Directions: Study the following table carefully and answer the questions that follow: Number of Candidates (in lakhs) Appearing In an Entrance Examination From Six Different Cities | City | A | B | C | D | E | F | | :--- | :--- | :--- | :--- | :--- | :--- | :--- | | **Number of candidates** | 1.25 | 3.14 | 1.08 | 2.27 | 1.85 | 2.73 | Ratio of Candidates Passing And Failing Within the City | City | Ratio of Passing and Failing | | :--- | :--- | | **A** | 7 : 3 | | **B** | 5 : 3 | | **C** | 4 : 5 | | **D** | 1 : 3 | | **E** | 3 : 2 | | **F** | 7 : 5 | What is the ratio of the number of candidates failing the exam from City $D$ to that of those failing the exam from City $A$ ?

### Concept & Strategy To find the number of candidates failing from a specific city, multiply the total number of candidates appearing from that city by the fraction of students who failed. The failing fraction is $\frac{\text{Failing Ratio}}{\text{Total Ratio}}$. Both values are in lakhs, so we can ignore the unit when computing their ratio. ### Step-by-Step Solution 1. **Candidates failing from City D:** - Total candidates = $2.27$ lakhs - Passing to Failing Ratio = $1 : 3$ - Failing fraction = $\frac{3}{1 + 3} = \frac{3}{4}$ - Number of failing candidates = $2.27 \times \frac{3}{4}$ 2. **Candidates failing from City A:** - Total candidates = $1.25$ lakhs - Passing to Failing Ratio = $7 : 3$ - Failing fraction = $\frac{3}{7 + 3} = \frac{3}{10}$ - Number of failing candidates = $1.25 \times \frac{3}{10}$ 3. **Calculate the Ratio:** - Ratio = $\frac{2.27 \times \frac{3}{4}}{1.25 \times \frac{3}{10}}$ - The $3$ in the numerators cancels out: Ratio = $\frac{\frac{2.27}{4}}{\frac{1.25}{10}}$ - Simplify the denominator: $\frac{1.25}{10} = \frac{1}{8}$ - Rewrite the expression: $\frac{2.27}{4} \times 8$ - Simplify: $2.27 \times 2 = 4.54$ 4. **Convert to Ratio Format:** - $4.54$ can be written as $\frac{454}{100}$. - Divide both by $2$ to simplify: $\frac{227}{50}$. - The ratio is $227 : 50$. ### Exam Strategy & Shortcut When computing the ratio $\frac{2.27 \times \frac{3}{4}}{1.25 \times \frac{3}{10}}$, immediately cancel the common factor of $3$. You are left with $\frac{2.27}{4} \times \frac{10}{1.25}$. Notice that $\frac{10}{1.25}$ is exactly $8$. Then, $2.27 \times \frac{8}{4} = 2.27 \times 2 = 4.54$. Expressing $4.54$ as a ratio gives $454:100$, which reduces instantly to $227:50$. ### Common Pitfall A common error is confusing the failing ratio for the passing ratio or dividing by just the failing part instead of the total ratio parts (e.g., using $\frac{3}{1}$ instead of $\frac{3}{4}$ for City D). ### Final Answer Therefore, the correct answer is **227 : 50**.