Question

The sum of the reciprocals of two consecutive integers is 19/90. Find the two integers.

Answer 1

Two consecutive integers are x and x+1
Their reciprocals are 1/x and 1/(x+1)

1/x + 1/(x+1) = 19/90

Multiply both sides by 90x(x+1)

90(x+1) + 90x = 19x(x+1)
90x + 90 + 90x = 19x² + 19x
19x² + 19x - 180x - 90 = 0
19x² - 161x - 90 = 0
19x² - 171x + 10x - 90 = 0
19x (x - 9) + 10 (x - 9) = 0
(x - 9) (19x + 10) = 0

x = 9, x = -10/19
But -10/19 is not an integer

So the two integers are 9 and 10

Answer 2

Medal?