let x and y be the two positive number and x>y

one positive number is 42 less than another positive number

y = x - 42

the reciprocal of the smaller number is added to three times the reciprocal of the larger number the sum is 1/8

(1/y) + 3(1/x) = 1/8

plug y = x - 42 in the above equation

(1/(x - 42)) + 3(1/x) = 1/8

by solving the above equation we find two roots

x1 = 18

x2 = 56

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for x = 18 we have y = x - 42 = 18 - 42 = - 24 < 0 thus x = 18 is not the solution

for x = 56 we have y = 56 - 42 = 14 >0

the two numbers are 56 and 14.