SOLUTION: The sum of the reciprocals of two consecutive positive numbers is 17/12. Find the numbers.

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Question 1034425: The sum of the reciprocals of two consecutive positive numbers is 17/12. Find the numbers.
Found 3 solutions by stanbon, Theo, ikleyn:
Answer by stanbon(75887)   (Show Source):
You can put this solution on YOUR website!
The sum of the reciprocals of two consecutive positive numbers is 17/12. Find the numbers.
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Integers:: x ; x+1
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Reciprocals:: 1/x ; 1/(x+1)
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Equation:
1/x + 1/(x+1) = 17/12
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12(x+1) + 12x = 17x(x+1)
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24x + 12 = 17x^2 + 17x
17x^2 - 7x - 12 = 0
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x =1.07... (That is not an integer; please check your post)
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Cheers,
Stan H.
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Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website!
let the first positive number equal x.
then the second positive number equals x+1.

the sum of the reciprocals of those numbers is equal to 17/12.

you get 1/x + 1/(x+1) = 17/12.

multiply both sides of this equation by x * (x+1) to get:

(x+1) + x = 17/12 * (x+1) * x

simplify to get 2x + 1 = 17/12 * (x^2 + x)

multiply both sides of this equaiton by 12 to get:

12 * (2x + 1) = 17 * (x^2 + x)

simplify to get 24x + 12 = 17x^2 + 17x

subtract the left side of the equation from both sides of the equaiton to get:

0 = 17x^2 + 17x - 24x - 12.

simplify to get 0 = 17x^2 - 7x - 12.

this is the same as 17x^2 - 7x - 12 = 0.

solve this quadratic equation using the quadratic formula to get:

x = (7 + sqrt(865))/34 or x = (7 - sqrt(865))/34

this results in x = 1.070908304 or x = -.6591435982.

x can't be negative, so x has to be equal to 1.070908304.

this makes x + 1 = 2.070908304.

if you did this correctly, you will get 1/1.070908304 + 1/2.070908304 = 17/12.

evaluate the left side of this equation to get 1.416666667 = 17/12.

multiply both sides of this equation by 12 to get 17 = 17.

this confirms the solution is correct.

your positive consecutive numbers are 1.070908304 and 2.070908304.

these are the same as (7+sqrt(865))/34 and 1 + (7 + sqrt(865))/34.

1 + (7 + sqrt(865))/34 can also be written as (41+sqrt(865))/34.

you can solve the quadratic equation graphically as well.

i used the quadratic formula to get an exact answer.
the graphing software i used gave me an approximate answer.
that graph is shown below:

Answer by ikleyn(52776)   (Show Source):
You can put this solution on YOUR website!
.
The formulation of the problem is INCORRECT since the solutions are not integer numbers.