SOLUTION: Ages of three persons are now in the proportion of 2:3:4 and in five years from now, the proportion will be 5:7:8. What is the present age of the youngest person??

The 8 should have been a 9.  Then there would have been a solution.
But with an 8 there, there is no solution.  I'll show you first,
why with the 8 there, there is no solution, and also if we change the
8 to 9 there will be a solution.

First let's try for a solution with the 8 there.

Since 2, 3, 4, have no common factor, the only ages
they could be are multiples of these, or

2x, 3x, 4x,  where n is a positive integer

Similarly, 

Since 5, 7, and 8 have no common factor, the only ages
they could be in 5 years are multiples of these, or

5y, 7y, 8y, where y is a positive integer.

So in 5 years we must have 

2x+5=5y, 3x+5=7y, 4x+5=8y or

For there to be a solution to the problem these 3 equations must
have a common integer solution, however they do not!

The first two equations have a common integer solution (x,y) = (10,5)
However the first and third do not! Their only common solution is 
(3.75,2.5), which not only is not (10,5), it is not an integer 
solution. Also the second and third do not either! Their only common
solution is (1.25,1.25).

Since the first two have a common integer solution, the culprit has to
be the 3rd equation!
----------------------------------------

If we change the 8 to 9, there is a solution.  So let's start all over 
and pretend your problem is:

Since 2, 3, 4, have no common factor, the only ages
they could be are multiples of these, or

2x, 3x, 4x,  where n is a positive integer

Similarly, 

Since 5, 7, and 9 have no common factor, the only ages
they could be in 5 years are multiples of these, or

5y, 7y, 9y, where y is a positive integer.

So in 5 years we must have 

2x+5=5y, 3x+5=7y, 4x+5=9y or

For there to be a solution to the problem the 3 equations must
have a common integer solution.

The first two equations have a common integer solution (x,y) = (10,5)
The first and second equations NOW have a common integer solution (x,y) = (10,5)
The second and third equations NOW have a common integer solution (x,y) = (10,5)

So x=10 and y=5
 
So their ages now are 2x, 3x, and 4x or 20, 30, and 40

So the present age of the youngest is 20.

Now let's check:

Certainly 20,30 and 40 are in the ratio 2:3:4,
since dividing them by 10 gives that ratio.

In 5 years they will be 25, 35, and 45.
This also checks with 5y, 7y, 9y.  And
certainly 25,35 and 45 are in the ratio 5:7:9,
since dividing them by 5 gives that ratio.

So apparently your 8 was a typo and should have been 9.

Edwin