The sum of two numbers is 18. The sum of their reciprocals is . Find the numbers.

Explanation

Let the two numbers be x and y.
`therefore` According to the I condition,
x + y = 18 …(1)
According to the II condition, `(1)/(x) + (1)/(y) = (1)/(4) …(2)`
implies `(y + x)/(xy) = (1)/(4)`
implies xy = 4(x + y) implies xy = 4 `xx` 18 [from (1)]
implies xy = 72
Now, we know that
`(x-y)^(2)=(x+y)^(2) - 4xy`
implies `(x-y)^(2)=(18)^(2)-4(72) = 324 - 288 = 36`
`therefore " "x - y = pm 6`
`{:("If x - y = 6 ...(3)"),("Now, from equations (3) and (1),"),(" x - y = 6"),( " "ul("x + y = 18")),("On adding, 2x = 24"),(therefore" x = 12"),(therefore " y = 18 - 12"),(" = 6 [from (1)]"):}:|{:("If x - y = - 6 ...(4)"),("Now, from equations (4) and (1),"),(" x - y = - 6"),(" "ul("x + y = 18")),("On adding, 2x = 12"),(therefore" x = 6"),(therefore " y = 18 - 6 [from (1)]"),(" = 12"):}`
`therefore` Two numbers are 12 and 6 in both cases.