The number of people in an audthorium is between 80 and 100, knowimg that the number of men is 7/8 the number of women, what is the number of people?

Question

I would like some help solving that problem. My approach: I've shown that x is between 128/3 and 160/3 but that didn't help me and now I'm stuck. Thanks in advance

Answer 1

The total number of people is the number of women plus $7/8$ the number of women.

\begin{align*} m&=\dfrac{7w}{8}\\ \implies w+\dfrac{7w}{8}&=total\\ \implies8w+7m&=15\\ \implies16w+14m&=20\\ \implies24w+21m&=45\\ \implies32w+26m&=60\\ \implies40w+35m&=75\\ \implies48w+42m&=90\\ \end{align*}

It appears that your $\quad x=48=\dfrac{144}{3}$

Answer 2

Here's how I'd do it. $7+8=15$, so out of every $15$ people, seven are men, and eight are women. $90$ is six fifteens (five fifteens is only $75$ people, too few, and seven fifteens is $105$ people, too many), so $6\times7$ men, and $6\times8$ women.