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If it is 143 miles from New York to Albany, 147 from Albany to Syracuse and 80 miles from Syracuse to Rochester, how far is it from New York to Rochester?

This question assumes you are travelling along Interstates I-97 and I-90, north along the Hudson to Albany and then east through Syracuse to Rochester, NOT in a straight line. Therefore just add the mileages of the three links:
143 + 147 + 80 = 370 miles.

The question should have made that clearer. The route described IS the quickest, but not the shortest, way to get to New York City to Rochester by car, even though it is far from a straight line. For proof, see http://www.sitesatlas.com/Maps/Maps/NY1.htm

I did the "shorter" way through Binghamton once, and regretted the choice.

The route described is via I-87 and I-90, not I-97 and I-90.

Sorry...I should have clarified. This is a geometry question that I could not figure out.
Thanks!

From the way the problem was stated, you cannot calculate the distance from New York City to Rochester "as the crow flies" from the highway mileages you were given. The route is not a straight line. All you can do is add the lengths of each segment of the route described, which happens to be the fastest way to drive there. If you want to get into the geometry of the problem, i.e. the "air miles", you need latitudes and longitudes of the beginning and end points.

The actual mileage from New York City to Rochester NY by air is 256 miles. See
http://www.indo.com/cgi-bin/dist?place1=New+York%2C+NY&place2=Rochester%2C+NY

You need to add the miles for each part of the trip between New York and Rochester:

143 + 147 + 80 = ?

That answer gives you the total miles between the two cities.

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