A hard question on tree planting for P4–P5, with a step-by-step solution, diagrams and the answer. Common in the Hua Xia Cup's P2 to P5 papers. Sawing logs, striking clocks and lining up lamp posts are all the same problem.
The question
A footpath in a housing estate is 144 m long.
Along one side there is a lamp post every 8 m from end to end, with posts at both ends.
The posts are to be re-spaced to one every 6 m, still with posts at both ends.
An old post that already stands on a new spot stays; every other new spot gets a new post.
How many new lamp posts are put up?
Show the hints (3)
💡Hint 1: Measuring from the start, the old posts are at 0, 8, 16, … m and the new spots at 0, 6, 12, … m. Which spots appear in both lists?
💡Hint 2: Work out how many posts the new plan needs, then take away the ones that can stay.
💡Hint 3: The spots that can stay are multiples of both 8 and 6: one every 24 m, both ends included. The new plan needs 25 posts.
Worked solution
Have a go first, then read the solution step by step.
1.New plan: 144 ÷ 6 = 24 gaps, both ends, so 24 + 1 = 25 posts.
2.Posts that stay are at common multiples of 8 and 6, every 24 m: 144 ÷ 24 = 6 gaps, so 6 + 1 = 7 posts.
3.New posts: 25 − 7 = 18. That is 3 new in each 24 m stretch, 6 stretches: 3 × 6 = 18.
⚠️Common mistake: leaving out the post at the very start (0 m) when counting those that stay, which gives 19.
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