M and N can finish a piece of work in $15$ and $20\text{ days}$ respectively. If they complete the entire work working together, find the ratio of their income shares.

When individuals work together for the same amount of time, their total earnings are divided in the direct ratio of their work efficiencies (work done per unit day).
Efficiency of M = $\frac{1}{15}$
Efficiency of N = $\frac{1}{20}$

Ratio of their daily efficiencies (and consequently, their incomes):

$$\text{Ratio}=\frac{1}{15}:\frac{1}{20}$$

Multiplying both terms by the LCM of $15$ and $20$ (which is $60$):

$$\text{Ratio}=\left(\frac{1}{15}\times 60\right):\left(\frac{1}{20}\times 60\right)=4:3$$