Difference between revisions of "1964 AHSME Problems/Problem 26"
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Let the speeds of the runners in miles per hour be <math>a, b, c</math>, with <math>a>b>c</math>. If person <math>a</math> reaches the finish line in <math>h</math> hours, then we have <math>10 = ah, 8 = bh, 6 = bh</math>. | Let the speeds of the runners in miles per hour be <math>a, b, c</math>, with <math>a>b>c</math>. If person <math>a</math> reaches the finish line in <math>h</math> hours, then we have <math>10 = ah, 8 = bh, 6 = bh</math>. | ||
− | Thus, the speeds of the second and third place people are <math>\frac{8}{h}</math> and <math>|frac{6}{h}</math>. The ratio of those speeds is <math>\frac{\frac{8}{h}}{\frac{6}{h}} = \frac{4}{3}</math>, meaning person <math>b</math> runs <math>\frac{4}{3}</math> times as fast as person <math>c</math>, or conversely that person <math>c</math> is <math>\frac | + | Thus, the speeds of the second and third place people are <math>\frac{8}{h}</math> and <math>|frac{6}{h}</math>. The ratio of those speeds is <math>\frac{\frac{8}{h}}{\frac{6}{h}} = \frac{4}{3}</math>, meaning person <math>b</math> runs <math>\frac{4}{3}</math> times as fast as person <math>c</math>, or conversely that person <math>c</math> is <math>\frac{3}{4}</math> times slower than person <math>b</math>. |
If <math>c</math> runs <math>0.75</math> times as fast as <math>b</math>, then when <math>b</math> has finished the race at <math>10</math> miles, <math>c</math> will run <math>10 \cdot 0.75 = 7.5</math> miles. This is a difference of <math>10 - 75. = 2.5</math> miles, which is answer <math>\boxed{\textbf{(C)}}</math> | If <math>c</math> runs <math>0.75</math> times as fast as <math>b</math>, then when <math>b</math> has finished the race at <math>10</math> miles, <math>c</math> will run <math>10 \cdot 0.75 = 7.5</math> miles. This is a difference of <math>10 - 75. = 2.5</math> miles, which is answer <math>\boxed{\textbf{(C)}}</math> |
Revision as of 21:43, 24 July 2019
Problem
In a ten-mile race beats by miles and beats by miles. If the runners maintain constant speeds throughout the race, by how many miles does beat ?
Solution
Let the speeds of the runners in miles per hour be , with . If person reaches the finish line in hours, then we have .
Thus, the speeds of the second and third place people are and . The ratio of those speeds is , meaning person runs times as fast as person , or conversely that person is times slower than person .
If runs times as fast as , then when has finished the race at miles, will run miles. This is a difference of miles, which is answer
See Also
1964 AHSC (Problems • Answer Key • Resources) | ||
Preceded by Problem 25 |
Followed by Problem 27 | |
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All AHSME Problems and Solutions |
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