In the diagram, the sum of the numbers in each quarter circle is the same. The value of $x+y+z$ is
(A) 75
(D) 171
(B) 64
(C) 54
(E) 300

![](https://cdn.mathpix.com/cropped/2024_04_20_e85ef69091c57cb633f3g-122.jpg?height=301&width=296&top_left_y=1758&top_left_x=1359)
Solution 1

In the top right quarter, we have no unknowns, so we can calculate the sum of the three numbers to be $13+17+45=75$. Therefore, the sum of the numbers (including the unknowns) in each of the other 3 quarters is 75 as well, and so

$$
\begin{aligned}
(z+28+8)+(x+19+50)+(y+3+63) & =3(75) \\
x+y+z+36+69+66 & =225 \\
x+y+z & =54
\end{aligned}
$$

## Solution 2

In the top right quarter, we have no unknowns, so we can calculate the sum of the three numbers to be $13+17+45=75$.

In the top left quarter, we then have $z+28+8=75$ and so $z=39$.

Similarly, $x=6$ and $y=9$.

Therefore, $x+y+z=6+9+39=54$.