In a group of $$48$$ children, the ratio of boys to girls is $$3:5$$. How many boys must join the group to make the ratio of boys to girls $$5:3$$? 
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Initially there are $$48$$ children of whom $$\frac{3}{8}$$ are boys and $$\frac{5}{8}$$ are girls, so there are $$18$$ boys and $$30$$ girls. When more boys join, there are still $$30$$ girls but now they form $$\frac{3}{8}$$ of the total. So the total number of pupils is now $$\frac{8}{3}\times30= 80$$, of whom $$80-30=50$$ are boys. Hence the number of boys joining is $$50-18=32$$. 