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The two consecutive numbers can be called x, x+1
Reciprocal of x is 1/x
The reciprocal of x+1 is 1/(x+1)
So, 1/x + 1/(x+1) = 1/12

Lets get a common denominator on the left side of x(x+1) [(x+1) - x] / [x(X+1)] = 1/12
1/(x^2+x) = 1/12
Lets cross multiply
x^2+x = 12
x^2+x-12 = 0

So, lets factorize the expression
(x+4)(x-3)= 0 

Therefore, x=-4 or 3
So x+1 = -3 or 4

Therefore, the two consecutive numbers are -4,-3 OR 3,4

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