Find the ages of the three Mabvuto is half as old again as Matthew, Matthew is half as old again as Ngilazi. Given that the sum of their ages is 114, find their ages. My work just messes up, I need someone to help me!
The key word is "again."  In English parlance, to say that Mabvuto is "half again as old" as Matthew means that Mabvuto is Matthew's age plus half Matthew's age.  That is, if we let $Mb$ denote Mabvuto's age and $Ma$ Matthew's, then $Mb = Ma + {1\over2}Ma= {3\over2}Ma$.  Likewise, the statement that Matthew is half again as old as Ngilazi implies $Ma = {3\over2}Ng$, where $Ng$ denotes Ngilazi's age.  Thus

$$114=Mb+Ma+Ng = {3\over2}Ma+{3\over2}Ng+Ng = {3\over2}\cdot{3\over2}Ng+{3\over2}Ng+Ng = {19\over4}Ng,$$

from which ones gets $Ng=24$, $Ma=36$, and $Mb=54$.
Denote the respective ages as $x,y,z$. Then you are given 
$$ x=\frac12y$$
$$ y=\frac12z$$
$$x+y+z=114$$
You should be able to solve this.