Let the two stage durations be variables. Use the overall rate of $25.5$ to find a ratio between those durations.

After finding the duration ratio, multiply each part of that ratio by its corresponding upload rate before finding the requested fraction.

Algebra is faster for this question, but Desmos can verify the time ratio. Let the second-stage duration be $1$ and let $x$ represent the first-stage duration. Enter $y_1=\frac{18x+30}{x+1}$ and $y_2=25.5$.

Click the intersection of the two graphs. Its $x$-coordinate is $0.6$, so the first-stage-to-second-stage time ratio is $0.6:1$, which is equivalent to $3:5$.

Enter $\frac{18(3)}{18(3)+30(5)}$. Desmos verifies the fraction of the total data transferred during the first stage.

Let $t$ be the duration of the first stage and let $s$ be the duration of the second stage. Total data uploaded is $18t+30s$, so the overall rate equation is $\frac{18t+30s}{t+s}=25.5$.

Multiply by $t+s$: $18t+30s=25.5t+25.5s$. Rearranging gives $4.5s=7.5t$, so $t:s=3:5$.

Use $3$ and $5$ as proportional stage durations. The first stage uploads $18(3)=54$ proportional units of data, and the second stage uploads $30(5)=150$ proportional units. Therefore, the fraction uploaded during the first stage is $\frac{54}{54+150}=\frac{9}{34}$.