1973 AHSME Problems/Problem 27
Problem
Cars A and B travel the same distance. Car A travels half that distance at miles per hour and half at miles per hour. Car B travels half the time at miles per hour and half at miles per hour. The average speed of Car A is miles per hour and that of Car B is miles per hour. Then we always have
Solution
Let be the total number of time in hours that Car B took to drive the distance. This means that for have the time Car B traveled miles and for half the time Car B traveled miles. That means the total distance traveled is miles, so Because both cars traveled the same distance, for half the distance Car A took hours and for half the distance Car A took hours. That means In order to compare and , we need to compare the values that and are equal to. Since the denominators of both numbers are positive, cross-multiplying won’t change the comparison sign. Because is positive, the comparison sign does not need to be changed either. By the Trivial Inequality, . All of the steps are reversible, so . This can be confirmed by testing values of and .
See Also
1973 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 26 |
Followed by Problem 28 | |
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