Question: Rationalise the denominator of $\frac{10 + \sqrt{5}}{4\sqrt{5}}$
Give your answer in its simplest form.
1. **State the problem:** Rationalise the denominator of the expression $$\frac{10 + \sqrt{5}}{4\sqrt{5}}$$ and simplify the result.
2. **Recall the formula and rule:** To rationalise a denominator containing a square root, multiply numerator and denominator by the conjugate or the radical itself to eliminate the root from the denominator.
3. **Multiply numerator and denominator by $\sqrt{5}$:**
$$\frac{10 + \sqrt{5}}{4\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}} = \frac{(10 + \sqrt{5})\sqrt{5}}{4\sqrt{5}\sqrt{5}}$$
4. **Simplify the denominator:**
$$4\sqrt{5}\sqrt{5} = 4 \times 5 = 20$$
5. **Expand the numerator:**
$$ (10 + \sqrt{5})\sqrt{5} = 10\sqrt{5} + (\sqrt{5})^2 = 10\sqrt{5} + 5 $$
6. **Rewrite the expression:**
$$ \frac{10\sqrt{5} + 5}{20} $$
7. **Factor numerator if possible:**
$$ 10\sqrt{5} + 5 = 5(2\sqrt{5} + 1) $$
8. **Simplify the fraction by canceling common factor 5:**
$$ \frac{\cancel{5}(2\sqrt{5} + 1)}{4\cancel{5}} = \frac{2\sqrt{5} + 1}{4} $$
**Final answer:**
$$ \boxed{\frac{2\sqrt{5} + 1}{4}} $$