To determine how much money Hilary initially had, let's break down the given problem step-by-step:
1. Let [tex]\( x \)[/tex] represent the initial amount of money Hilary had.
2. She spent [tex]\(\frac{1}{3}\)[/tex] of her money on a book:
[tex]\[
\frac{1}{3} x
\][/tex]
3. She spent [tex]\(\frac{1}{4}\)[/tex] of her money on felt-tip pens:
[tex]\[
\frac{1}{4} x
\][/tex]
4. The total amount of money spent on both the book and the pens is:
[tex]\[
\frac{1}{3} x + \frac{1}{4} x
\][/tex]
5. To combine these fractions, find a common denominator (which is 12):
[tex]\[
\frac{1}{3} x = \frac{4}{12} x \quad \text{and} \quad \frac{1}{4} x = \frac{3}{12} x
\][/tex]
Therefore,
[tex]\[
\frac{1}{3} x + \frac{1}{4} x = \frac{4}{12} x + \frac{3}{12} x = \frac{7}{12} x
\][/tex]
6. Hilary was left with £5, meaning the amount of money she had after spending is:
[tex]\[
x - \frac{7}{12} x = 5
\][/tex]
7. Simplify the left-hand side of the equation:
[tex]\[
\left(1 - \frac{7}{12}\right) x = 5
\][/tex]
[tex]\[
\frac{12}{12} x - \frac{7}{12} x = 5
\][/tex]
[tex]\[
\frac{5}{12} x = 5
\][/tex]
8. To solve for [tex]\( x \)[/tex], isolate [tex]\( x \)[/tex] by multiplying both sides of the equation by the reciprocal of [tex]\(\frac{5}{12}\)[/tex]:
[tex]\[
x = 5 \times \frac{12}{5}
\][/tex]
[tex]\[
x = 12
\][/tex]
Therefore, the amount of money Hilary must have had to start with is £12.
So, the correct answer is:
[tex]\[ \boxed{£ 12} \][/tex]
