James is in charge of sending out boxes from a distribution centre. The contents of the boxes are identified by shapes stamped on them: a heart, a moon, or a sun. All boxes with the same stamp have the same mass.
The following diagrams show what James observed when arranging some of the boxes and standard weights on a scale.
Given that each scale is balanced, determine the mass of each box.
From the diagrams we notice the following.
One moon box has the same mass as the sum of the mass of a heart box and the mass of a sun box.
Three heart boxes have a total mass of \(15\) kg.
Four heart boxes have the same total mass as one sun box.
Since \(3\) heart boxes have a total mass of \(15\) kg, then the mass of \(1\) heart box must be \(\frac{1}{3}\) of \(15\) kg. Therefore, \(1\) heart box has a mass of \(5\) kg.
Since 4 heart boxes have the same mass as \(1\) sun box, then \(1\) sun box must have a mass of \(5 \times 4 = 20\) kg.
Since \(1\) moon box has the same mass as the sum of the mass of \(1\) heart box and the mass of \(1\) sun box, then \(1\) moon box must have a mass of \(20 + 5 = 25\) kg.
Therefore, \(1\) moon box has a mass of \(25\) kg, \(1\) heart box has a mass of \(5\) kg, and \(1\) sun box has a mass of \(20\) kg.