Question 24
A merchant received four sacks of potatoes and wants to weigh each one. He knows their weights in kilograms are distinct integers. Let the weights be $a,b,c,d$, with $a\lt b\lt c\lt d$.
(a) Show that when the sacks are weighed in pairs, the largest result is $c+d$ and the second largest is $b+d$. Also show that the smallest result is $a+b$ and the second smallest is $a+c$.
(b) The scale has broken and only indicates weights greater than or equal to $100$ kilograms. The merchant weighs the sacks in pairs, making six weighings. Four results, in kilograms, are $101,112,116,127$. For the other two, he could only determine that the sums were less than or equal to $100$ kilograms. Find the weights of the four sacks.
(a) Show that when the sacks are weighed in pairs, the largest result is $c+d$ and the second largest is $b+d$. Also show that the smallest result is $a+b$ and the second smallest is $a+c$.
(b) The scale has broken and only indicates weights greater than or equal to $100$ kilograms. The merchant weighs the sacks in pairs, making six weighings. Four results, in kilograms, are $101,112,116,127$. For the other two, he could only determine that the sums were less than or equal to $100$ kilograms. Find the weights of the four sacks.