Bunuel wrote:
Over the course of 8 races, a runner's finishing time was 5 seconds faster each race than the race prior. If her combined total for all 8 races was 44:20, what was her finishing time for the second race?
A. 5:30
B. 5:35
C. 5:40
D. 5:45
E. 5:50
Official solution from
Veritas Prep.
This word problem is essentially a sequence problem. You don't know what her time was for each individual race, but you can use the same variable (for example, x) and relate each race to the others. If her first race took her x seconds, then the second was (x - 5), the third was (x - 10), etc. So for 8 total races, that means that the sequence would go:
x, (x - 5), (x - 10), ..., (x - 35)
You can then sum the times quickly by recognizing that each of 8 races has its own x, so that means 8x. And then you'll subtract from all but the first race. That means that 7 races will involve subtraction, with the smallest number subtracted being 5 and the largest being 35. Since this is an evenly-spaced set, you can multiply those 7 terms by the average of 20, and know that her overall time can be represented as 8x - 140.
You know that \(8x - 140 = 2660\) (NOTE: You'll calculate 2660 by multiplying 44 minutes by 60 and adding the remaining 20 seconds. 40 times 60 is 2400, plus 4 times 60 is another 240, plus 20 gives you 2660).
That then means that \(8x = 2800\), which means that x (the time of her first race) equals 350 seconds. 350 seconds is 5:50, but keep in mind that that is the time of her FIRST race, and the question asks for her second. Subtracting 5, that means that her second race took her 5:45.
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