The broken weight
A merchant had a forty pound measuring weight that broke into four pieces as the result of a fall. When the pieces were subsequently weighed, it was found that the weight of each piece was a whole number of pounds and that the four pieces could be used to weigh every integral weight between 1 and 40 pounds. What were the weights of the pieces?
Note that he used a balance scale to weigh things. So, for example, he could use a 1-pound weight and a 4-pound weight to weigh a 3-pound object, by placing the 3-pound object and 1-pound weight on one side of the scale, and the 4-pound weight on the other side.
Let the weights be \(a,b,c\) and \(d\), where \(a,b,c\) and \(d\) are in increasing order. Find \(\dfrac ba + \dfrac dc \).