SOLUTION: one integer is 5 less than the another. The sum of twice the reciprocal of the lesser integer and thrice the reciprocal of the larger is 5/6. What are the two integers?

Question 1190758: one integer is 5 less than the another. The sum of twice the reciprocal of the lesser integer and thrice the reciprocal of the larger is 5/6. What are the two integers?
Found 2 solutions by Theo, greenestamps:
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
let y = the smaller number.
let x = the larger number.
you get:
2/y + 3/x = 5/6
multiply both sides by 6 to get:
12/y + 18/x = 5
multiply both dised by xy to get:
12x + 18y = 5xy
you are given that the smaller number is 5 less than the larger number.
this gets you:
y = x - 5
replace y with that in the equation of 12x + 18y = 5xy to get:
12x + 18(x-5) = 5x(x-5)
simplify to get:
12x + 18x - 90 = 5x^2 - 25x
combine like terms to get:
30x - 90 = 5x^2 - 25x
subtract 30x and add 90 to both sides to get:
0 = 5x^2 - 25x - 30x + 90
combine like terms to get:
0 = 5x^2 - 55x + 90
divide both sides by 5 to get:
0 = x^2 - 11x + 18
factor the quadratic on the right to get:
0 = (x-9) * (x-2)
solve for x to get:
x = 9 or x = 2
when x = 2, y = x - 5 = -3
when x = 9, y = x - 5 = 4
replace these values in the original equation of 2/y + 3/x = 5/6 to get:
2/-3 + 3/2 = 5/6
2/4 + 3/9 = 5/6
2/-3 + 3/2 becomes -2/3 + 3/2 which becomes -4/6 + 9/6 which becomes 5/6.
2/4 + 3/9 becomes 18/36 + 12/36 which becomes 30/36 which becomes 5/6.
this confirms the solution is correct.
your solution is:
the two integers are either 2 and -3 or 9 and 4.

Answer by greenestamps(13200) (Show Source): You can put this solution on YOUR website!

x = smaller integer
x+5 = larger integer

The sum of twice the reciprocal of the lesser integer and thrice the reciprocal of the larger is 5/6:

Multiply everything by the least common denominator 6(x)(x+5) to clear fractions:

or

Both solutions satisfy the conditions of the original problem:

ANSWERS: 4 and 9, or -2 and 3

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