Question

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