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A municipality wants to light three parallel streets 150 m, 210 m and 300 m long, with streetlights placed at regular intervals on both sides of each street. In addition, the Municipality wants the distance between two consecutive streetlights to be the same in all three streets and that both at the beginning and end of each street there should be two streetlights (one on each side). What is the minimum number of streetlights needed? At what distance should they be placed?

Being $$\text{gcd}(150, 300, 210) = 2 \times 3 \times 5 = 30$$ we had streetlights are to be placed every 30 m. For each street we will have: $$150 : 30 = 5 $$ $$300 : 30 = 10 $$ $$210 : 30 = 7 $$ A total of 22 streetlights will therefore be needed, placed 30 m apart.

The solution is 50 streetlights. Why?

Bill Dubuque
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Sebastiano
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1 Answers1

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On the first street, we need twelve streetlights. Two at $0$ m (there has to be one on either side of the street), two at $30$ m, $60$ m, $90$ m, $120$ m each, and finally two at $150$m. That's $2(5 + 1) = 12$. Similarly, we need $22 = 2(10 + 1)$ streetlights for the second street and $16 = 2(7 + 1)$ for the third one. In total, that makes $50$.

Sebastiano
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Tzimmo
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  • Why do you use $(5 + 1)$ or $(7+1)$ or $(10+1)$? Why add you $+1$? – Sebastiano Dec 16 '23 at 21:44
  • 1 is for the streetlights at 0m. Then add a pair of streetlights after each interval of 30m. – Tzimmo Dec 16 '23 at 21:59
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    @Sebastiano You've got a "fencepost error" as you're not counting the streetlight at the first endpoint. How many fenceposts spaced 3 ft apart do you need to make a 30 ft fence? (Hint: the answer's not 10!) – Jam Dec 16 '23 at 22:01