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Default Combining sumIF statements with other functions?

I am wondering can a 'sumif statement and vlookup formula' be combined to
provide a total of a range of numbers or maybe even combined with using
'greater than a number but less than another number' functions. For example,
I have a range of 5 digit numbers in column 'A', and have dollar amounts that
correspond to those 5 digit numbers in column 'B', in a seperate tab I need a
formula that would look in coulmn 'A' for a range of numbers (for ex..
greater than 10000 and less than 19999) then give me the total of the
corresponding dollar amounts in column 'B'. Is this possible or am I SOL????

 
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