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Tyneside Average Speed

Age 14 to 16 Short Challenge Level:

Answer: $34 \frac{2}{7}$


Using distance = 120 miles

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

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

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


Using distance = $X$
$\text{Newcastle  }\xrightarrow[X\text{ miles}]{\text{30 mph}}\text{  South Shields         time: }\frac X{30}\\
\hspace{5mm}\\
\text{Newcastle  }\xleftarrow[X \text{ miles}]{\text{40 mph} }\text{  South Shields         time: }\frac X{40}\\
\hspace{5mm}\\
\hspace{10mm}\text{          total time: } \frac{X}{30}+\frac{X}{40}\\
\hspace{2mm}\text{    total distance: } 2X\\
\hspace{2mm}\text{ average speed: } \dfrac{2X}{\frac{X}{30}+\frac{X}{40}}\\
\hspace{5mm}$
$\hspace{33mm} \begin{split} 
&=\tfrac{240X}{4X+3X}\\
\hspace{5mm}\\
&=\tfrac{240}7\\
\hspace{5mm}\\
& =34\tfrac27\end{split}$



This problem is taken from the UKMT Mathematical Challenges.
You can find more short problems, arranged by curriculum topic, in our short problems collection.