Hi,

The total number of people aged greater than or equal to 51 $$= 30$$
The total number of people aged less than 51 $$\le 39$$, let this be $$x$$

The total number of people are thus $$30+x$$ and their average age is $$38$$, hence the sum of their ages will be $$1140$$.


If we want to maximise the number of people under the age of 51, we should maximise the sum of their ages, and given a fixed total, this will happen when the sum of ages of all people older than or of 51 years of age is minimum, which in turn will happen when all the people who are not older than 51 years are actually exactly 51 years old.

Thus, sum of ages of all people not less than 51 years of age $$= 51\times 30= 1530$$.

Hence the sum of ages of all individuals less than 51 years of age $$= 38(30+x)-1530= 38x-390$$.


Thus the average age of all individuals less than 51 years of age will be;


$$\frac{38x-390}{x} = 38 - \frac{390}{x}$$


Since $$x= 39$$ is the maximum possible value, the average age will be $$38-10=28$$.

If you notice the formula just above, the more the number of people will be, the lesser the quantity $$\frac{390}{x}$$ will be and then in turn, the larger the quantity $$38-\frac{390}{x}$$ will be, which is average age, so we took $$x=39$$.

Hope this helps.