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Mr. Earl E. Bird gets up every day at 8:00 AM to go to work. If he drives at an average speed of 40 miles per hour, he will be late by 3 minutes. If he drives at an average speed of 60 miles per hour, he will be early by 3 minutes. How many miles per hour does Mr. Bird need to drive to get to work exactly on time? ' |
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Brenda and Sally run in opposite directions on a circular track, starting at diametrically opposite points. They first meet after Brenda has run 100 meters. They next meet after Sally has run 150 meters past their first meeting point. Each girl runs at a constant speed. What is the length of the track in meters? ' |
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Shelby drives her scooter at a speed of miles per hour if it is not raining, and miles per hour if it is raining. Today she drove in the sun in the morning and in the rain in the evening, for a total of miles in minutes. How many minutes did she drive in the rain? ' |
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Eric plans to compete in a triathlon. He can average miles per hour in the -mile swim and miles per hour in the -mile run. His goal is to finish the triathlon in hours. To accomplish his goal what must his average speed in miles per hour, be for the -mile bicycle ride? ' |
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Andrea and Lauren are kilometers apart. They bike toward one another with Andrea traveling three times as fast as Lauren, and the distance between them decreasing at a rate of kilometer per minute. After minutes, Andrea stops biking because of a flat tire and waits for Lauren. After how many minutes from the time they started to bike does Lauren reach Andrea? ' |
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Cagney can frost a cupcake every 20 seconds and Lacey can frost a cupcake every 30 seconds. Working together, how many cupcakes can they frost in 5 minutes? ' |
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Solution 1Let the time he needs to get there in be t and the distance he travels be d. From the given equations, we know that and . Setting the two equal, we have and we find of an hour. Substituting t back in, we find . From , we find that r, and our answer, is . Solution 2Since either time he arrives at is 3 minutes from the desired time, the answer is merely the harmonic mean of 40 and 60. The harmonic mean of a and b is . In this case, a and b are 40 and 60, so our answer is , so . Solution 3A more general form of the argument in Solution 2, with proof: Let be the distance to work, and let be the correct average speed. Then the time needed to get to work is . We know that and . Summing these two equations, we get: . Substituting and dividing both sides by , we get , hence . (Note that this approach would work even if the time by which he is late was different from the time by which he is early in the other case - we would simply take a weighed sum in step two, and hence obtain a weighed harmonic mean in step three.) |
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Solution 1Call the length of the race track . When they meet at the first meeting point, Brenda has run meters, while Sally has run meters. By the second meeting point, Sally has run meters, while Brenda has run meters. Since they run at a constant speed, we can set up a proportion: . Cross-multiplying, we get that . Solution 2The total distance the girls run between the start and the first meeting is one half of the track length. As the girls run at constant speeds, the interval between the meetings is twice as long as the interval between the start and the first meeting. Thus between the meetings Brenda will run meters. Therefore the length of the track is meters |
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Solution 1Doug can paint of a room per hour, Dave can paint of a room in an hour, and the time they spend working together is . Since rate times time gives output, Solution 2If one person does a job in hours and another person does a job in hours, the time it takes to do the job together is hours. Since Doug paints a room in 5 hours and Dave paints a room in 7 hours, they both paint in hours. They also take 1 hour for lunch, so the total time hours. Looking at the answer choices, is the only one satisfied by . |
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Since we know that she drove both when it was raining and when it was not and that her total distance traveled is miles. We also know that she drove a total of minutes which is of an hour. We get the following system of equations, where is the time traveled when it was not raining and is the time traveled when it was raining: Solving the above equations by multiplying the second equation by 30 and subtracting the second equation from the first we get: We know now that the time traveled in rain was of an hour, which is minutes |
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Since , Eric takes hours for the swim. Then, he takes hours for the run. So he needs to take hours for the mile run. This is |
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Let their speeds in kilometers per hour be and . We know that and that . (The second equation follows from the fact that .) This solves to and . As the distance decreases at a rate of kilometer per minute, after minutes the distance between them will be kilometers. From this point on, only Lauren will be riding her bike. As there are kilometers remaining and , she will need exactly an hour to get to Andrea. Therefore the total time in minutes is . |
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Cagney can frost one in 20 seconds, and Lacey can frost one in 30 seconds. Working together, they can frost one in seconds. In 300 seconds (5 minutes), they can frost . |
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