Directions: Each of the questions below consists of a statement and/or a question and two statements labelled I and II given below it. You have to decide whether the data provided in the statements is/are sufficient to answer the question. Read both the statements and Give answer (a) if the data in Statement I alone are sufficient to answer the question, while the data in Statement II alone are not sufficient to answer the question; Give answer (b) if the data in Statement II alone are sufficient to answer the question, while the data in Statement I alone are not sufficient to answer the question; Give answer (c) if the data either in Statement I or in Statement II alone are sufficient to answer the question; Give answer (d) if the data even in both Statements I and II together are not sufficient to answer the question; and Give answer (e) if the data in both Statements I and II are necessary to answer the question. How many boys are there in the class? (M.A.T., 2007) I. The class has 45 children in all and ratio of boys to girls is $4 : 5$. II. The ratio of girls to boys is $4 : 5$ and boys are nine more than the girls.

### Concept & Proportional Division To find an exact quantity from a ratio, you need the ratio itself and one of the following absolute values: the total sum, the difference between the quantities, or the exact value of one of the quantities. ### Step-by-Step Solution * **Analyzing Statement I:** * Given: Total children = 45. Ratio of boys : girls = 4 : 5. * The total number of ratio parts = 4 + 5 = 9. * Number of boys = (4 / 9) * 45 = 20. * Statement I gives a definitive answer and is sufficient. * **Analyzing Statement II:** * Given: Ratio of girls : boys = 4 : 5. Difference (Boys - Girls) = 9. * Let girls = 4x and boys = 5x. * The difference is 5x - 4x = 1x. * We are told the difference is 9, so 1x = 9. * Number of boys = 5x = 5 * 9 = 45. * Statement II also gives a definitive answer and is sufficient. ### Exam Strategy & Shortcut Recognize the structural sufficiency of the data. Statement I gives a "Ratio + Sum" which is always sufficient. Statement II gives a "Ratio + Difference" which is also always sufficient. You do not actually need to calculate the final values (20 and 45) during the exam. ### Common Pitfall A common mistake is realizing that the two statements yield different numerical answers (20 vs 45) and assuming this means the data is inconsistent or insufficient. In Data Sufficiency, statements are evaluated purely on their independent ability to answer the prompt. They do not have to agree with each other unless you are forced to combine them (Option E). ### Final Answer Therefore, the correct answer is **if the data either in Statement I or in Statement II alone are sufficient to answer the question**.