There were 400 adults attending a concert in a stadium. During the break, $\frac{1}{3}$ of the men and $\frac{3}{5}$ of the women left the stadium. 224 adults remained in the stadium.
How many women were there in the stadium at first? Answer : ___

1. Express the remaining adults in terms of fractions of men and women.

Fraction of men remaining = $1 - \frac{1}{3} = \frac{2}{3}$
Fraction of women remaining = $1 - \frac{3}{5} = \frac{2}{5}$
Let $1$ unit represent $\frac{1}{3}$ of the men, and $1$ part represent $\frac{1}{5}$ of the women.
$2$ units of men + $2$ parts of women = $224$

2. Find the value of 1 unit of men plus 1 part of women.

$1$ unit of men + $1$ part of women = $224 \div 2 = 112$

3. Scale up to match the total number of men.

Multiply by $3$ to match the total number of men ($3$ units):
$3$ units of men + $3$ parts of women = $112 \times 3 = 336$

4. Compare with the total adults at first to find the number of women.

Total adults at first = $3$ units of men + $5$ parts of women = $400$
Difference = ($3$ units of men + $5$ parts of women) - ($3$ units of men + $3$ parts of women)
$2$ parts of women = $400 - 336 = 64$
$1$ part of women = $64 \div 2 = 32$
Total women at first ($5$ parts) = $32 \times 5 = 160$