Solve 152.5 · 0.5 · (x/54.9) + 152.5 · (1 − x/54.9) = 100 for x.

Step-by-step solution

1. 1

   Read the structure before touching the algebra. The fraction $\frac{x}{54.9}$ is the share of the whole that sits at half strength; $1-\frac{x}{54.9}$ is the remaining share at full strength $152.5$. That is why the same fraction appears twice with opposite signs — it makes the equation linear in one unknown, $t=\frac{x}{54.9}$.

2. 2

   Collapse the constant product and expand. First $152.5\cdot 0.5=76.25$. Distributing the second term,

   $152.5\left(1-\frac{x}{54.9}\right)=152.5-152.5\cdot\frac{x}{54.9}$

   so the equation reads

   $76.25\cdot\frac{x}{54.9}+152.5-152.5\cdot\frac{x}{54.9}=100$

3. 3

   Combine the two $\frac{x}{54.9}$ terms. Their coefficients subtract:

   $76.25-152.5=-76.25$

   leaving

   $152.5-76.25\cdot\frac{x}{54.9}=100$

   The drop from full strength to the target is exactly half the full value — that is the $-76.25$.

4. 4

   Isolate the fraction. Subtract $152.5$ and divide by $-76.25$:

   $-76.25\cdot\frac{x}{54.9}=-52.5\quad\Longrightarrow\quad \frac{x}{54.9}=\frac{52.5}{76.25}$

5. 5

   Reduce the ratio to a fraction instead of rounding. Multiply numerator and denominator by $100$ and divide by $125$:

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

   This is where the problem is prettier than it looks: $549=9\cdot 61$, so

   $x=54.9\cdot\frac{42}{61}=\frac{549}{10}\cdot\frac{42}{61}=\frac{9\cdot 42}{10}=\frac{378}{10}$

6. 6

   State the exact answer and check it. So $x=37.8$ exactly — the decimal terminates, it is not a rounded value. Substituting back:

   $76.25\cdot\frac{37.8}{54.9}+152.5-152.5\cdot\frac{37.8}{54.9}=152.5-76.25\cdot\frac{42}{61}=152.5-52.5=100\ \checkmark$