The product of two numbers is 20, and the sum of their reciprocals is 3/5. Find the numbers.

Let one number be x, and the other be y.

Product of two numbers = 20
x * y = 20

xy = 20

Sum of their reciprocals = 3/5



Through cross multiplication: 



Through substituting xy for 20 (see above, xy=20)



x+y = 12

Now we have a pair of simultaneous equations:

xy = 20 .........(1)
x+y = 12 .......(2)

Manipulate (2)

x +y = 12

y = 12 -x ..........(3)

Substitute (3) into (1):

xy = 20
x(12-x) = 20

12x - x^2 = 20

x^2 -12x +20 = 0

Through factorisation (x-10)(x-2) = 0. Hence, x= 10 or 2

Substitute x=10 or 2 into (3)

y = 12-x

If x = 10, y = 12-10 = 2
If x = 2,  y = 12-2 = 10

So we have two numbers, 10 and 2.