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2015-10-01T09:40:29+08:00

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Let first number  be x ; its reciprocal: 1/x
Second number x + 1; its reciprocal:   1   
                                                      x + 1

Equation:

1/x + 1/(x+1) = 5/6

   1(x)(x+1)(6)   + 1 (x) (x+1)(6) = 5 (x) (x+1) (6)
        x                     x + 1                 6

6 (x + 1) + 6 (x) = 5x (x+1)

6x + 6 + 6x = 5x² + 5x

Equate to zero:
5x² - 6x - 6x + 5x - 6 = 0

5x² - 7x - 6 = 0

(x + 3/5) = 0           (x - 2) = 0

x = -3/5         x = 2

Choose 2 because we're looking for two consecutive positive integers.

The numbers are:
first number: x = 2
second number: x + 1 = 2 + 1 = 3

The two consecutive positive numbers are 2 and 3.

To check:

1/2 + 1/3 = 5/6

3 + 2 = 5/6
   6

5/6 = 5/6

NOTE:  Keep practicing.
1 5 1