Patterns & Reasoning

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

The path's shape decides how many gaps there are

A path is 42.5 m42.5\ \text{m} long on each side, and 7070 sculptures are to be placed along both sides at equal intervals, including one at each end. How many meters apart are the sculptures? (Ignore the thickness of the sculptures.)

Number systemOperations
Your answer
How to solve
Strategy Draw a Diagram — Two traps sit in a row here. The count is for both sides, so halve it first; and the gaps between objects on a straight line are one fewer than the objects. Draw one side and both become obvious.
1STEP 1

Split the sculptures between the two sides

The 70 is the total for both sides, and the sides are the same, so each side gets half.

70 ÷ 2 = 35
2STEP 2

Count the gaps on one side

Along a straight path with a sculpture at each end, the gaps sit between sculptures, so there is one fewer gap than there are sculptures.

35 - 1 = 34
3STEP 3

Share the length among the gaps

The 42.5 m is split into 34 equal gaps.

42.5 ÷ 34 = 1.25
Answer
1.25 m
Multiply back: 1.25 × 34 = 42.5 m, the length of one side.
Takeaway

Objects at both ends of a straight path leave one fewer gap than there are objects.

  • Split the sculptures between the two sides
  • Count the gaps on one side
  • Share the length among the gaps
Where next?
Another one like thissuggested
12 practice problems
Step back and solidifysuggested
Cuts-to-pieces correspondence ruleObjects versus gaps when spacing along a line
2 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, Lattice / Grid Pattern from Small Cases. Why the shapes repeat →