if the ratio of kids who know how to swim is 6:10, that means that 60% of the schools population knows how to swim. So you can multiply 350 by 0.60 to get 210 kids.
You can show why the calculation is true since 6/10 = x/350. We can say the two ratios are equal since 6/10 ratio holds true for every group of 10 that leading up to a total number of 350. (you can have 35 groups of 10 each one having a 6 to 10 ratio. (6/10)x(35/35)=210/350