Pan Balance Problems: One Worksheet for Missing Weights

Algebra and Function Practice

A pan balance shows equality: when the pans level, the total mass on each side is the same. Treat each pan as one side of an equation. Known weights become numbers; an unknown weight becomes a letter such as x. Then undo operations in reverse order to isolate x. This lesson uses grams throughout and assumes a simple two-pan balance with no friction or arm-length differences. All numerical values are hypothetical teaching examples; measure your actual equipment if you build physical models.

Practice materials: Printable downloads are not available for this lesson yet. Use the instructions and four exercises below.

Reading a Pan-Balance Diagram

Each pan is a group. Count or add the labeled weights on the left pan and write that total. Do the same for the right pan. If one object has an unknown weight, label it x and include it in that pan's total. A level balance means left total equals right total.

For example, if the left pan holds a 20 g weight and an unknown block, and the right pan holds 50 g, write x + 20 = 50. The unknown block is not automatically 30 g until you subtract 20 from both sides.

1. List every item on the left pan and every item on the right pan.
2. Replace each known weight with its number in grams.
3. Replace each unknown object with the same letter if it is the same object; use different letters for different unknowns.
4. Write an equation with an equals sign between the two pan totals.
5. Check that the equation matches the diagram before solving.

Solving with Inverse Operations

To isolate x, undo addition or subtraction first, then undo multiplication or division. Whatever you do to one side, do to the other so the balance stays level. Keep units consistent: if weights are in grams, answers are in grams.

If the equation is 3x = 60, divide both sides by 3 to get x = 20. If the equation is x + 15 = 40, subtract 15 from both sides to get x = 25. Always substitute your answer back into the original pan totals to confirm equality.

1. Identify the operation applied to x.
2. Apply the inverse operation to both sides.
3. Simplify each side.
4. Repeat until x is alone.
5. Substitute the value into the original equation and verify both sides are equal.

Handling Two Unknowns or Repeated Objects

If the same unknown object appears more than once on a pan, combine it: two x blocks become 2x. If two different unknown objects appear, you need two balance statements to solve both; one diagram alone is not enough unless one unknown is given elsewhere.

When a diagram shows x + x + 10 = 30, combine to 2x + 10 = 30, subtract 10, then divide by 2. The answer is x = 10 g. Check: 10 + 10 + 10 = 30.

1. Count repeated copies of the same unknown and write them as a coefficient.
2. Combine known weights on each pan.
3. Solve the resulting equation using inverse operations.
4. Substitute back and confirm both pan totals match.

Checking and Recording Answers

A correct answer makes both pan totals equal. Write the original equation, the inverse-operation steps, and the final substitution. If the totals do not match, recheck the diagram for missed items or a misread weight.

Record units every time. A number without grams is incomplete in a weight problem. If a diagram uses kilograms, convert consistently before solving, or state the unit clearly in the equation.

1. Write the original equation from the diagram.
2. Show each inverse-operation step on both sides.
3. State the final value with units.
4. Substitute the value into the original equation.
5. Confirm left total equals right total.

Worked example

A level balance has a 30 g weight and one unknown block on the left pan. The right pan has a 75 g weight. Find the unknown block's mass.

Four practice exercises

Common mistakes

• Adding weights from both pans together instead of comparing them. Keep left and right totals separate; the equals sign compares them, it does not combine them.
• Forgetting to apply an operation to both sides of the equation. Write the operation under both sides or draw a vertical line through the equals sign to remind yourself.
• Treating two different unknown objects as the same letter. Use different letters for different objects unless the diagram states they are identical.
• Dropping units or mixing grams and kilograms. Convert all weights to one unit before writing the equation, and include the unit in the final answer.
• Stopping after finding x without checking the pan totals. Substitute x into the original equation and confirm both sides are equal.

Self-check rubric

• Writes an equation that matches the pan totals, using a letter for the unknown.
• Applies inverse operations to both sides and shows the steps.
• States the final value with correct units.
• Substitutes the answer back and confirms both pan totals are equal.
• Uses different letters for different unknown objects and combines repeated identical objects correctly.

Questions and answers