In a company there are 75% skilled workers and the remaining are unskilled. 80% of skilled workers and 20% of unskilled workers are permanent. If the number of temporary workers is 126, then what is the total number of workers?

### Concept & Logic This is a classic nested percentage problem. The best approach is the **Base-100 Assumption Method**. By assuming the total number of workers is $100$ units, we can navigate through the percentage splits easily without fractions. Once we find the temporary workers in terms of our "units", we can equate it to the real value given ($126$) to find the multiplier. ### Step-by-Step Solution Let the total number of workers in the company be $100$ units. * **Skilled Workers:** $75\%$ of $100 = 75$ units * **Unskilled Workers:** $100 - 75 = 25$ units Now, let's calculate the permanent workers: * **Permanent Skilled:** $80\%$ of $75 = \frac{4}{5} \times 75 = 60$ units * **Permanent Unskilled:** $20\%$ of $25 = \frac{1}{5} \times 25 = 5$ units Total Permanent Workers $= 60 + 5 = 65$ units. This means the remaining workers must be temporary: * **Temporary Workers:** $100 - 65 = 35$ units. We are given that the actual number of temporary workers is $126$. We can set up an equation: $$ 35 \text{ units} = 126 $$ $$ 1 \text{ unit} = \frac{126}{35} = 3.6 $$ Since the total number of workers is $100$ units: $$ \text{Total Workers} = 100 \times 3.6 = 360 $$ ### Exam Strategy & Shortcut Use fractions mentally to speed up the process. $75\%$ is $\frac{3}{4}$ and $25\%$ is $\frac{1}{4}$. Permanent is $80\%$ ($\frac{4}{5}$) of $\frac{3}{4} = \frac{3}{5}$. Permanent unskilled is $20\%$ ($\frac{1}{5}$) of $\frac{1}{4} = \frac{1}{20}$. Total permanent fraction $= \frac{3}{5} + \frac{1}{20} = \frac{13}{20}$. Therefore, the temporary fraction is $1 - \frac{13}{20} = \frac{7}{20}$. If $\frac{7}{20}$ of total $= 126$, then Total $= \frac{126 \times 20}{7} = 18 \times 20 = 360$. This algebraic fractional method is extremely fast if you are comfortable with percentage-to-fraction conversions. ### Common Pitfall Students often confuse "temporary" and "permanent" groups. Because the prompt gives percentages for the *permanent* staff, it is easy to accidentally set $65$ units $= 126$ rather than subtracting from $100$ to find the temporary staff. Always double-check what specific subgroup a raw number belongs to. ### Final Answer Therefore, the correct answer is **360**.