Wednesday, January 15, 2014

why did you get this wrong?

Yesterday we gave a school-wide test to many of our high school students.  It was part of a monthly, statewide voluntary "fun" competition called the Florida Math League.  There are always six questions and they range from pretty simple to pretty tough.  Perfect scores are rare.  My calculus students usually average about 3 correct, although there is no calculus on the test.

One question caught my eye.  Many, many of our bright students got it wrong, but it's a common type of word problem.  What do you think?  Would you get this one right?

In a 10-km race, First Runner beat second runner by 2 km, and First Runner beat Third Runner by 4 km.  If all 3 runners always ran at constant rates, by how many km did Second Runner beat Third Runner. 

The answer will be posted below in the comments.  If you're a math teacher teaching Algebra II or above, ask your own students and let me know how well they do.

Why do you think so many people get this kind of problem wrong? 

Photo Credit


  1. Answer: When First Runner had gone 10 km, Second had gone 8 km and Third had gone 6 km. Since the ratio of distances travelled by Second and Third is 8:6 = 4:3, Third traveled 3/4 of the distance traveled by Second. When Second finished the 10 km, Third had gone 3/4 the distance or 7.5 km. So to answer the question, Second beat Third by 2.5 km.

    1. I would think most get it wrong because they assume that the race ended when First Runner crossed the line. They fail to take into account that Second Runner and Third Runner still have to finish the race and only pay attention to the information in the problem itself that shows Third Runner behind by 2 km. Methinks.

  2. I agree with Jwilson - how many said it was by 4 km?
    Analyzing their answers is one way to see where their reasoning went wrong.
    The better question is: how can the question be written better?

  3. Thanks to you both for your comments! The most common wrong answer was 2, which in my mind makes sense, in that if the students assume as you suggested that the race ended when the First crossed the line, the Second and Third were indeed separated by 2 km. I don't think anyone said 4 km.

    I agree that questions need to written with precision. Certainly we run into trouble as math and science teachers when students do not make the same assumptions that we do. Perhaps it could have been written, at the end of the 10 km, by how many km did Second Runner beat Third Runner.

    I am actually also interested in whether students can actually solve this problem at all, even if the make all the correct assumptions. I'd love for some teachers to ask their students this questions and ensure that they understand that all runners must complete the 10 km. How many still can't solve it?

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