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# There and Back

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Age 14 to 16

Challenge Level

- Problem
- Student Solutions

In this problem, consider the following:

When Brian swims with the flow, his total speed is his speed in the
water plus the speed of the moving water. When he swims against the
flow, his total speed is the difference between his speed in still
water and the speed of the moving water.

This leads to the following solution:

Let the speed of the river be *v* , distance to the
moored boat = *d* , and the time to complete the downstream
journey = *t* .

Downstream |
Upstream |

Speed of man = 3 v |
Speed of man = v |

Distance = d |
Distance = d |

Time = t |
Time = 12 - t |

Again, using distance = speed x time, gives the following equations:

**(1)** d = 3 *vt*

**(2)** d = *v* (12- *t* )

combining **(1)** and **(2)**
gives

3 *vt* = *v* (12 - *t* )

which leads us to *t* = 3 minutes

Using this information tells us that the speed of the river is d/9

and since Brian swims at twice this speed, his speed in the still water is 2d/9.

He swims a distance of 2 *d* at this speed.

So it will take him 9 minutes.

Correct solutions were sent from
**Nicholas** and **Adrian**
(South Greenhoe Middle School).