Question 108707This question is from textbook Algebra Structure and Method Book 1
: At 9:30 a.m. Andrew left Exeter for Portsmouth, cycling at 12 mi/hr. At 10:00 a.m. Stacy left Portsmouth for Exeter, cycling at 16 mi/hr. The distance from Exeter to Portsmouth is 20 mi. Find the time when they met.
This question is from textbook Algebra Structure and Method Book 1
Answer by solver91311(24713)   (Show Source):
You can put this solution on YOUR website! Let's start with the basic expression for distance, rate, and time:
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Let Andrew's total distance be d1, total time be t1, and rate be r1.
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Let Stacy's total distance be d2, total time be t2, and rate be r2.
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We are given r1 and r2 directly, so let's plug those values into the basic equation.
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 and 
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We still have only two equations but four variables, so let's look for some relationships between d1 and d2, and between t1 and t2.
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Since Andrew and Stacy met at a point somewhere between Portsmouth and Exeter, we know that the distance Andrew travelled plus the distance Stacy travelled must equal the total distance between the two towns, namely 20 miles. So write:
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 or, in a manner that will be more convenient later:
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Now Andrew's time started at 9:30 AM, and stopped when he met Stacy. And Stacy's time started at 10:00 AM, and stopped when she met Andrew. So, Andrew's time was one half hour longer than Stacy's.
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 : Notice that we used 0.5 hour instead of 30 minutes. That is because the rates are both expressed in miles per hour. You could convert the given rates to miles per minute, and then use 30 minutes, but that would just make your numbers larger and therefore the arithmetic more difficult later. We mathematicians are a lazy lot.
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Let's rearrange our first two equations so they show time as a function of distance and rate:
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 and  :
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And then, substitute our expressions for d1 in terms of d2, and t1 in terms of t2 into the first equation:
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And then simplify to obtain:
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 : (Subtract 0.5 from both sides, combine the terms using the LCD of 12, and collect terms)
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Having done that step, we now have two different expressions for t2, which therefore must be equal:
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Now we know that Stacy travelled 8 miles by the time she met Andrew. So we know from that Stacy must have an elapsed time of 0.5 hour at the time she met Andrew.
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Therefore, they met at 10:00 AM plus 30 minutes, or 10:30 AM.
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Check:
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If Stacy's distance was 8 miles, then Andrew's distance must have been 12 miles. If Andrew was traveling at 12 miles per hour, he must have taken 1 hour to travel that 12 miles. 9:30 AM plus 1 hour is 10:30 AM. Check.
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