Combine the balances

Each shape has one positive weight within a question. Identical shapes weigh the same, and each balanced scale says its two groups have equal total weight. A count beside a shape tells you how many copies are on the pan.

Here, a circle and a triangle balance a square and a triangle. Remove one triangle from each side. The remaining circle and square still balance.

You can remove matching groups from both sides, replace a group with an equal group, or multiply both sides by the same amount. Keep every balance true as you work.

This is a new question, so the weights can change. Two circles now balance one square. The other scale says that one square balances three triangles.

Replace the square in the first balance with three triangles. Two circles therefore balance three triangles. Doubling both groups shows that four circles balance six triangles.

You can find the answer using the relationships alone. Check the number of copies when substituting: one square stands for three triangles, so two squares would stand for six.

Start a new question. One circle plus one triangle balances one square. One square balances three triangles. How many triangles balance two circles?

Replace the square using the second balance, then remove one triangle from both sides. Once you know what balances one circle, double both groups for the target.

This checkpoint offers two choices. Practice rounds contain six questions with five choices each. The eight levels vary the number of shape types, balances, copies and matching groups to cancel.