Question 3.EP.2: Which of the following compounds would you predict are solub...

(a) $KClO_{3}$ is potassium chlorate. From the solubility guidelines in Table 3.1, we see that compounds containing $K^{+}$ and $ClO_{3}^{-}$ tend to be soluble and that no common exceptions are mentioned. So we predict that $KClO_{3}$ should be soluble.
(b) $CaCO_{3}$ is calcium carbonate. Again consulting the table, we see that carbonates are generally insoluble and that $Ca^{2+}$ is not listed among the exceptions. Thus $CaCO_{3}$ should be insoluble.
(c) $BaSO_{4}$ is barium sulfate. Although most sulfates are soluble, $BaSO_{4}$ is listed in the table as an exception to that rule. Therefore we expect $BaSO_{4}$ to be insoluble.
(d) $KMnO_{4}$ is potassium permanganate. Although the permanganate ion ($MnO_{4}^{-}$) is not listed in Table 3.1, the table does tell us that all compounds of $K^{+}$ are soluble. So we would predict that $KMnO_{4}$ should be soluble.

Discussion
Here we simply consulted the table to check the solubility of each compound. Chemists generally gain familiarity with many of these solubility rules and learn to recognize soluble and insoluble salts without consulting such a table. You should check with your instructor to see whether you are expected to memorize any of these rules.