let

v = the rate of each chef in carrots per min

3 chefs, all working at the same speed, can grate 6 1/2 = 6.5 carrots in 2 1/2 = 2.5 min

the rate of the 3 chefs together is: 6.5/2.5 = 13/5 carrots per min

1/v + 1/v + 1/v = 13/5

by solving we find:

v = 15/13 carrots per min

click here to see the step by step solution of the equation:

5 chefs have a rate of 5*v = 5*(15/13) = 75/13 carrots per min

let t = the time needed to grate 20 carrots

75/13 = 20/t

by solving we find:

t = 52/15 min

t = 3 min (7/15)*60 sec

t = 3 min 18 sec

click here to see the step by step solution of the equation:

the 5 chefs will grate 20 carrots in 3 min 18 seconds time.