【Add】 , 【sum】 , 【total】 , 【tot】 , 【cast】 , 【figure】 and 【foot】 share the meaning to find or represent the amount reached by putting together arithmetically a series of numbers or quantities, and are commonly followed by up.

【Add】 is both the common and the technical word; it commonly implies strict adherence to the traditional arithmetical operation. Even in figurative use it implies a similar operation.

• taken as a whole the vignettes and the stories 【add】 up to a single effect
  —Aldridge
• the whole undertime trend adds up to a major consideration for businessmen and employees
  —Lack

【Sum】 stresses the result attained rather than the method followed. In figurative use 【sum】 up implies a gathering and consolidation into a new whole, especially for the production of a single telling effect.

• a lawyer in summing up summarizes in brief and logical form the evidence favorable to his case or client that has been given
• I summed up all the systems in a phrase and all existence in an epigram
  —Wilde
• values they can 【sum】 up in a few simple formulas
  —Croly

【Total】 tends to replace 【sum】 up in literal use.

• determined the cost by totaling all expenditures

It may also mean to reach the 【sum】 or number of.

• absences due to colds totaled 253 last week

【Tot】 , 【cast】 , 【figure】 , and 【foot】 are used especially of commercial matters (as accounts and bookkeeping devices).

【Tot】 and 【cast】 often imply facility in reckoning.

• the waiter quickly totted the bill
• if you 【tot】 up all the items that we owed against all the items that foreigners owed us
  —Hutton
• 【cast】 up an account

【Figure】 usually suggests the task or burden involved in reckoning.

• 【figure】 the costs of operating an automobile

【Foot】 connotes bookkeeping and totals at the bottom of each column of figures.

• his debts will 【foot】 up to more than he can ever pay