a quarter, that is $\frac14\cdot\frac14=\frac{1}{16}.$ Hence the sum is

$$\frac18+\frac{1}{16}=\frac{2}{16}+\frac{1}{16}=\frac{3}{16}. $$

Answer 2

You get $3/16$ when you look at the whole square as $1$. You get $3/4$ when you look at a quarter of the whole square as $1$.

Let's look at your first method with the whole square as $1$. Then for the first quarter you should get $1/8$, since there are totally $8$ triangles like that. For the second, $0$. For the third, $1/16$. The 4th, $0$. That will add up to $3/16$, same as the other method gives you.