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Algebra / Systems of two linear equations in two variables Difficulty: Hard

During a month, Morgan ran r miles at 5 miles per hour and biked b miles at 10 miles per hour. She ran and biked a total of 200 miles that month, and she biked for twice as many hours as she ran. What is the total number of miles that Morgan biked during the month?

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Explanation

Choice D is correct. The number of hours Morgan spent running or biking can be calculated by dividing the distance she traveled during that activity by her speed, in miles per hour, for that activity. So the number of hours she ran can be represented by the expression $the fraction r over 5$ , and the number of hours she biked can be represented by the expression $the fraction b over 10$ . It’s given that she biked for twice as many hours as she ran, so this can be represented by the equation $the fraction b over 10, end fraction, equals, 2 times the fraction r over 5$ , which can be rewritten as $b equals 4 r$ . It’s also given that she ran r miles and biked b miles, and that she ran and biked a total of 200 miles. This can be represented by the equation $r plus b, equals 200$ . Substituting $4 r$ for b in this equation yields $r plus 4 r, equals 200$ , or $5 r equals 200$ . Solving for r yields $r equals 40$ . Determining the number of miles she biked, b, can be found by substituting 40 for r in $r plus b, equals 200$ , which yields $40 plus b, equals 200$ . Solving for b yields $b equals 160$ .

Choices A, B, and C are incorrect because they don’t satisfy that Morgan biked for twice as many hours as she ran. In choice A, if she biked 80 miles, then she ran 120 miles, which means she biked for 8 hours and ran for 24 hours. In choice B, if she biked 100 miles, then she ran 100 miles, which means she biked for 10 hours and ran for 20 hours. In choice C, if she biked 120 miles, then she ran for 80 miles, which means she biked for 12 hours and ran for 16 hours.