1. The problem asks to write a quarter of a billion in expanded notation. 2. A billion is $1,000,000,000$. 3. A quarter of a billion is $\frac{1}{4} \times 1,000,000,000 = 250,000,000$. 4. Expanded notation means expressing the number as the sum of each digit times its place value. 5. For $250,000,000$, the expanded notation is: $$200,000,000 + 50,000,000 + 0 + 0 + 0 + 0 + 0 + 0 + 0$$ 6. Simplifying, we write: $$2 \times 100,000,000 + 5 \times 10,000,000$$ 7. Therefore, the quarter of a billion in expanded notation is: $$2 \times 10^{8} + 5 \times 10^{7}$$