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Problem of the Week
Problem A and Solution
Balancing Act

Problem

Bailey 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, and the cost of sending a box depends on its mass. Bailey has a balance scale and a few standard weights to help with the job. The following diagrams show what Bailey observed when arranging some of the boxes and standard weights on the scales.

A balanced scale has a box with a heart stamp on the left side and a 2 kilogram weight on the right side.

A balanced scale has a box with a moon stamp along with a box with a sun stamp on the left side, and a 24 kilogram weight on the right side.

A balanced scale has a box with a moon stamp along with a box with a sun stamp on the left side, and two boxes with a heart stamp along with one box with a moon stamp on the right side.

Find the mass of each box.

Solution

From the diagrams we notice the following.

From this, we can conclude that one moon box and two heart boxes have a total mass of \(24\) kg. Also, two heart boxes have the same mass as one sun box.

Since one heart box has a mass of \(2\) kg, then two heart boxes have a mass of \(4\) kg. Therefore, one sun box has a mass of \(4\) kg.

This means \(4 \text{kg} + \text{(mass of a moon box)} = 24\) kg. Since \(4 + 20 = 24\), we can determine that one moon box must have a mass of \(20\) kg.

Teacher’s Notes

The idea of a balance scale is a nice analogy for an algebraic equation. We can represent the information in the problem using equations with variables to represent the masses of the different types of boxes. Here is one way to solve the problem algebraically.

Let \(x\) represent the mass of a heart box.
Let \(y\) represent the mass of a sun box.
Let \(z\) represent the mass of a moon box.

From the information in the diagrams, we can write the following equations: $$ \begin{align} x &= 2 \tag{1}\\ y + z &= 24 \tag{2}\\ y + z &= 2x + z \tag{3} \end{align} $$

From equation \((1)\), we already know that a heart box has a mass of \(2\) kg. From equations \((2)\) and \((3)\), we notice that the left sides are the same, so the right sides of the equations must be equal to each other. This means we know: \[2x + z = 24 \tag{4}\]

Now, substituting \(x = 2\) into equation \((4)\), we get \[2(2) + z = 24\]

Subtracting \(4\) from each side of this equation, we get \[z = 20\]

Finally, substituting \(z = 20\) into equation \((2)\), we get \[y + 20 = 24\]

Subtracting \(20\) from each side of this equation, we get \[y = 4\]

So, a heart box has a mass of \(2\) kg, a sun box has a mass of \(4\) kg, and a moon box has a mass of \(20\) kg.