If a person's expenditure increased by 15% and their savings decreased from 24% to 22.5% of their income, by what percentage did their income increase?

Let the original income be $I_1$.
Original savings $S_1 = 24\%$ of $I_1 = 0.24 I_1$.
Original expenditure $E_1 = I_1 - S_1 = I_1 - 0.24 I_1 = 0.76 I_1$.

Let the new income be $I_2$.
The expenditure increased by $15\%$.
New expenditure $E_2 = E_1 \times (1 + 0.15) = 0.76 I_1 \times 1.15 = 0.874 I_1$.
New savings $S_2 = 22 \frac{1}{2}\%$ of $I_2 = 0.225 I_2$.

The new income is the sum of the new expenditure and new savings:

Substitute the expressions from Step 2:

Subtract $0.225 I_2$ from both sides:

Divide by 0.775:

Percentage increase $= \frac{I_2 - I_1}{I_1} \times 100\%$

The percentage increase of his income is $\boxed{12.77\%}$.

The position vectors are $\vec{OB} = \begin{pmatrix} 8 \\ -4 \end{pmatrix}$ and $\vec{OL} = \begin{pmatrix} 4 \\ 6 \end{pmatrix}$.

The position vectors are $\vec{OU} = \begin{pmatrix} -6 \\ -8 \end{pmatrix}$ and $\vec{OL} = \begin{pmatrix} 4 \\ 6 \end{pmatrix}$.

The magnitude of a vector $\begin{pmatrix} x \\ y \end{pmatrix}$ is $\sqrt{x^2 + y^2}$.

The magnitude of TI is $\boxed{\sqrt{53}}$.

The man takes an exact number of steps, whether his stride is 60 cm or 64 cm. This means the total width of the road must be a common multiple of 60 cm and 64 cm. To find the least width, we need to find the Least Common Multiple (LCM).

To find the LCM, take the highest power of all prime factors present in either number.

Since 1 metre = 100 cm:

The least width of the road is $\boxed{9.6 \text{ metres}}$.