In a party there are both male and female members initially. If 15 female members leave the party, then the remaining number of females becomes exactly double the number of males present. However, if instead 45 male members leave the party, then the number of females becomes five times the number of males. Based on this information, find the original number of female members who were present at the party.

Introduction / Context:
This aptitude question tests linear equation formation and solving with two variables. The situation describes a party with male and female members and gives two different conditions about members leaving the party. From these conditions we have to form equations in terms of the original number of males and females and then determine the initial female population. Such questions are common in bank exams, SSC, and other competitive tests that assess algebraic reasoning.

Given Data / Assumptions:

• Let the original number of female members be F.
• Let the original number of male members be M.
• If 15 females leave, the remaining females become twice the number of males.
• If instead 45 males leave, the remaining males become one fifth of the number of females, that is the females are five times the males.

Concept / Approach:
The approach is to convert each verbal condition into a linear equation in F and M. With two independent equations and two unknowns, we can solve the system using substitution or elimination. Once M is eliminated, we obtain a direct value for F. Finally, we check the solution in both original conditions to ensure consistency.

Step-by-Step Solution:
From the first condition: F - 15 = 2M.From the second condition: F = 5(M - 45), because females are five times the new number of males.Expand the second equation: F = 5M - 225.From the first equation, express F as F = 2M + 15.Equate both expressions for F: 2M + 15 = 5M - 225.Rearrange: 15 + 225 = 5M - 2M, so 240 = 3M.Therefore M = 240 / 3 = 80.Substitute M = 80 into F = 2M + 15 to get F = 2 * 80 + 15 = 175.

Verification / Alternative check:
Check the first condition: if F = 175, then after 15 females leave, females = 160 and males are still 80, so females are 2 times males, which is correct. Check the second condition: if 45 males leave, remaining males = 80 - 45 = 35. Females remain 175, and 175 is 5 times 35, so the second condition is also satisfied. Thus the solution is consistent and correct.

Why Other Options Are Wrong:
145 gives F - 15 = 130 and cannot be twice an integer M that fits the second condition. 165 does not satisfy the second situation when 45 males leave. 135 leads to smaller numbers that fail both conditions when checked. 155 also breaks at least one of the two ratio conditions, so these values cannot be correct.

Common Pitfalls:
Candidates sometimes misread the statements and reverse the ratios, for example taking males as double females. Another common error is to change the numbers when people leave instead of changing only the respective group. Arithmetic slips while solving 2M + 15 = 5M - 225 can also lead to an incorrect M, so careful algebraic manipulation is important. Writing clear equations before solving prevents these mistakes.

Final Answer:
The original number of female members at the party is 175.