In a graduate class of $200$, $40\%$ are women and $\frac{1}{5}$ become lecturers. If the number of men who become lecturers is twice that of women, calculate approximate percentage of men who became lecturers.

### Concept & Logic To find the percentage of a specific subset, we need two absolute numbers: the total size of the group in question (total men) and the size of the subset (men who became lecturers). $$\text{Percentage} = \left(\frac{\text{Men Lecturers}}{\text{Total Men}}\right) \times 100$$ ### Step-by-step Solution **Given:** Total students = $200$ Women = $40\%$ of total **Step 1: Calculate total women and men** Total Women = $40\%$ of $200 = 80$ Total Men = $200 - 80 = 120$ **Step 2: Calculate women who became lecturers** Women Lecturers = $\frac{1}{5}$ of $80$ Women Lecturers $= 16$ **Step 3: Calculate men who became lecturers** The problem states this is twice the number of women lecturers. Men Lecturers $= 2 \times 16 = 32$ **Step 4: Calculate the required percentage** We need the percentage of *men* who became lecturers (out of the total men). Percentage $= \left(\frac{32}{120}\right) \times 100$ $= \left(\frac{4}{15}\right) \times 100$ $= \frac{400}{15} = 26.66\%$ Rounding to the nearest whole number gives an approximate percentage of $27\%$. ### Exam Strategy & Shortcut Work purely in proportions or use a base of $100$ to shrink the mental math. If the class is $100$, women = $40$, men = $60$. Women lecturers = $\frac{1}{5}$ of $40 = 8$. Men lecturers = $2 \times 8 = 16$. Percentage of men = $\frac{16}{60} = \frac{4}{15}$. Memorizing common fractions helps here: $\frac{1}{15} \approx 6.66\%$, so $\frac{4}{15} \approx 26.66\%$, which rounds to $27\%$. ### Common Pitfall A major trap is calculating the percentage of men lecturers out of the *entire* class ($32$ out of $200 = 16\%$) instead of out of the total number of *men* ($32$ out of $120$). The examiners specifically included $16\%$ as option (a) to catch this mistake! ### Final Answer **Therefore, the correct answer is 27%.**