Match units before comparing path lengths

4.MD.A.14.MD.A.22.MD.A.4 · adapt · grade 4

Archetype: Length as Sum of Parts with Unit Matching · step in a 7-type progression

On the map, how many kilometers and meters shorter is the shortest route from home to the library than the longest way around? (You may not pass through the same place twice.)

The map below connects Home, the Bookstore, the Library, the Post Office, the School, and the Bank by roads. The road distances are:

Home ~ Bookstore: $1300\ \text{m}$
Bookstore ~ Library: $1\ \text{km}\ 100\ \text{m}$
Home ~ Post Office: $850\ \text{m}$
Post Office ~ Bank: $540\ \text{m}$
Bank ~ Library: $1840\ \text{m}$
Home ~ School: $1\ \text{km}\ 250\ \text{m}$
School ~ Bank: $1\ \text{km}$

Show solution

Understand

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.

Givens

Home~Bookstore: 1300 m
Bookstore~Library: 1 km 100 m = 1100 m
Home~Post Office: 850 m
Post Office~Bank: 540 m
Bank~Library: 1840 m
Home~School: 1 km 250 m = 1250 m
School~Bank: 1 km = 1000 m

Unknowns

The length of the shortest Home-to-Library route
The length of the longest Home-to-Library route
The difference between them in km and m

Constraints

A route may not pass through the same place twice
All lengths must be in the same unit (meters) before adding or comparing

Plan

#2 Make a Systematic List · also uses: #1 Draw a Diagram#8 Analyze the Units

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.

Execute

#8 Analyze the Units 4.MD.A.1

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}

Putting every road in meters lets us add and compare without mixing units.

#2 Make a Systematic List 2.MD.A.4

Library connects only to Bookstore and to Bank. Reaching Bookstore from Home is direct; reaching Bank from Home goes through Post Office or through School. Listing them: (a) Home-Bookstore-Library, (b) Home-Post Office-Bank-Library, (c) Home-School-Bank-Library. No other route avoids repeating a place.

\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}

Because Library has just two neighbors, every route must arrive via Bookstore or via Bank, which makes the list short and complete.

#1 Draw a Diagram 4.MD.A.2

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

Adding the road lengths along each path gives that path's total distance.

#8 Analyze the Units 4.MD.A.2

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}

The difference of the biggest and smallest totals answers 'how much shorter', and 1690 m regroups into 1 km and 690 m.

Answer: 1 km 690 m

Review

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.

Eliminate possibilities (tool 3): since Library only touches Bookstore and Bank, the direct Home-Bookstore-Library path is clearly shortest and the School-Bank detour (longest single roads) is clearly longest, so only those two need exact totals before subtracting.

Standards · min grade 4

4.MD.A.1 Know relative sizes of measurement units and convert larger to smaller units — Converting the km-and-m road labels into meters
4.MD.A.2 Solve word problems involving distances, time, liquid volumes, and money — Totaling each route and subtracting to compare distances
2.MD.A.4 Measure to determine how much longer one object is than another — Comparing the longest and shortest routes to find how much shorter

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