Patterns & Reasoning

Depth · one concept from first principles · step 2 of 4

Group by numerator plus denominator, then count

The positive rational numbers are listed, without reducing any fraction, in the following order:

(11),(21,12),(31,22,13),⋯\left(\frac{1}{1}\right), \left(\frac{2}{1}, \frac{1}{2}\right), \left(\frac{3}{1}, \frac{2}{2}, \frac{1}{3}\right), \cdots

In which position in the list is 57\frac{5}{7}? (For example, 13\frac{1}{3} is the 66th number.)

OperationsNumber system
Your answer
How to solve
Strategy Look for a Pattern — First find the rule behind the brackets: the kth bracket holds the k fractions whose numerator plus denominator is k + 1. Then the position of 5/7 is the number of fractions in all earlier brackets plus its place within its own bracket.
1STEP 1

Find the rule

In each bracket numerator plus denominator is constant: 2, then 3, then 4. Inside, the denominator goes 1, 2, 3, ….

1 + 1 = 2, 2 + 1 = 1 + 2 = 3, 3 + 1 = 2 + 2 = 1 + 3 = 4
2STEP 2

Locate the bracket of 5/7

For 5/7 the sum is 12, so it is in the bracket with sum 12, the 11th bracket.

5 + 7 = 12
3STEP 3

Count the earlier fractions

Those brackets have 1, 2, …, 10 fractions.

1 + 2 + 3 + … + 10 = 55
4STEP 4

Position inside its bracket

The sum-12 bracket starts at denominator 1 and goes up by one each time.

11/1, 10/2, 9/3, 8/4, 7/5, 6/6, 5/7
5STEP 5

Add

Add the earlier count and the place in the bracket.

55 + 7 = 62
Answer
62nd
Test the method on the given example: 1/3 has sum 4; brackets with sums 2 and 3 hold 1 + 2 = 3 fractions, and 1/3 has denominator 3, so it is number 3 + 3 = 6, matching the problem.
Takeaway

Big lists are easy to search once you know what each group has in common.

  • Find the rule
  • Locate the bracket of 5/7
  • Count the earlier fractions
  • Position inside its bracket
  • Add
Where next?
Level upsuggested
Run a counting rule backward to every input
Another one like thissuggested
12 practice problems
Same look, different logicsuggested
Match numerators to bound the denominatorRewrite both ends over one denominator
Step back and solidifysuggested
Find the rule and write it as an expressionConstant increase yields a correspondence equation
13 reasoning problems grow out of this type — the bridge toward competition →

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▶ Practice — 12 problems

The same shape shows up later on the AMC, as Arithmetic Expression Decomposition, Geometric Series Share, Lattice / Grid Pattern from Small Cases. Why the shapes repeat →