To solve $1 \frac{1}{3} \div \frac{3}{5}$, let's break it down step by step:

Step 1: Convert the mixed fraction to an improper fraction

$1 \frac{1}{3} = \frac{4}{3}$ This is because: $1 \frac{1}{3} = 1 + \frac{1}{3} = \frac{3}{3} + \frac{1}{3} = \frac{4}{3}$

Step 2: Division of fractions

When dividing fractions, you multiply the first fraction by the reciprocal (flip the second fraction).

So: $\frac{4}{3} \div \frac{3}{5} = \frac{4}{3} \times \frac{5}{3}$

Step 3: Multiply the fractions

Now, multiply the numerators and denominators: $\frac{4 \times 5}{3 \times 3} = \frac{20}{9}$

Step 4: Simplify (if necessary)

The fraction $\frac{20}{9}$ is already in its simplest form, but we can also express it as a mixed number: $\frac{20}{9} = 2 \frac{2}{9}$

So, the quotient of $1 \frac{1}{3} \div \frac{3}{5}$ is either $\frac{20}{9}$ or $2 \frac{2}{9}$.

Let me know if you'd like any further explanation or if you have any questions!

Follow-up questions:

1. How do you multiply fractions?
2. What is the difference between a proper and improper fraction?
3. How do you convert an improper fraction to a mixed number?
4. What are the rules for dividing fractions?
5. Can you solve $\frac{5}{6} \div \frac{1}{2}$ using the same method?

Tip: When dividing fractions, always remember to flip the second fraction (take its reciprocal) before multiplying!