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Problem of the Week
Problem B and Solution
A Balancing Act

Problem

If a scale is balanced, then the total mass on each side of the scale is the same. Consider the following balanced scale, where the number on an object represents its mass, in grams, and two identical objects with question marks on them have the same unknown mass.

A balanced scale. The left side has two identical squares with question marks on them. The right side has a single weight with the number 10 on it.

Since the right side has a mass of \(10\) g, it follows that the two squares must also have a total mass of \(10\) g. Since the square objects are identical, they must each have a mass of \(10\div 2=5\) g.

  1. Find the mass of the indicated shape for each of the three balanced scales.

    1. A balanced scale. The left side has two identical circles with question marks on them. The right side has two weights: one has the number 8 on it and the other has the number 6.

    2. A balanced scale. The left side has two identical hexagons with question marks on them. The right side has a third hexagon, identical to the two on the left side, with a question mark on it, and also a weight with the number 8.

    3. A balanced scale. The left side has two identical pentagons with question marks on them and one weight with the number 2. The right side has two weights: one has the number 8 and the other has the number 10.

  2. Using the same idea as in part (a), determine the mass of each symbol in the balanced scales shown. Note that here, the information from the previous scale is used in solving the next one.

    A balanced scale. The left side has four identical stars and the right side has one weight with the number 14.   

    A balanced scale. The left side has two identical stars that are the same as the stars on the previous scale, and a weight with the number 2. The right side has two identical triangles.   

    A balanced scale. The left side has four identical triangles that are the same as the triangles on the previous scale. The right side has two identical ovals and a weight with the number 13.

Solution

    1. Since \(2\) circles have a mass of \(8+6=14\) g, it follows that one circle has a mass of \(14\div 2=7\) g.

    2. If we remove one hexagon from each side of the scale, the scale will remain balanced because the hexagons have equal mass. Then we see that one hexagon has a mass of \(8\) g.

    3. If we remove \(2\) g from each side of the scale, then it follows that \(2\) pentagons have a mass of \(8+10-2=16\) g. Then one pentagon has a mass of \(16\div 2= 8\) g.

  2. From the first scale, we see that \(4\) stars have a mass of \(14\) g, so it follows that one star has a mass of \(14\div 4=7\div 2=3\frac{1}{2}\) g.

    In the second scale there are two stars and an object with a mass of \(2\) g on the left side. These have a total mass of \(3\frac{1}{2}+3\frac{1}{2}+2=9\) g. Then, two triangles have a mass of \(9\) g, so one triangle has a mass of \(9\div 2=4\frac{1}{2}\) g.

    In the third scale there are four triangles on the left side. These have a total mass of \(4\times 4\frac{1}{2}=18\) g. If we subtract \(13\) g from each side of this scale, then each side will have a mass of \(18-13=5\) g. Thus, \(2\) ovals have a mass of \(5\) g, so one oval has a mass of \(5\div 2=2 \frac{1}{2}\) g.