$\text{Newcastle } \begin{eqnarray} \xrightarrow{\text{30 mph}} \\ \xleftarrow[\text{40 mph}]{} \end{eqnarray} \text{ South Shields}$

Suppose the distance is 120 miles (the distance won't affect the average speed so choose an easy number)

$\text{Newcastle } \begin{eqnarray} \xrightarrow{\text{30+30+30+30}} \\ \xleftarrow[\text{40+40+40}]{} \end{eqnarray} \text{ South Shields average speed}=\frac{30\times4+40\times3}7=\frac{240}{7}=34\frac27$

$\text{Newcastle }\xrightarrow[X\text{ miles}]{\text{30 mph}}\text{ South Shields time: }\frac X{30}\\

\hspace{5mm}\\

\text{Newcastle }\xleftarrow[X \text{ miles}]{\text{40 mph} }\text{ South Shields time: }\frac X{40}\\

\hspace{5mm}\\

\hspace{10mm}\text{ total time: } \frac{X}{30}+\frac{X}{40}\\

\hspace{2mm}\text{ total distance: } 2X\\

\hspace{2mm}\text{ average speed: } \dfrac{2X}{\frac{X}{30}+\frac{X}{40}}\\

\hspace{5mm}$

$\hspace{33mm} \begin{split}

&=\tfrac{240X}{4X+3X}\\

\hspace{5mm}\\

&=\tfrac{240}7\\

\hspace{5mm}\\

& =34\tfrac27\end{split}$

This problem is taken from the UKMT Mathematical Challenges.

You can find more short problems, arranged by curriculum topic, in our short problems collection.