Grades 5-6 Video Solutions 2022-2022_5-6

Video Transcription

Question number 28. Four villages, A, B, C, and D, lie along a road, in that order.  The distance between neighboring villages is 10 kilometers. There are 10 students who live in village A, 20 students who live in village B, 30 students who live in village C, and 40 students who live in village D.  The villagers want to build a school so that the total distance traveled by the students when going to school is as small as possible. Where should they build the school?  So the fastest way to solve this problem is just to brute force, and that is done by multiplying the number of students from each village by the number of kilometers they would need to travel, and then doing this for each option.  You can also do this algebraically, but for the purpose of the competition, that would take much longer, just because there are quite a few different possibilities that you just need to work out in the calculations.  With that being said, getting to it. Option A, if they're in A, then first we note that the students in A don't have to travel at all, so there's a hidden 10 times 0 kilometers over here.  Obviously that's just 0, it doesn't matter. So starting from B, 20 times 10, 30 times 20, 40 times 30, it gets us to 2,000 kilometers.  Option B, so now the students from A are back, 10 times 10, students from B are gone this time, 30 times 10, 40 times 20, we get to 1,200, so definitely a little less distance.  If they're between B and C, we add this extra 5 kilometers, because the distance between neighboring villages is 10 kilometers, so if they're halfway in between, it's going to be a 5 kilometer addition.  So 10 times 15, 20 times 5, 30 times 5, 40 times 15, it gets us to 1,000 kilometers.  Option C, 10 times 20, 20 times 10, 40 times 10, we get 800 kilometers, and then option D, 10 times 30, 20 times 10, 30 times 10, it gets us to 1,000 kilometers.  So the shortest distance is 800 kilometers, so the answer is C.  This is a typo over here, but this is a great point to remember over here that the answer choice is C, but the actual answer choice is D in C.  Got tripped up on this, so make sure to remember exactly what your value is and correctly mark that on your answer sheet.

Video Summary

To minimize the total distance traveled by students from four villages (A, B, C, and D) when attending a new school, a calculation determines the optimal location is village C, resulting in the shortest travel distance of 800 kilometers. Each village is 10 km apart along a road, with differing student populations. By calculating travel distances for each potential school location with student numbers from each village, C provides the minimum total travel distance despite an initial mention of a typo regarding answer choice labeling, emphasizing the importance of accurate answer marking.

Keywords

optimal location

travel distance

village C

student populations

school placement