$1100$ boys and $700$ girls are examined in a test; $42\%$ of the boys and $30\%$ of the girls pass. The percentage of the total who failed is

### Concept & Logic To find the overall failure percentage, calculate the absolute number of boys who failed and girls who failed. Sum these values to get the total number of failures, and find what percentage this forms of the total combined student population. $$ \text{Total Failed \%} = \frac{\text{Failed Boys} + \text{Failed Girls}}{\text{Total Students}} \times 100 $$ ### Step-by-Step Solution * **Given:** * Total boys = $1100$, Total girls = $700$. (Total students = $1800$) * Pass percentage: Boys = $42\%$, Girls = $30\%$ * **Calculation:** * **Step 1: Find the failure percentages.** If $42\%$ of boys pass, then $(100 - 42)\% = 58\%$ fail. If $30\%$ of girls pass, then $(100 - 30)\% = 70\%$ fail. * **Step 2: Calculate the absolute number of failures.** Failed boys = $58\%$ of $1100 = 0.58 \times 1100 = 638$. Failed girls = $70\%$ of $700 = 0.70 \times 700 = 490$. * **Step 3: Calculate the total failure percentage.** Total failed students = $638 + 490 = 1128$. Failure Percentage = $\left( \frac{1128}{1800} \right) \times 100\%$ $= \frac{1128}{18}$ $= 62.66...\%$ which is $62\frac{2}{3}\%$. ### Exam Strategy & Shortcut Use the weighted average method with simplified population ratios. The ratio of boys to girls is $1100 : 700$, which simplifies to $11 : 7$. The failure rates are $58\%$ and $70\%$ respectively. Weighted Average = $\frac{(11 \times 58) + (7 \times 70)}{11 + 7}$ $= \frac{638 + 490}{18}$ $= \frac{1128}{18} = 62\frac{2}{3}\%$. This bypasses calculating the total large population figures. ### Common Pitfall Students often calculate the overall *pass* percentage first and forget to subtract it from $100\%$ at the end. Another trap is taking the simple average of the failure rates $\frac{58 + 70}{2} = 64\%$, completely ignoring the fact that there are more boys than girls, which skews the true average closer to the boys' failure rate. ### Final Answer **Therefore, the correct answer is $62\frac{2}{3}\%$.**