Question 1204922: A positive integer is 5 less than another. If the reciprocal of the smaller
integer is subtracted from 3 times the reciprocal of the larger,
, then the
result is 1/12. Find the two integers
Found 2 solutions by josgarithmetic, Theo:
Answer by josgarithmetic(39617)   (Show Source):
Answer by Theo(13342)  (Show Source):
You can put this solution on YOUR website! let x equal the larger number
let y equal the smaller number
your equation is 3/x - 1/y = 1/12
that equation states that 3 times the reciprocal of the larger number minus the reciprocal of the smaller number is equal to 1/12.
another way of saying that is that the reciprocal of the smaller number is subtracted from 3 time the reciprocal of the larger number.
you start with 3/x - 1/y = 1/12
multiply both sides of the equation by xy to get:
3y - x = xy/12
multiply both sides of the equation by 12 to get:
12 * (3y - x) = xy
simplify to get:
36y - 12x = xy
you are given that y = x-5, so replace y with x-5 in the equation to get:
36 * (x-5) - 12x = x * (x-5)
simplify to get:
36x - 180 - 12x = x^2 - 5x
combine like terms to get:
24x - 180 = x^2 - 5x
subtract 24x from both sides of the equation and add 180 to both sides of the equation to get:
0 = x^2 - 29x + 180
factor this quadratic equation to get:
(x - 20) * (x - 9) = 0
solve for x to get:
x = 20 or x = 9
when x = 20, y = 15
when x = 9, y = 4
your initial equation was 3/x - 1/y = 1/12
when x = 20 and y = 15, the equation becomes 3/20 - 1/15 = 1/12
multiply both sides of this equation by 300 to get 45 - 20 = 25
this results in 25 = 25, confirming the equation is true.
when x = 9 and y = 4, the equation becomes 3/9 - 1/4 = 1/12
multiply both sides of this equation by 36 to get 12 - 9 = 36/12
this results in 3 = 3, confirming the equation is true.
your value pairs of x = 20 and y = 15 or x = 9 and y = 4 both satisfy the requirements of the problem.
you actually have two answers.
the first answer is your integers are 20 and 15.
the second answer is your integers are 9 and 4.
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