Incorrect scales A set of scales does not always show the correct mass. If something is less than 100g they show the exact mass. When something weighs 1000g or more, they show some mass over 1000g. You have 5 balls with the masses A g, B g, C g, D g, and E g each less than 1000g. When you weigh these in pairs the scales show the following: B + D = 1200, C + E = 2100, B + E = 800, B + C = 900, A + E = 700. Which ball is the heaviest?
B + D > 1000
C + E > 1000
B + E = 800
B + C = 900
A + E = 700
I am able to get few lower boundaries and upper boundaries but nothing significant to move forward with solution. I looked up some problems on this forum but the explanations were not clear enough and the guidance was to ask this as a question.
 A: As you've already found, you have
$$B + D \gt 1000 \tag{1}\label{eq1A}$$
$$C + E \gt 1000 \tag{2}\label{eq2A}$$
$$B + E = 800 \tag{3}\label{eq3A}$$
$$B + C = 900 \tag{4}\label{eq4A}$$
$$A + E = 700 \tag{5}\label{eq5A}$$
You have \eqref{eq3A} minus \eqref{eq5A} giving
$$B - A = 100 \implies B = A + 100 \tag{6}\label{eq6A}$$
You have \eqref{eq4A} minus \eqref{eq3A} giving
$$C - E = 100 \implies C = E + 100 \tag{7}\label{eq7A}$$
Substituting \eqref{eq7A} into \eqref{eq2A} gives
$$2E + 100 \gt 1000 \implies E \gt 450 \tag{8}\label{eq8A}$$
From \eqref{eq7A}, you have $C \gt E$ and, more specifically,
$$C \gt 550 \tag{9}\label{eq9A}$$
With \eqref{eq9A}, \eqref{eq4A} along with \eqref{eq6A} gives
$$A \lt 250, \; B \lt 350 \tag{10}\label{eq10A}$$
Using $B = 900 - C$ from \eqref{eq4A}, substituting that into \eqref{eq1A} and using \eqref{eq9A} gives
$$900 - C + D \gt 1000 \implies D \gt C + 100 \implies D \gt 650 \tag{11}\label{eq11A}$$
Using the equation above and \eqref{eq7A}, you have $D \gt C \gt E$. Also, using \eqref{eq10A} shows $D \gt B \gt A$ as well. Thus, $D$ is the largest weight.
