An institute organised a fete and $\frac{1}{5}$ of the girls and $\frac{1}{8}$ of the boys participated in the same. What fraction of the total number of students took part in the fete?

### Concept & Logic This problem introduces the concept of weighted averages and Data Sufficiency in algebra. To find a combined fraction for a total population, you must know the ratio or absolute counts of the individual subgroups. The overall participation fraction is a weighted average of $\frac{1}{5}$ and $\frac{1}{8}$, meaning its exact value depends entirely on how many boys versus girls there are in the institute. ### Step-by-Step Solution * **Given:** Fraction of girls participating = $\frac{1}{5}$. Fraction of boys participating = $\frac{1}{8}$. * **Calculation:** Let the total number of girls be $G$ and the total number of boys be $B$. Number of participating girls = $\frac{1}{5}G$ Number of participating boys = $\frac{1}{8}B$ * Set up the expression for the total fraction of participating students: $$ \text{Total Fraction} = \frac{\text{Participating Girls} + \text{Participating Boys}}{\text{Total Girls} + \text{Total Boys}} $$ $$ \text{Total Fraction} = \frac{\frac{1}{5}G + \frac{1}{8}B}{G + B} $$ * Because we do not have an equation linking $G$ and $B$ (i.e., we don't know the ratio of boys to girls), we cannot simplify this expression into a single numerical fraction. The value will shift depending on the gender ratio of the school. ### Exam Strategy & Shortcut Recognize the **Missing Weight Condition** immediately. You have rates ($\frac{1}{5}$ and $\frac{1}{8}$) but no absolute values or relative weights (ratios) connecting the two different groups (Boys and Girls). Without knowing the ratio of Boys to Girls, it is mathematically impossible to find the combined average rate. Mark "Data inadequate" and move on within 5 seconds. ### Common Pitfall A standard trap is assuming there is an equal number of boys and girls ($G = B$), which would lead you to calculate a simple average: $\frac{\frac{1}{5} + \frac{1}{8}}{2} = \frac{13}{80}$, and then likely guess an incorrect option. Never assume equal weights unless explicitly stated. ### Final Answer Therefore, the correct answer is **Data inadequate**.