Question

if the reciprocal of the smaller of two consecutive integers is subtracted from three times the reciprocal of the larger, the result is 3/10. find
the integers

Answer 1

The question led me to a quadratic equation: let 1st nos be n,and 2nd be n+1 3/n+1 - 1/n=3/10,collect lcm:

Answer 2

3n-1(n+1)/n(n+1)=3/10. 10(2n-1)=3(n*n)+3n,20n-10=3(n*n)+3n,take like terms,find n,choose the integer as n's value.

Answer 3

smaller number --- x
next consecutive number ---- x+1

3(1/(x+1)) - 1/x = 3/10
multiply each term by 10x(x+1), the LCM

30x - 10(x+1) = 3x(x+1)
30x - 10x - 10 = 3x^2 + 3x
3x^2 - 17x + 10 = 0
(x - 5)(3x - 2) = 0
x = 5 or x = 2/3
but our number was to be an integer, so
x = 5

the integers are 5 and 6

check:
3(1/6) - 1/5
= 1/2 - 1/5
= 3/10
My answer is correct