Tyneside average speed
Can you work out the average speed of the van?
Age
14 to 16
Challenge level
Problem
A van travels from Newcastle to South Shields at an average speed of 30 mph and returns by the same route at an average speed of 40 mph.
What is the van's average speed for the whole journey?
If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.
Student Solutions
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 = $X$
$\text{Newcastle }\xrightarrow[X\text{ miles}]{\text{30 mph}}\text{ South Shields time: }\frac X{30}\\
\hspace{5mm}\\
\text{Newcastle }\xleftarrow[X \text{ miles}]{\text{40 mph} }\text{ South Shields time: }\frac X{40}\\
\hspace{5mm}\\
\hspace{10mm}\text{ total time: } \frac{X}{30}+\frac{X}{40}\\
\hspace{2mm}\text{ total distance: } 2X\\
\hspace{2mm}\text{ average speed: } \dfrac{2X}{\frac{X}{30}+\frac{X}{40}}\\
\hspace{5mm}$
$\hspace{33mm} \begin{split}
&=\tfrac{240X}{4X+3X}\\
\hspace{5mm}\\
&=\tfrac{240}7\\
\hspace{5mm}\\
& =34\tfrac27\end{split}$