Men and women ratio changes after people leave After 15 women leave a group, there are 2 men for each woman. Then 45 men leave and the ratio becomes 5 women for each man. Find twice the initial number of women.

Introduction / Context:
Ratio problems with people entering or leaving convert naturally into linear equations. By expressing final ratios after changes, you can solve for initial counts of men and women, then compute the requested derived quantity.

Given Data / Assumptions:

• Initial men = m; initial women = w.
• After 15 women leave: women = w − 15; men = m; ratio men : women = 2 : 1 ⇒ m = 2(w − 15).
• Then 45 men leave: men = m − 45; women unchanged at w − 15; ratio women : men = 5 : 1 ⇒ w − 15 = 5(m − 45).
• All counts are nonnegative integers with w ≥ 15 and m ≥ 45 (consistent with leaving).

Concept / Approach:
Use the two ratio equations to eliminate m and solve for w. Then compute the requested double (2w). Keep algebra tidy to avoid sign mistakes.

Step-by-Step Solution:
From first ratio: m = 2(w − 15) = 2w − 30.From second ratio: w − 15 = 5(m − 45).Substitute m: w − 15 = 5((2w − 30) − 45) = 5(2w − 75) = 10w − 375.Rearrange: w − 15 = 10w − 375 ⇒ −9w = −360 ⇒ w = 40.Therefore twice the initial number of women = 2w = 80.

Verification / Alternative check:
If w = 40, then m = 2(40 − 15) = 50. After the first change: women = 25, men = 50 (ratio 2:1). After 45 men leave: men = 5, women = 25 (ratio 5:1). Everything matches perfectly.

Why Other Options Are Wrong:
25, 23, 35: These do not equal 2w with w = 40 and would fail the ratio checks when back-substituted.

Common Pitfalls:
Mixing the order of changes; flipping the ratios (men:women vs women:men); or missing that women remain the same in the second step. Carefully encode each stage before solving.

Final Answer:
80