Dance ensembles

4 dance ensembles were dancing at the festival. None had less than 10 and more than 20 members. All dancers from some of the two ensembles were represented in each dance. First, 31 participants were on the stage, then 32, 34, 35, 37, and 38. How many dancers did the individual ensembles have?

Correct result:

a =  14
b =  17
c =  18
d =  20

Solution:

$$\begin{gathered} 10<a,b,c,d<20 \\ a=14 \\ b=17 \\ c=18 \end{gathered}$$

$$\begin{gathered} d=20 \\ \text{ Verifying Solution: } \\ {s}_1=a+b=14+17=31 \\ {s}_2=a+c=14+18=32 \\ {s}_3=a+d=14+20=34 \\ {s}_4=b+c=17+18=35 \\ {s}_5=b+d=17+20=37 \\ {s}_6=c+d=18+20=38 \end{gathered}$$

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