Let $ a_1 \geq a_2 \geq a_3 \in \mathbb{Z}^+$ be given and let N$ (a_1, a_2, a_3)$ be the number of solutions $ (x_1, x_2, x_3)$ of the equation \[ \sum^3_{k=1} \frac{a_k}{x_k} = 1.\] where $ x_1, x_2,$ and $ x_3$ are positive integers. Prove that \[ N(a_1, a_2, a_3) \leq 6 a_1 a_2 (3 + ln(2 a_1)).\]