document.write( "Question 1193694: Cameron went on a bike ride of 40 miles. He realized that if he had gone 8 mph faster, he would have arrived 16 hours sooner. How fast did he actually ride? \n" ); document.write( "

Algebra.Com's Answer #825741 by greenestamps(13198) $\"\"$ $\"About$

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\n" ); document.write( "Formally, you would set this problem up by writing an equation that says the difference between 40 miles at the actual rate and 40 miles at a rate 8mph faster is 16 hours.

\n" ); document.write( "x = actual rate; 40/x = actual time
\n" ); document.write( "x+8 = other rate; 40/(x+8) = other time

\n" ); document.write( " $\"%2840%2Fx%29-40%2F%28x%2B8%29=16\"$

\n" ); document.write( "Solve by multiplying through by the common denominator and solving the resulting quadratic equation.

\n" ); document.write( "But I agree with tutor @ikleyn that the way to get the most benefit from working this problem is to solve it using logical reasoning and simple mental arithmetic.

\n" ); document.write( "I will expand on her idea a bit to help you see how to find the solution.

\n" ); document.write( "The distance is an integer; and the differences in rates and times are integers. That means the solution is almost certainly in integers.

\n" ); document.write( "So look at the possible whole numbers of rates and times for which distance (rate * time) is 40 miles; then look for two of the times that differ by the required 16 hours.

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document.write( "  rate    time\r\n" );
document.write( " --------------\r\n" );
document.write( "    1       40\r\n" );
document.write( "    2       20\r\n" );
document.write( "    4       10\r\n" );
document.write( "    5        8\r\n" );
document.write( "    8        5\r\n" );
document.write( "   10        4\r\n" );
document.write( "   20        2\r\n" );
document.write( "   40        1

\n" ); document.write( "There are only two times in that list that have a difference of 16 hours: 20 and 4. Check to see that the difference in rates for those two times is the required 8mph.

\n" ); document.write( "40 miles in 20 hours = 2mph; 40 miles in 4 hours = 10mph; 10-2 = 8.

\n" ); document.write( "ANSWER: The actual rate is 2mph (almost impossibly slow for riding a bike!)....

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