Total length is the sum of its parts

4.MD.A.14.MD.A.22.MD.B.5 · adapt · grade 4

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

If the distance from Gyuri's house to the fire station is $3\ \mathrm{km}\ 200\ \mathrm{m}$ , then what is the distance from the bank to the fire station, in km and m?

(Figure) Gyuri's house, the bank, the community center, and the fire station lie on a straight line in this order. The distance from Gyuri's house to the community center is $1\ \mathrm{km}\ 480\ \mathrm{m}$ , and the distance from the bank to the community center is $670\ \mathrm{m}$ .

Show solution

Understand

Four places lie on a straight line in order: Gyuri's house, the bank, the community center, and the fire station. The house-to-community-center distance is 1 km 480 m, the bank-to-community-center distance is 670 m, and the house-to-fire-station distance is 3 km 200 m. I need the distance from the bank to the fire station in km and m.

Givens

Order along the line: house (A), bank (B), community center (C), fire station (D).
House to community center (A to C) = 1 km 480 m = 1480 m.
Bank to community center (B to C) = 670 m.
House to fire station (A to D) = 3 km 200 m = 3200 m.

Unknowns

The distance from the bank to the fire station (B to D), in km and m.

Constraints

1 km = 1000 m (convert before adding or subtracting).
Because the points are in order, segment lengths add along the line.

Plan

#1 Draw a Diagram · also uses: #7 Identify Subproblems

The figure is a number line of four ordered points, so drawing/reading it shows how the segments fit together. I break the bank-to-fire-station distance into pieces I can find: first locate the bank from the house, then take the rest of the way to the fire station. Working in meters avoids unit mistakes until the final conversion.

Execute

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

From the figure the points are in order A (house), B (bank), C (center), D (fire station). Write each known distance in meters: A to C = 1 km 480 m = 1480 m, B to C = 670 m, A to D = 3 km 200 m = 3200 m.

1 \text{ km } 480 \text{ m} = 1480 \text{ m}, \quad 3 \text{ km } 200 \text{ m} = 3200 \text{ m}

Converting km+m into all meters lets me add and subtract distances with plain whole numbers.

#7 Identify Subproblems 2.MD.B.5

The bank (B) sits between the house (A) and the center (C). Since A to C is 1480 m and B to C is 670 m, the part from the house to the bank is the difference.

1480 - 670 = 810 \text{ m}

On a line, the near part equals the whole stretch minus the far part.

#7 Identify Subproblems 4.MD.A.2

The whole house-to-fire-station distance (A to D = 3200 m) is the house-to-bank part plus the bank-to-fire-station part. So the bank-to-fire-station distance is the total minus the house-to-bank piece.

3200 - 810 = 2390 \text{ m}

Removing the front piece (house to bank) from the full line leaves exactly the bank-to-fire-station stretch.

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

Regroup 2390 m using 1000 m = 1 km: 2390 = 2000 + 390, so 2390 m = 2 km 390 m.

2390 \text{ m} = 2 \text{ km } 390 \text{ m}

Every 1000 m is one kilometer, so trade thousands of meters for km to report the distance.

Answer: 2 km 390 m

Review

Check by another route: center-to-fire-station = A to D minus A to C = 3200 - 1480 = 1720 m, and bank-to-center = 670 m, so bank-to-fire-station = 670 + 1720 = 2390 m = 2 km 390 m. It also must be less than the full 3 km 200 m, and 2 km 390 m is - both checks pass.

Work backwards / add the parts (tool 11): start at the fire station and move toward the bank by adding C-to-D (1720 m) and C-to-B (670 m), reaching the same 2390 m.

Standards · min grade 4

4.MD.A.1 Know relative sizes of measurement units and convert larger to smaller units — Converting between km+m and meters before and after the arithmetic.
4.MD.A.2 Solve word problems involving distances, time, liquid volumes, and money — Subtracting the house-to-bank piece from the total to find the bank-to-fire-station distance.
2.MD.B.5 Solve word problems involving lengths using same units — Finding the house-to-bank length as a difference of same-unit lengths on the line.

💡 Lay the places on a line, turn km+m into meters, add and subtract the parts, then turn meters back into km+m!