In a class, $\frac{3}{5}$ of the students are girls and rest are boys. If $\frac{2}{9}$ of the girls and $\frac{1}{4}$ of the boys are absent, what part of the total number of students is present?

### Concept & Strategy The problem asks for the fraction of the total students present. We can solve this by calculating the individual fractions of girls and boys who are present, and then finding their weighted sum relative to the total population. Alternatively, assuming a convenient plug-in number for the total number of students simplifies the calculation significantly. ### Step-by-Step Solution * **Given:** * Fraction of girls = $\frac{3}{5}$ * Fraction of boys = $1 - \frac{3}{5} = \frac{2}{5}$ * Fraction of absent girls = $\frac{2}{9} \implies$ Fraction of present girls = $1 - \frac{2}{9} = \frac{7}{9}$ * Fraction of absent boys = $\frac{1}{4} \implies$ Fraction of present boys = $1 - \frac{1}{4} = \frac{3}{4}$ * **Calculation:** * Total fraction of present students = (Fraction of girls $\times$ Present girls) + (Fraction of boys $\times$ Present boys) $$ \text{Present} = \left(\frac{3}{5} \times \frac{7}{9}\right) + \left(\frac{2}{5} \times \frac{3}{4}\right) $$ $$ \text{Present} = \frac{7}{15} + \frac{3}{10} $$ * Find a common denominator for 15 and 10, which is 30: $$ \text{Present} = \frac{14}{30} + \frac{9}{30} = \frac{23}{30} $$ ### Exam Strategy & Shortcut **Plug-in Number Method:** Choose a total number of students that is a multiple of the denominators: 5, 9, and 4. Let the total number of students be $5 \times 9 \times 4 = 180$. * Number of girls = $\frac{3}{5} \times 180 = 108$ * Number of boys = $180 - 108 = 72$ * Present girls = $108 \times \left(1 - \frac{2}{9}\right) = 108 \times \frac{7}{9} = 84$ * Present boys = $72 \times \left(1 - \frac{1}{4}\right) = 72 \times \frac{3}{4} = 54$ * Total present students = $84 + 54 = 138$ * Required fraction = $\frac{138}{180} = \frac{23}{30}$ ### Common Pitfall Students often calculate the fraction of *absent* students ($\frac{3}{5} \times \frac{2}{9} + \frac{2}{5} \times \frac{1}{4} = \frac{7}{30}$) and forget to subtract it from 1 to find the present students. Always double-check whether the question asks for the "present" or "absent" share. ### Final Answer **Therefore, the correct answer is 23/30.**