The Ants

Geometry Level 1

Two ants start to move from vertex $A$ at the same speed at the same time. One moves along the square with side length $3$ , and the other along the rectangle with dimensions $3\times6$ . The rectangle and square do not intersect other than at $A$ .

Find the minimum distance that each ant must cover before they meet again with each other.

6 12 18 36

4 solutions

Chung Kevin
Dec 2, 2016

The only possible meeting point is back at A.

The first ant comes back to A after every 12 units.
The second ant comes back to A after every 18 units.

Hence, the first time that they meet again will be the least common multiple of these 2 terms, which is 36.

Steven Chase
Dec 1, 2016

It's the least common multiple of 12 and 18 (the two perimeters). Answer is 36

It doesn't state the ants can't turn around. Therefore if ant 1 travels 3 and comes back 3 equaling 6. Then ant 2 travels 6 and back 6 equaling 12. Then 12+6=18.

Kevin O'Brien - 4 years, 6 months ago

Jun Soriao
Jan 29, 2017

The first ant, Emmet, crawling along the blue line of the square shall have covered 4 sides or for a total of 12 points; while Emmet 2, crawling along as well along the pink line of the rectangle shall have covered 4 sides as well or for a total of 24 points. The pink triangle is twice the size of the blue square or 24 points or for a total of 36 points. Therefore, it will take 36 points before they see each other again.

what are the points as you defined

Ashish Srivastava - 4 years, 4 months ago

Roy Bertoldo
Jan 28, 2017

Red perimeter = 18

Blue perimeter = 12

LCM = Distance traveled when they meet = 36

The Ants

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