Grades 7-8 Video Solutions 2021-video 2021 7-8/20

Video Transcription

Question 20. Three villages are connected by paths as shown. From Downend to Uphill, the detour via Middleton is 1 km longer than the direct wiggly path shown.  From Downend to Middleton, the detour via Uphill is 5 km longer than the direct wiggly path.  From Uphill to Middleton, the detour via Downend is 7 km longer than the direct wiggly path.  How long is the shortest of the three direct wiggly paths between the villages?  Let's take a closer look at the three villages and the paths. We can label each of the paths as X, Y, and Z.  Since we know that from Downend to Uphill, or distance X as we've labeled it, the detour via Middleton is 1 km longer than the direct path.  So we can write down the formula X plus 1 is equal to Y plus Z.  We can do the same for Downend to Middleton, where the detour via Uphill is 5 km longer than the direct path. We get Z plus 5 equals X plus Y.  Finally, from Uphill to Middleton, the detour via Downend is 7 km longer. So we get Y plus 7 is equal to X plus Z.  Now we can simplify these formulas a little bit, so that we only have one variable on one side.  This will give us X is equal to Y plus Z minus 1, Z is equal to X plus Y minus 5, and Y is equal to X plus Z minus 7.  If we take a look at our first formula, X is equal to Y plus Z minus 1, we can input Y equals X plus Z minus 7, where the Y is.  And with that, we will get X is equal to X plus Z minus 7 plus Z minus 1. Simplify this to 2Z equals 8, where Z is equal to 4.  Next, we look at this formula, Z equals X plus Y minus 5. Since we know that Z is equal to 4, we can set it as 4 equals X plus Y minus 5.  Next, we simplify this and get 9 is equal to X plus Y.  With this, we take a look at our first formula again, X is equal to Y plus Z minus 1, and we substitute that into our current formula.  That we get 9 is equal to Y plus Z minus 1 plus Y. We can simplify this, since Z is 4, into Y plus 4 minus 1 plus Y is equal to 9.  Or, 6 is equal to 2Y. When we divide both sides, we get Y equals 3.  Now that we know both Z and Y, we can take the first formula and substitute those, and we get X equals 3 plus 4 minus 1, or X equals 6.  Now the question wanted to know, which of these is the shortest, and how long is the shortest? We see that Y is the smallest number, being 3. So our answer will be C, 3 kilometers.

Video Summary

Three villages are connected by paths labeled X, Y, and Z. Each direct path has an equivalent detour path with known distance differences. The problem requires finding the shortest direct path. By setting up equations based on the detour information and solving them, the respective direct path lengths are determined as X = 6 km, Y = 3 km, and Z = 4 km. The shortest path is Y, measuring 3 kilometers. Thus, the shortest direct wiggly path between the villages is 3 kilometers.

Keywords

villages

paths

shortest path

distance

equations