Problem

Match units before comparing path lengths

A map connects Home, Bookstore, Library, Post Office, School, and Bank by roads with given lengths. We must find every route from Home to Library that never repeats a place, then compare the shortest route with the longest route and report how much shorter the shortest one is, in km and m.

Measurement & data

Your answer

How to solve

Strategy Make a Systematic List — We list every simple path from Home to Library on the map, total each in meters (after converting the km-and-m labels), then pick the smallest and largest totals and subtract. The map (diagram) keeps the connections clear, and unit-matching to meters makes the sums and the final difference clean.

1STEP 1

Convert all labels to meters

Rewrite the mixed labels in meters: 1 km 100 m = 1100 m, 1 km 250 m = 1250 m, 1 km = 1000 m. The rest are already in meters.

1\text{ km }100\text{ m} = 1100\text{ m}, 1\text{ km }250\text{ m} = 1250\text{ m}, 1\text{ km} = 1000\text{ m}

2STEP 2

List the simple routes from Home to Library

The only no-repeat routes are (a) Home-Bookstore-Library, (b) Home-Post Office-Bank-Library, (c) Home-School-Bank-Library.

\text{(a) Home}\to\text{Bookstore}\to\text{Library}; \text{(b) Home}\to\text{Post Office}\to\text{Bank}\to\text{Library}; \text{(c) Home}\to\text{School}\to\text{Bank}\to\text{Library}

3STEP 3

Total each route

Route (a): 1300 + 1100 = 2400 m. Route (b): 850 + 540 + 1840 = 3230 m. Route (c): 1250 + 1000 + 1840 = 4090 m.

a=1300+1100=2400; b=850+540+1840=3230; c=1250+1000+1840=4090

4STEP 4

Compare shortest and longest

The shortest route is (a) at 2400 m; the longest is (c) at 4090 m. Subtract to find how much shorter the shortest is: 4090 - 2400 = 1690 m, which is 1 km 690 m.

4090 - 2400 = 1690 \text{ m} = 1 \text{ km } 690 \text{ m}

Answer

1 km 690 m

4090 m − 2400 m = 1690 m = 1 km 690 m

The three route totals (2400, 3230, 4090 m) are all in a sensible few-kilometer range for a neighborhood map, and the shortest is the direct two-road path while the longest takes the most roads — exactly what we'd expect. The difference 1690 m is less than the longest route, so it is reasonable, and 1690 m correctly regroups as 1 km 690 m.

Takeaway

Match every road to the same unit, list every no-repeat path, then subtract the shortest from the longest — careful listing beats guessing!

Convert all labels to meters
List the simple routes from Home to Library
Total each route
Compare shortest and longest

Where next?

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