# Difference between revisions of "1954 AHSME Problems/Problem 36"

Katzrockso (talk | contribs) (Created page with "== Problem 36== A boat has a speed of <math>15</math> mph in still water. In a stream that has a current of <math>5</math> mph it travels a certain distance downstream and r...") |
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== Solution 1 == | == Solution 1 == | ||

WLOG, let the distance the boat travel be <math>1</math> mile. Then the boat takes <math>\frac{1}{15+5}=\frac{1}{20}=3</math> minutes, to travel down a mile, then to travel back up the river, the boat travels <math>15-5=10</math> miles per hour, taking <math>\frac{1}{10}=\frac{1}{10}=6</math> minutes to travel up the river. This gives an average speed of <math>\frac{20(3)+10(6)}{3+6}=\frac{120}{9}=\frac{40}{3}</math> which as a ratio to <math>15</math> is <math>\frac{\frac{5\cdot 8}{3}}{3\cdot 5}\implies\frac{8}{9}</math> <math>\boxed{(\textbf{C})}</math> | WLOG, let the distance the boat travel be <math>1</math> mile. Then the boat takes <math>\frac{1}{15+5}=\frac{1}{20}=3</math> minutes, to travel down a mile, then to travel back up the river, the boat travels <math>15-5=10</math> miles per hour, taking <math>\frac{1}{10}=\frac{1}{10}=6</math> minutes to travel up the river. This gives an average speed of <math>\frac{20(3)+10(6)}{3+6}=\frac{120}{9}=\frac{40}{3}</math> which as a ratio to <math>15</math> is <math>\frac{\frac{5\cdot 8}{3}}{3\cdot 5}\implies\frac{8}{9}</math> <math>\boxed{(\textbf{C})}</math> | ||

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+ | == Solution 2 == | ||

+ | Let the distance the boat traveled be <math>d</math>. | ||

+ | Then since <math>rate=\frac{distance}{time}</math>, the average speed of the round trip is <math>\frac{2d}{\frac{d}{10}+\frac{d}{20}}</math>, or <math>\frac{40}{3}</math>. Note that this means that the round trip speed is the same no matter how long the trip. The speed of the boat in still water is <math>15</math> mph, so the ratio is <math>\frac{\frac{40}{3}}{15}</math>, or <math>\frac{8}{9} \implies </math>\boxed{(\textbf{C})}$ |

## Revision as of 17:16, 15 April 2017

## Problem 36

A boat has a speed of mph in still water. In a stream that has a current of mph it travels a certain distance downstream and returns. The ratio of the average speed for the round trip to the speed in still water is:

## Solution 1

WLOG, let the distance the boat travel be mile. Then the boat takes minutes, to travel down a mile, then to travel back up the river, the boat travels miles per hour, taking minutes to travel up the river. This gives an average speed of which as a ratio to is

## Solution 2

Let the distance the boat traveled be . Then since , the average speed of the round trip is , or . Note that this means that the round trip speed is the same no matter how long the trip. The speed of the boat in still water is mph, so the ratio is , or \boxed{(\textbf{C})}$