smc

Let $A$ do $r_A$ of the job per day, $B$ do $r_B$ of the job per day, and $C$ do $r_C$ of the job per day. These three quantities have unit $\frac{\text{job}}{\text{day}}$. Therefore our three conditions give us the three equations: $$\begin{array}{rl}(2\text{ days})(r_A+r_B)& =1\text{ job},\\ (4\text{ days})(r_B+r_C)& =1\text{ job},\\ (2.4\text{ days})(r_C+r_A)& =1\text{ job}.\end{array}$$ We divide the three equations by the required constant so that the coefficients of the variables become 1: $$\begin{array}{rl}{r}_A+r_B& =\frac{1}{2}\cdot \frac{\text{job}}{\text{day}},\\ r_B+r_C& =\frac{1}{4}\cdot \frac{\text{job}}{\text{day}},\\ r_C+r_A& =\frac{5}{12}\cdot \frac{\text{job}}{\text{day}}.\end{array}$$ If we add these three new equations together and divide the result by two, we obtain an equation with left-hand side $r_A+r_B+r_C$, so if we subtract $r_B+r_C$ (the value of which we know) from both equations, we obtain the value of $r_A$, which is what we wish to determine anyways. So we add these three equations and divide by two: $$r_A+r_B+r_C=\frac{1}{2}\cdot \left(\frac{1}{2}+\frac{1}{4}+\frac{5}{12}\right)\cdot \frac{\text{job}}{\text{day}}=\frac{7}{12}\cdot \frac{\text{job}}{\text{day}}.$$ Hence: $$\begin{array}{rl}{r}_A& =(r_A+r_B+r_C)-(r_B+r_C)\\ & =\frac{7}{12}\cdot \frac{\text{job}}{\text{day}}-\frac{1}{4}\cdot \frac{\text{job}}{\text{day}}\\ & =\frac{1}{3}\cdot \frac{\text{job}}{\text{day}}.\end{array}$$ This shows that $A$ does one third of the job per day. Therefore, if $A$ were to do the entire job himself, he would require $\boxed{3}$ days.