In a school of 432 pupils, the ratio of the number of boys to the number of girls is 5 : 4. After some new admissions, the number of boys increases by 12 and the ratio of boys to girls becomes 7 : 6. How many new girls were admitted?

Introduction / Context:
This problem involves ratios and changes in a population after new admissions. You are given the initial ratio and total strength of students, then told how the ratio changes after some additional boys and girls are admitted. The task is to determine how many new girls joined. This blends ratio, basic algebra and understanding of how changes affect proportions.

Given Data / Assumptions:

• Initial total number of pupils = 432.
• Initial ratio of boys : girls = 5 : 4.
• After some admissions, number of boys increases by 12.
• New ratio of boys : girls becomes 7 : 6.
• The number of new girls admitted is to be found.

Concept / Approach:
First, use the initial ratio and total strength to find the original numbers of boys and girls. Then incorporate the changes: boys increase by a fixed number, and girls increase by some unknown x. Apply the new ratio to form an equation in x. Solving this equation gives the number of new girls admitted. This is a typical approach whenever a ratio changes after some additions or subtractions.

Step-by-Step Solution:
1) Let initial numbers be B boys and G girls with B : G = 5 : 4 and B + G = 432. 2) From the ratio, let B = 5k and G = 4k. 3) Then 5k + 4k = 432, so 9k = 432. 4) Solve for k: k = 432 / 9 = 48. 5) Therefore, B = 5 * 48 = 240 and G = 4 * 48 = 192 initially. 6) After admissions, the number of boys becomes B' = 240 + 12 = 252. 7) Let the number of new girls admitted be x. Then G' = 192 + x. 8) New ratio is B' : G' = 7 : 6, so 252 : (192 + x) = 7 : 6. 9) Cross-multiply: 252 * 6 = 7 * (192 + x). 10) This gives 1,512 = 1,344 + 7x. 11) So 7x = 1,512 - 1,344 = 168, hence x = 168 / 7 = 24.
Verification / Alternative check:
After admissions, the number of girls becomes 192 + 24 = 216. Now check the final ratio: B' : G' = 252 : 216. Divide both terms by 36 to simplify, we get 7 : 6, which matches the given new ratio. This confirms that taking 24 new girls is consistent with both the initial conditions and the final ratio after admissions.

Why Other Options Are Wrong:
If we try option A (12), the new number of girls would be 204 and the ratio 252 : 204 would simplify to 21 : 17, not 7 : 6. Similarly, option B (14), option D (20) and option E (18) lead to simplified ratios that do not match 7 : 6. Only when x = 24 does the ratio of boys to girls precisely become 7 : 6, so 24 is the only correct choice.

Common Pitfalls:
One common mistake is to apply the ratio to the increase instead of to the final numbers. Another common error is forgetting that the total number of pupils also changes, so using the old total of 432 after admissions is incorrect. Always recompute the new numbers and then apply the new ratio. Careful algebraic setup is the key to handling such multi-step ratio problems correctly.

Final Answer:
The number of new girls admitted to the school is 24.