Find two positive numbers that differ by 12 and have harmonic mean 5...
I started out with 1/2(1/x+12 + 1/x-12) = 1/5 but then it got all jumbled
The two positive integers:
Let x = one of them
And y = the other, and greater than x.
So,
y -x = 12
y = x +12 ---------**
Their Harmonic Mean,
H = 2 / [1/x +1/(x+12)] = 5
2 / [1/x +1/(x+12)] = 5
2/5 = 1/x +1/(x+12)
Clear the fractions, multiply both sides by 5(x)(x+12),
2(x)(x+12) = 1(5)(x+12) +1(5)(x)
2x^2+24x = 5x +60 +5x
2x^2 +24x -10x -60 = 0
2x*2 +14x -60 = 0
x^2 +7x -30 = 0
(x +10)(x -3) = 0
x = -10 or 3
Positive integer, so x = 3
y = 3 +12 = 15
Check,
2 / [1/3 +1/15] =? 5
2 / [(15 +3)/(3*15)] =? 5
2 / [18 / 45] =? 5
2(45/18) =? 5
90/18 =/ 5
5 =? 5
Yes, so, OK.
Therefore, the two positive integers are 3 and 15. --------answer.