Thursday, December 13, 2007

An 'Improper' Question for Middle Schoolers?

From various Google searches reaching this site I often discover some wonderful problems. These often inspire me to extend the question further. Part (a) below was from the Google search. I modified it slightly and then developed it. Thank you to whomever typed in the original search!

Note: For all of these questions, p/q represents an "improper fraction" in which p>q>0 and p,q are positive integers.

(a) How many values of p/q are there in which the sum of the numerator and denominator is 29 and 2 1/3 > p/q > 2 1/4.
Notes: The numerical quantities in the inequality are mixed numerals.
How might one approach this without algebra?

(b) Show there are exactly 16 values of p/q such that p+q = 500 and 5 > p/q > 4.
Is algebra the preferred method here?

(c) Show there are exactly 33 values of p/q such that p+q = 1000 and
5 > p/q > 4.

(d) For Advanced Algebra students: Derive a formula for the number of values of p/q such that p+q = N and B > p/q > A.
N is a positive integer greater than 2 and A,B are positive reals with B >A >2 . Express your formula in terms of A, B and N. The greatest integer function may be needed.

1 comment:

Andrew S. said...

a) There are no values that fit the conditions. Without resorting to algebra, I just tried values. There aren't that many pairs of positive integers that add to 29, so I tried them. The important ones to try are 20 & 9 and 21 & 8.

On part b) and c) I preferred using algebra to trying out numbers like I did in part a). This may be because I'm an algebra 2 teacher. But it seemed the least clunky method that occurred to me.