Question

The sum of two numbers is 20 and the sum of their reciprocal is 5/24.find the numbers

Answer 1

let the two numbers be x&y

so

x+y=20

1/x+1/y=5/24

x+y/xy=5/24

20=5/24×xy

xy=20×24/5

xy=96

(x+y)^2=x^2+y^2+2xy

400=x^2+y^2+192

x^2+y^2=208

(x-y)^2=x^2+y^2-2xy

(x-y)^2=208-192

(x-y)^2=16

x-y=4

by adding equations we get

2x=24

x=12

y=8

Answer 2

Let the two numbers be x and y
Sum of numbers = 20
⇒ x + y = 20 ⇒ y = 20-x
Sum of their reciprocals = 5/24
⇒ 1/x + 1/y = 5/24
(x + y)/xy = 5/24
24x + 24y = 5xy
24x + 24(20-x) = 5x(20-x)
24x + 480-24x = 100x - 5x²
5x² - 100x + 480 = 0
x² - 20x + 96 = 0
x² -12x -8x +96 =0
x(x-12)-8(x-12) =0
(x-12)(x-8) = 0
x = 12 or 8
If x =12 then y =8 or if x =8 then y = 12
Therefore the two numbers are 12 and 8