Let's solve the problems shown in the image:

a) Change $7 \frac{1}{2}$ to a top-heavy fraction.

A top-heavy fraction (also known as an improper fraction) has a numerator larger than the denominator.

$7 \frac{1}{2} = \frac{7 \times 2 + 1}{2} = \frac{14 + 1}{2} = \frac{15}{2}$

So, the top-heavy fraction is $\frac{15}{2}$.

b) Change $3 \frac{2}{3}$ to a top-heavy fraction.

$3 \frac{2}{3} = \frac{3 \times 3 + 2}{3} = \frac{9 + 2}{3} = \frac{11}{3}$

So, the top-heavy fraction is $\frac{11}{3}$.

c) Work out $7 \frac{1}{2} \times 3 \frac{2}{3}$. Give your answer as a mixed number.

First, convert the mixed numbers to improper fractions:

$7 \frac{1}{2} = \frac{15}{2}, \quad 3 \frac{2}{3} = \frac{11}{3}$

Now multiply the fractions:

$\frac{15}{2} \times \frac{11}{3} = \frac{15 \times 11}{2 \times 3} = \frac{165}{6}$

Next, simplify $\frac{165}{6}$:

$\frac{165 \div 3}{6 \div 3} = \frac{55}{2}$

Convert $\frac{55}{2}$ back to a mixed number:

$\frac{55}{2} = 27 \frac{1}{2}$

So, the answer is $27 \frac{1}{2}$.

d) Work out $7 \frac{1}{2} + 3 \frac{2}{3}$. Give your answer as a mixed number.

Again, convert the mixed numbers to improper fractions:

$7 \frac{1}{2} = \frac{15}{2}, \quad 3 \frac{2}{3} = \frac{11}{3}$

Find a common denominator for addition:

$\text{Common denominator of 2 and 3 is 6.}$

Convert each fraction:

$\frac{15}{2} = \frac{45}{6}, \quad \frac{11}{3} = \frac{22}{6}$

Now add the fractions:

$\frac{45}{6} + \frac{22}{6} = \frac{67}{6}$

Convert $\frac{67}{6}$ back to a mixed number:

$\frac{67}{6} = 11 \frac{1}{6}$

So, the answer is $11 \frac{1}{6}$.

Do you need further details or have any questions?

Here are 5 questions related to this topic:

1. How do you simplify improper fractions to mixed numbers?
2. What are some methods to find a common denominator when adding fractions?
3. Can you explain the process of converting a mixed number to an improper fraction?
4. Why is it important to simplify fractions after multiplication or addition?
5. What is the difference between a top-heavy fraction and a proper fraction?

Tip: When multiplying fractions, always simplify before multiplying if possible to make calculations easier.