3)   What is 5% of 5? ____1/4______

        Percent means parts per 100. So 5% = 0.05 or 5 / 100. "of" means times.
         5 / 100 * 5 = 25 / 100  = 1/4
  
       Every eighth grader got this, but less than half of sixth graders.
       Most sixth graders seem to have trouble with percents.

4)   If  a # b = (a/b + b)/a ,   then what is  3 # 2?  ____7/6_or 1_1/6______

       The # here is a mystery operator that is defined as shown.
       Just as a + b means to add a and b,
       a # b means to combine a and b according to the shown expression
       so we get,
              (3/2 + 2) / 3 = 3  (7/2) / 3  = 7/6 or  1  1/6  or 1.1666666666.
       the best answer is  1 1/6, but the other two are fine.
       
       The # symbol confused a lot of students.
       Better get used to it because it is a popular sort of problem in math competitions.
       About half got this question wrong across the board.

5)   What is  -2 ( 4 – ( 3 – 2)) ?  ___-6_____

       Evaluate the from the innermost parenthesis to the outermost.
            -2 ( 4 - 1) = -2 (3 ) = -6

       Most got this I'm happy to report. 5th graders would have had a lot of trouble.

6)   If a watermelon weighs 3 pounds more than a cantaloupe, and together they weigh 11 pounds.
     How much does the cantaloupe weigh? __4_Pounds__

      I hope you were able to write the two algebraic equations
           W = C + 3
           W + C = 11
      Replacing  C + 3 for W in the second equation we have
           (C + 3 ) + C = 11
            2C + 3 = 11
            2C = 8
            C = 4 lbs

        The 8th graders extra knowledge of algebra helped on this one.
        80% of 8th graders got is, while only 50% of 6th and 7th graders got it.
        I'm Surprised that there was not even more of a difference actually.
        Maybe the review we did beforehand helped.

7)   What is the prime factorization of 63? ____7 * 3 * 3_______

        The fundamental theorem of arithmetic states that every positive integer can be
        decomposed into a unique product of primes (the prime factorization).
        for 63 it is  7 * 3 * 3
 
        Very few students missed this, but if you did, you should review
        http://www.mathsisfun.com/prime-factorization.html
     

8)    What is the average (or mean)  of  -3, 4, 5, 0, 2, 4? _____2_______

          Sum the numbers to get 12, then divide by 2 to get 6.
          I'm happy to report that almost everyone got this.

9)   My Prius gets 60 miles per gallon. How many gallons of gas will I burn driving 100 miles? 1 2/3 gal

          Try using unit cancellation to solve.
          100 miles / (60 miles/gallon) = 100/60 gallons = 5/3 gallons or 1 2/3 g.

          Unit cancellation is an important topic. It must be covered in 6th grade because there was a big gap
          between 6th graders (45%) and 7/8th (80%) graders for correct responses.

10)   What is 7³/7²?  _____7______

        7³/7^{2 =} 7^(3-2) = 7¹ = 7   but if you did not know to do it that way, you should notice that

        7 * 7 * 7 / 7 * 7    so two of the sevens on top cancel the two on the bottom, leaving 7.

        The second approach should be easier to understand, but the first has broader applicability.
        All 8th graders got this correct, but only 60% of 6th graders got it.
   

11)   What is  4⁴ – 4³ – 4² ?  ____176______

        Most people worked out that 256 - 64 - 16 = 176
        but slightly easier is to factor out 4²  to get
            16 ( 16 - 4 - 1) = 16 (11) = 176

        This problem is interesting because 8th graders did not do better than the rest.
        Every grade level got it correct only half the time.

12)   If a watermelon weighs 6 pounds plus half a watermelon,
       how much does a watermelon and a half weigh?  ___18 lb_____

       This is exactly like the brick problem we did on the board
         W = 6 + 1/2 W  =>  1/2 W = 6  =>  W = 12 lb
         so  3/2 W = 18 lb

        Equidistant gaps in performance between the grades, as you would expect for increased
        algebra experience.

13)   What is 2^-3? ___1/8____

        I didn't really expect you to know that  x^-a = 1 / x^a 
        that is why I wrote it on the board at the beginning of class. -8 was a very common wrong answer.
       
          2^-3 = 1/ 2³ = 1/8

      All but one 8th grader got this question correct, however, not a single 6th or 7th grader got it!
      It must be covered extensively in 7th grade. In future weeks, we will probably get exponent problems
      that are a lot harder that this. We will be discussing how to do them, but 6th and 7th graders may
      have a difficult time keeping up.            

14)   What is x if   3x + 3 = 5x -2 ? ___5/2 or 2.5____

       This is a basic algebraic equation with one variable. 6th graders can be forgiven for not
       getting it because they are not expected to know algebra yet.
       Start by subtracting 3x from both sides to get 3 = 2x - 2
       Next, add 2 to both sides to get  5 = 2x
       Lastly, divide by 2 to get x = 5/2

        I'm happy to report that every 8th grader got this correct. Very few 6th or 7th graders though.

15)   If g(x) = x² – 3, what is g(3)? _____6______

        I thought this was pretty easy, but quite a few missed it. Perhaps the g(x) notation was confusing.
        g(x) means that we are defining a function, g, of x.
        g(3) means you should plug 3 in for x and get  3² - 3 = 9 - 3 = 6.

       All but one 8th grader got this, but very few 6th or 7th graders.

16)   If f(a) = a + 3,  and g(b) = 3b -1,  what is  f(g(2)) ? ___8____

       This was harder, but several people got it. f(g(x)) implies a composite function and is
       read "f of g of x". f(g(2)) can be written like this

           f(g(2)) =  [ ( 3 (2)  - 1)  + 3 ] = [ 5 + 3 ] = 8
 
        OK, I admit, composite functions may be a little beyond the normal curriculum, but I thought it
        was something they could figure out. More than half the 8th graders got it, but none in the
        lower grades did.  

17)   Five is 20% of what? ___25_____

        20% = 20/100 or 1/5, so
        x/5 = 5.  Now solve for 5.
        x = 25.  Lastly verify that 20% of 25 is 5.

        Percentages again. Most 8th graders got it (90%), but few from the lower grades did (40%).  
        100 was a common wrong answer among 6th graders.

18)   Four consecutive integers sum to 34. What is the smallest of these integers? __7___

        This is really typical of the sorts of problems you will see on the contests.
        One way to solve goes like this:
           x  + (x + 1) + (x + 2) + (x + 3) = 34
           so  4x + 6 = 34 =>  4x = 28  => x = 7

        All but one 8th grader got this. Far fewer from lower grades got it (~20%).

19)   Curtis rides uphill for 30 minutes at 6 miles per hour.  He then comes down the same hill at 18 miles per hour. What was his average speed?  ___9 miles/hour______

     This is a rate type problem. You cannot average rates like you can other numbers.
     12 mph was a very common wrong answer.
     The average rate (speed in this case) is always the total change over the total time.
     Since it takes Curtis 1/2 hour to ride up the hill at 6 miles/hour, the hill is 3 miles.
     Since he road down 3 times faster, it will take him only 10 minutes (or 1/6 hour) to go down.
     Hence his average speed is
        6 miles [distance up and down hill] / (1/2 + 1/6 ) hours [time to go up then down]
         = 6 / (4/6) = 36 / 4 = 9 m/hr
  
     Alternatively, he harmonic mean 2 * (6 * 18 ) / (6 + 18) can be applied to determine the answer.

      Rate type problems are very popular on competitions because they are hard and do not require algebra.
      You should know how to do unit cancellation though. Only 2 eighth graders go this and none from the
      lower grades did. Congratulations to those 8th graders that got it!