1. The problem involves simplifying the expression $6 \times \frac{4}{5}$.\n\n2. Start by multiplying the whole number by the numerator of the fraction:\n$$6 \times \frac{4}{5} = \frac{6 \times 4}{5} = \frac{24}{5}.$$\n\n3. The expression $\frac{24}{5}$ is already in simplest form because 24 and 5 have no common factors other than 1.\n\n4. The other expressions you wrote, such as $\frac{25}{4} \times \frac{4}{5}$, simplify as follows:\n$$\frac{25}{4} \times \frac{4}{5} = \frac{25 \times 4}{4 \times 5} = \frac{100}{20} = 5.$$\n\n5. The last expression $\frac{25 \times A}{A \times 5}$ simplifies by canceling $A$ (assuming $A \neq 0$):\n$$\frac{25 \times A}{A \times 5} = \frac{25}{5} = 5.$$\n\n6. So, the key rule is that when multiplying fractions, multiply numerators and denominators respectively, and cancel common factors to simplify.\n\nFinal answer for the first expression: $\frac{24}{5}$.