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# Balancing Scales

How many stars would balance the bottom scale?

These next four quizzes will explore various types of balance puzzles and strategies for approaching them. Most of these puzzles can be solved quickly with a bit of creativity and mental math, and this toolbox of techniques is an excellent foundation for setting forth into variable algebra.

What is the combined weight of one diamond and two circles?

What is the weight of a single square?

What is the weight of a single circle?

Scale puzzles are a great way to visualize how to solve variable algebraic equations. Let's take the following balance scale:

If we let \(t\) stand for triangle we can represent the balance scale with the following equation: \[2t + 3 = 13.\]

With single variable equations our goal is to isolate the unknown variable, in this case \(t.\)

So we can subtract the weight of 3 from both sides and solve for \(t\): \[\begin{align} 2t &= 13 -3\\ 2t &= 10. \end{align}\] Taking three away from each side, we realize that two triangles equal 10, so one triangle must equal 5: \(t = 5.\)

Which combination of shapes would balance the scale?

Sometimes when we solve balance puzzles, we are trying to determine what one shape equals in terms of other shapes. For example, we might not know the numerical value of a square, but we might know that it is equivalent to 3 stars.

When working with these types of puzzles, look for places where you can pull the same shapes from both sides of a scale, such as 1 square from each side. In addition, look for places where you can substitute one or two shapes for another.

Let's use the example below to determine ways to express the value of one triangle in terms of circles.

From the first scale, we can see that one triangle is equivalent to one circle and two squares. If we replace the triangle on the bottom scale, we have two squares and two circles balanced with three squares.

After we cancel two red squares from each side, we can see that one square equals two circles.

If we substitute each square on the top balance with two circles, we can see that five circles balance one triangle.

According to the following scales, what could balance three diamonds?

What should be placed on the right side of the bottom scale in order to make it balance?

What should be placed on the left side of the bottom scale in order to make it balance?

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