Difficulty: Easy
Correct Answer: 416
Explanation:
Introduction / Context:
This problem tests the ability to apply a simple ratio to find total numbers when one category count is known. It is a direct and common use of ratios in everyday contexts such as class composition and population data.
Given Data / Assumptions:
Concept / Approach:
If boys : girls = 8 : 5, we can represent the numbers as 8k and 5k for some positive factor k. The actual count of girls is 5k, which is given as 160. Solving for k allows us to compute the total number of students as 8k + 5k.
Step-by-Step Solution:
Step 1: Let number of boys = 8k and number of girls = 5k.
Step 2: It is given that the number of girls is 160, so 5k = 160.
Step 3: Solve for k by dividing 160 by 5: k = 160 / 5 = 32.
Step 4: Now find the number of boys: boys = 8k = 8 * 32 = 256.
Step 5: Total number of students = boys + girls = 256 + 160.
Step 6: Total = 416.
Verification / Alternative check:
We can verify by checking the ratio of 256 to 160. Compute 256 / 160 = 1.6 which is the same as 8 / 5 (since 8 / 5 = 1.6). This confirms that the boy to girl ratio remains 8 : 5 and the total 416 is consistent with the given data.
Why Other Options Are Wrong:
Totals like 100, 250, 260 or 520 do not yield 160 girls when the ratio 8 : 5 is applied. For example, if total is 260, the girls count would be 5 / 13 * 260 = 100, which contradicts the condition of 160 girls. Thus these options are inconsistent with the given ratio and the provided count of girls.
Common Pitfalls:
Learners sometimes treat 8 and 5 as actual counts rather than as parts of a ratio. Another mistake is to divide 160 by 8 instead of 5, or to add 8 and 5 incorrectly. Remember that the known count corresponds to its ratio part (5k in this case), and you must solve for k before calculating the total.
Final Answer:
The total number of students in the college is 416.
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