In a housing colony the number of adults and the number of children are in the ratio 7 : 3. The mean age of the adults is 40 years and the mean age of the children is 10 years. What is the mean age of all the residents?

Take the group sizes as 7$k$ and 3$k$. The total age is $7k(40)+3k(10)=280k+30k=310k$ years spread over 10$k$ residents, so the mean age is 31 years — and $k$ cancels, so the actual head counts are never needed. The plain average $(40+10)/2=25$ ignores that adults outnumber children; 19 comes from attaching each mean to the wrong group, $(7\times 10+3\times 40)/10$; and 40 is the adults' own mean, which the children must pull down. A quick sanity check: the answer has to sit nearer to 40 than to 10.