SOLUTION: Find two consecutive odd integers such that the sum of their reciprocals is 12/35.I have been trying for three days to write a problem out for this question and I am for some reaso

Algebra ->  Problems-with-consecutive-odd-even-integers -> SOLUTION: Find two consecutive odd integers such that the sum of their reciprocals is 12/35.I have been trying for three days to write a problem out for this question and I am for some reaso      Log On


   

Question 294753: Find two consecutive odd integers such that the sum of their reciprocals is 12/35.I have been trying for three days to write a problem out for this question and I am for some reason not able to come up with an answer. Thank you
Found 2 solutions by richwmiller, jim_thompson5910:
Answer by richwmiller(17219) About Me  (Show Source):
You can put this solution on YOUR website!
So where is your work?
Let x be first odd integer
x+2 be the second odd integer
What do we know?
The reciprocals add up to 12/35
1/x+1/(x+2)=12/35
We have one equation with one unknown
We get two answers
-7/6 which isn't an odd number and 5
check
1/5+1/7=12/35
7/35+5/35=12/35
ok

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Consecutive odd integers follow the pattern: x, x+2, ...


So adding their reciprocals to get 12%2F35 means that 1%2Fx%2B1%2F%28x%2B2%29=12%2F35


1%2Fx%2B1%2F%28x%2B2%29=12%2F35 Start with the given equation


35%28x%2B2%29%2B35x=12x%28x%2B2%29 Multiply EVERY term by the LCD 35x%28x%2B2%29 to clear out the fractions.


35x%2B70%2B35x=12x%5E2%2B24x Distribute



70x%2B70=12x%5E2%2B24x Combine like terms.


0=12x%5E2%2B24x-70x-70 Get every term to one side.


0=12x%5E2-46x-70 Combine like terms.


Solved by pluggable solver: Quadratic Formula
Let's use the quadratic formula to solve for x:


Starting with the general quadratic


ax%5E2%2Bbx%2Bc=0


the general solution using the quadratic equation is:


x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29




So lets solve 12%2Ax%5E2-46%2Ax-70=0 ( notice a=12, b=-46, and c=-70)





x+=+%28--46+%2B-+sqrt%28+%28-46%29%5E2-4%2A12%2A-70+%29%29%2F%282%2A12%29 Plug in a=12, b=-46, and c=-70




x+=+%2846+%2B-+sqrt%28+%28-46%29%5E2-4%2A12%2A-70+%29%29%2F%282%2A12%29 Negate -46 to get 46




x+=+%2846+%2B-+sqrt%28+2116-4%2A12%2A-70+%29%29%2F%282%2A12%29 Square -46 to get 2116 (note: remember when you square -46, you must square the negative as well. This is because %28-46%29%5E2=-46%2A-46=2116.)




x+=+%2846+%2B-+sqrt%28+2116%2B3360+%29%29%2F%282%2A12%29 Multiply -4%2A-70%2A12 to get 3360




x+=+%2846+%2B-+sqrt%28+5476+%29%29%2F%282%2A12%29 Combine like terms in the radicand (everything under the square root)




x+=+%2846+%2B-+74%29%2F%282%2A12%29 Simplify the square root (note: If you need help with simplifying the square root, check out this solver)




x+=+%2846+%2B-+74%29%2F24 Multiply 2 and 12 to get 24


So now the expression breaks down into two parts


x+=+%2846+%2B+74%29%2F24 or x+=+%2846+-+74%29%2F24


Lets look at the first part:


x=%2846+%2B+74%29%2F24


x=120%2F24 Add the terms in the numerator

x=5 Divide


So one answer is

x=5




Now lets look at the second part:


x=%2846+-+74%29%2F24


x=-28%2F24 Subtract the terms in the numerator

x=-7%2F6 Divide


So another answer is

x=-7%2F6


So our solutions are:

x=5 or x=-7%2F6





Since we're only looking for integer values of 'x', we can ignore x=-7%2F6.


So the only answer is x=5 which means that the two consecutive odd integers are 5 and 7.


Notice how 1%2F5%2B1%2F7=7%2F35%2B5%2F35=%287%2B5%29%2F35=12%2F35 which confirms our answer.