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There and Back
Brian swims at twice the speed that a river is flowing, downstream from one moored boat to another and back again, taking 12 minutes altogether. How long would it have taken him in still water?
Escalator
At Holborn underground station there is a very long escalator. Two people are in a hurry and so climb the escalator as it is moving upwards, thus adding their speed to that of the moving steps. ... How many steps are there on the escalator?
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.
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NRICH is part of the family of activities in the Millennium Mathematics Project.
NRICH is part of the family of activities in the Millennium Mathematics Project.