Here are the solutions to the problems: Problem 1: Convert $3\frac{2}{7}$ into an improper fraction. Step 1: Multiply the whole number by the denominator. $$3 \times 7 = 21$$ Step 2: Add the numerator to the result. $$21 + 2 = 23$$ Step 3: Place this sum over the original denominator. $$\frac{23}{7}$$ The improper fraction is $\boxed{\frac{23}{7}}$. Problem 2: There are 600 apples in a basket. If $\frac{1}{4}$ of them are rotten, how many apples are good? Step 1: Calculate the number of rotten apples. $$\text{Rotten apples} = \frac{1}{4} \times 600 = 150$$ Step 2: Subtract the number of rotten apples from the total to find the number of good apples. $$\text{Good apples} = 600 - 150 = 450$$ There are $\boxed{450}$ good apples. Problem 3: Subtract: $12\frac{4}{7} - 9\frac{1}{7}$ Step 1: Subtract the whole number parts. $$12 - 9 = 3$$ Step 2: Subtract the fractional parts. $$\frac{4}{7} - \frac{1}{7} = \frac{4-1}{7} = \frac{3}{7}$$ Step 3: Combine the whole number and fractional parts. $$3 + \frac{3}{7} = 3\frac{3}{7}$$ The result is $\boxed{3\frac{3}{7}}$. Problem 4: Simplify: $49 + 64 \div 8 \times 3 - 10$ Step 1: Perform division. $$49 + (64 \div 8) \times 3 - 10$$ $$49 + 8 \times 3 - 10$$ Step 2: Perform multiplication. $$49 + (8 \times 3) - 10$$ $$49 + 24 - 10$$ Step 3: Perform addition from left to right. $$(49 + 24) - 10$$ $$73 - 10$$ Step 4: Perform subtraction. $$63$$ The simplified value is $\boxed{63}$.