Problem
Each denominator d holds d-1 fractions, so the running totals are 1, 3, 6, 10, 15, 21, 28, 36, 45.
Each new denominator adds one more fraction than the previous group, a clear growing pattern.
3.OA.D.9Look For A PatternDenominator 9 ends at position 36. The next group, denominator 10, covers 37-45 — so 41 falls in the denominator-10 group.
Using the running totals reduces a far-off 41st term to 'which short group is it in'.
3.OA.D.9Solve An Easier Related ProblemThe 41st fraction in the list belongs to the group whose denominator is 10.
The groups for denominators 2 through 9 fill exactly the first 36 positions, and the next group, denominator 10, fills positions 37 through 45, so position 41 lands inside it.
Each group holds one less fraction than its denominator, so denominators 2 through 9 make groups of size 1, 2, 3, up to 8, which add to 36.
Inside a group the numerators run 1, 2, 3, up to one less than the denominator, and there is exactly one fraction for each numerator, so counting the numerators counts the fractions.
The groups sit one right after another with no gaps and no overlaps, so adding up the group sizes gives the total positions used, which tells where each group ends.
Position 41 is the (41 - 36) = 5th fraction in the denominator-10 group, so its numerator is 5.
Within a group the numerator equals the place number, so the 5th term has numerator 5.
3.NF.A.1Look For A PatternThis only needs Grade 3 pattern sense: add up the group sizes until you reach 41, then count inside that group!