In a certain algebra classroom, if three round tables were removed and replaced by three square tables, the ratio of round tables to square tables would be 1:1. If instead three square tables were removed and replaced by three round tables, the ratio of round tables to square tables would be 13:7. How many square tables are in the algebra classroom?

If you start with the first scenario, where the ratio is 1:1, you must add 3 round tables and subtract 3 square tables to get to the initial condition. To get to the second scenario, you must add 3 more round tables and subtract 3 more square tables. Altogether, the number of round tables is now 12 more than the number of square tables.

The ratio for the second scenario is ...

round : square = 13 : 7

In this ratio, the difference of ratio units is 13-7 = 6, half as many as the difference in the number of tables. So, multiplying these ratio units by 2, we get the ratio of round to square tables in the second scenario:

round : square = 13 : 7 = 26 : 14

The initial condition is 3 square tables more, hence ...

the number of square tables is 17.

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Check

Initially, the table count is round : square = 23 : 17. In the first scenario, the table exchange results in the ratio 20 : 20, or 1 : 1. In the second scenario, the table exchange results in the ratio 26 : 14, or 13 : 7.