let 'x', 'y' and 'z' be the numbers and let x < y < z

the sum of three numbers is 81

x + y + z = 81

the largest of the numbers is one more than ten times the smallest number

z = 1 + 10x

the other number is 5 times the smallest number

y = 5x

lets plug y = 5x and z = 1 + 10x into x + y + z = 81 we get

x + 5x + (1 + 10x) = 81

by solving we find

x = 5

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y = 5*5 = 25

z = 1 + 10*5 = 51

the three numbers are 5, 25 and 51.