Solve for x: 152.5 * 0.5 * (x / 54.9) + 152.5 * (1 - x / 54.9) = 100.

Step-by-step solution

1. 1

   Substitute for the repeated fraction. The quantity $\tfrac{x}{54.9}$ appears twice, so let

   $u=\frac{x}{54.9}$

   and the equation becomes a plain linear equation in $u$:

   $76.25u+152.5(1-u)=100$

   The structure is a weighted blend: a fraction $u$ of the total runs at half rate ($76.25$) and the remaining $1-u$ at full rate ($152.5$).

2. 2

   Expand and collect.

   $76.25u+152.5-152.5u=100\;\Longrightarrow\;-76.25u=-52.5$

   The coefficient $-76.25$ is exactly the difference between the two rates, which is what makes this an alligation-style problem.

3. 3

   Solve for $u$.

   $u=\frac{52.5}{76.25}=\frac{5250}{7625}=\frac{42}{61}\approx 0.68852$

   So about $68.9\%$ of the total is on the half rate — a sensible figure, since the target $100$ sits closer to $76.25$ than to $152.5$.

4. 4

   Recover $x$.

   $x=54.9u=54.9\cdot\frac{52.5}{76.25}=\frac{2882.25}{76.25}=37.8$

   The result is exact, not rounded: $76.25\times 37.8=2882.25$ precisely.

5. 5

   Verify in the original equation. With $\tfrac{37.8}{54.9}=0.688525$:

   $76.25(0.688525)+152.5(0.311475)=52.5+47.5=100\;\checkmark$

   The two contributions $52.5$ and $47.5$ add exactly to the target, confirming the split.