Divid 15 into two parts such that the sum of their reciprocal is 3/10

From the information obtained from the question, two equations can be created:
 
Let x and z be the two numbers (parts)

.  .  .  . (1)

 .  .  .  . (2)

By transposing (2), make 'z' the subject of the equation
 .  .  .  . (3)

By substituting (3) into equation (1) to find a value for x








⇒ 

∴ either         OR    

Thus x = 5 or x = 10

By substituting the values of x into (2) to find z

     z + (5) = 15       OR      z + (10) = 15

    ⇒      z = 10      OR                 z = 5

So, the two numbers or two parts into which fifteen is divided to yield the desired results are 5 and 10.