one positive number is 32 less than another number. if the reciprocal of the smaller number is added to five times the reciprocal of the larger number, the sum is 1/6. find the two numbers

Answer 1

n and n+32

1/n + 5/(n+32) = 1/6

multiply all terms by 6n(n+32)

6(n+32) + 5(6n) = n(n+32)

6n + 192 + 30 n = n^2 + 32 n

n^2 - 4n - 192 = 0

(n-16)(n+12) = 0
n = 16
n+32 = 48
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check
1/16 + 5/48 = 1/6 ???
3/48 + 5/48
8/48
1/6 sure enough

Answer 2

one number --- x
the other ------ x-32

1/(x-32) + 5(1/x) = 1/6
times 6x(x-32)
6x + 30(x-32) = x(x-32)
6x + 30x - 960 = x^2 - 32x
x^2 - 68x + 960 = 0
(x-20)(x-48) = 0
x = 20 or x = 48

if x = 20 then the other is negative, but both our numbers are to be positive

so x = 48

and the two number are 48 and 16

check:
"the reciprocal of the smaller number is added to five times the reciprocal of the larger number"
= 1/16 + 5(1/48)
= 1/6 , as required