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In an exhibition hall, the ratio of the number of females to the number of males was 5:6. The ratio of the number of girls to the number of women was 2:3. The ratio of the number of boys to the number of men was 3:4. If there were 60 more men than women, how many people were there altogether?

f = female, m = male, G = girls, W = women, B = Boy, M = men

$\frac{f}{m} = \frac56 \rightarrow \frac{G+W}{B+M}=\frac56 \\ \frac{G}{W} = \frac23 \rightarrow W=\frac32 G \\ \frac{B}{M} = \frac34 \rightarrow B=\frac34 M \rightarrow B= \frac34 (60+W) = \frac34 (60+\frac32 G) = 45 + \frac{9}{8} G\\ M=60+W \rightarrow M = 60 + \frac32 G$

$G+W = G+\frac32 G = \frac52 G\\ B+M = 45 + \frac98 G + 60 + \frac32 G = 105 + \frac{21}{8} G\\ \begin{array}{rcl} \hline \frac{G+W}{B+M}&=&\frac56\\ \frac{ \frac52 G}{105 + \frac{21}{8} G} &=& \frac56 \\ 15G &=& 5\cdot(105 + \frac{21}{8} G ) \\ 15G &=& 525 + 13.125 G\\ 1.875G &=& 525\\ G&=& \frac{525}{1.875} \\ \mathbf{G}& \mathbf{=} & \mathbf{280}\\ \end{array}$

$W=\frac32 G =\frac32 \cdot 280 \\ \mathbf{W} \mathbf{=} \mathbf{420}\\ $

M = 60 + W

M = 60 + 420

M = 480

$B=\frac34 M \\ B=\frac34 480\\ \mathbf{B=360}$

280 + 420 + 480 + 360 = 1540 people were there altogether