First situation

A and E work together for ten days

A and E’s one day work $(= \frac{1}{{50}} + \;\frac{1}{{75}} = \frac{1}{{30}})$

Work done $(= 10 \times \frac{1}{{30}} = \frac{1}{3})$

Work left $(= 1 - \frac{1}{3} = \frac{2}{3})$

Then C joins, A, C and E work together for five days

A, C and E’s one day work $(= \frac{1}{{50}} + \frac{1}{{80}} + \frac{1}{{75}} = \frac{{11}}{{240}})$

Work done $(= 5 \times \frac{{11}}{{240}} = \frac{{11}}{{48}})$

Work left $(= \frac{2}{3}\;-\;\frac{{11}}{{48}} = \frac{7}{{16}})$

After that A and C work together 

A and C’s one day work $(= \frac{1}{{50}} + \frac{1}{{80}} = \frac{{13}}{{400}})$

Work left = $(\frac{7}{{16}})$

∴ Days taken to complete the REMAINING work $(= \frac{7}{{16}} \times \frac{{400}}{{13}} = \frac{{175}}{{13}})$

Total days = $(10 + 5 + \frac{{175}}{{13}} = 28\frac{6}{{13}})$ days

Second situation

B and D work together for SIX days

B and D’s one day work $(= \frac{1}{{60}} + \frac{1}{{150}} = \frac{7}{{300}})$

Work done $(= 6 \times \frac{7}{{300}} = \frac{7}{{50}})$

∴ Work left $(= 1 - \frac{7}{{50}} = \frac{{43}}{{50}})$

Then A and E joins and B LEAVES, means A, D and E work together

A, D and E’s one day work $(= \frac{1}{{50}} + \frac{1}{{150}} + \frac{1}{{75}} = \frac{1}{{25}})$

Work done $(= 2 \times \frac{1}{{25}} = \frac{2}{{25}})$

Work left $(= \frac{{43}}{{50}} - \frac{2}{{25}} = \frac{{39}}{{50}})$

 After TWO days A leaves, now D and E work

D and E’s one day work $(= \frac{1}{{150}} + \frac{1}{{75}} = \frac{1}{{50}})$

Work left = $(\frac{{39}}{{50}})$

∴ Days taken to complete the remaining work $(= \frac{{39}}{{50}} \times \frac{{50}}{1} = 39)$

Total days = 6 + 2 + 39 = 47