Practice · Total length is the sum of its parts · Sensim Math

Variant 1 answer: 2 km 180 m

If the distance from Gyuri's house to the fire station is $3 km 50 m$ , 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 $1 km 350 m$ , and the distance from the bank to the community center is $480 m$ .

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 350 m, the bank-to-community-center distance is 480 m, and the house-to-fire-station distance is 3 km 50 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 350 m = 1350 m.
Bank to community center (B to C) = 480 m.
House to fire station (A to D) = 3 km 50 m = 3050 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 350 m = 1350 m, B to C = 480 m, A to D = 3 km 50 m = 3050 m.

1 \text{ km } 350 \text{ m} = 1350 \text{ m}, \quad 3 \text{ km } 50 \text{ m} = 3050 \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 1350 m and B to C is 480 m, the part from the house to the bank is the difference.

1350 - 480 = 870 \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 = 3050 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.

3050 - 870 = 2180 \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 2180 m using 1000 m = 1 km: 2180 = 2000 + 180, so 2180 m = 2 km 180 m.

2180 \text{ m} = 2 \text{ km } 180 \text{ m}

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

Answer: 2 km 180 m

Review

Check by another route: center-to-fire-station = A to D minus A to C = 3050 - 1350 = 1700 m, and bank-to-center = 480 m, so bank-to-fire-station = 480 + 1700 = 2180 m = 2 km 180 m. It also must be less than the full 3 km 50 m, and 2 km 180 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 (1700 m) and C-to-B (480 m), reaching the same 2180 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!

Variant 2 answer: 1 km 220 m

If the distance from Gyuri's house to the fire station is $2 km 10 m$ , 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 $1 km 80 m$ , and the distance from the bank to the community center is $290 m$ .

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 80 m, the bank-to-community-center distance is 290 m, and the house-to-fire-station distance is 2 km 10 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 80 m = 1080 m.
Bank to community center (B to C) = 290 m.
House to fire station (A to D) = 2 km 10 m = 2010 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 80 m = 1080 m, B to C = 290 m, A to D = 2 km 10 m = 2010 m.

1 \text{ km } 80 \text{ m} = 1080 \text{ m}, \quad 2 \text{ km } 10 \text{ m} = 2010 \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 1080 m and B to C is 290 m, the part from the house to the bank is the difference.

1080 - 290 = 790 \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 = 2010 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.

2010 - 790 = 1220 \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 1220 m using 1000 m = 1 km: 1220 = 1000 + 220, so 1220 m = 1 km 220 m.

1220 \text{ m} = 1 \text{ km } 220 \text{ m}

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

Answer: 1 km 220 m

Review

Check by another route: center-to-fire-station = A to D minus A to C = 2010 - 1080 = 930 m, and bank-to-center = 290 m, so bank-to-fire-station = 290 + 930 = 1220 m = 1 km 220 m. It also must be less than the full 2 km 10 m, and 1 km 220 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 (930 m) and C-to-B (290 m), reaching the same 1220 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!

Variant 3 answer: 1 km 900 m

If the distance from Gyuri's house to the fire station is $2 km 600 m$ , 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 $1 km 200 m$ , and the distance from the bank to the community center is $500 m$ .

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 200 m, the bank-to-community-center distance is 500 m, and the house-to-fire-station distance is 2 km 600 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 200 m = 1200 m.
Bank to community center (B to C) = 500 m.
House to fire station (A to D) = 2 km 600 m = 2600 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 200 m = 1200 m, B to C = 500 m, A to D = 2 km 600 m = 2600 m.

1 \text{ km } 200 \text{ m} = 1200 \text{ m}, \quad 2 \text{ km } 600 \text{ m} = 2600 \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 1200 m and B to C is 500 m, the part from the house to the bank is the difference.

1200 - 500 = 700 \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 = 2600 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.

2600 - 700 = 1900 \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 1900 m using 1000 m = 1 km: 1900 = 1000 + 900, so 1900 m = 1 km 900 m.

1900 \text{ m} = 1 \text{ km } 900 \text{ m}

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

Answer: 1 km 900 m

Review

Check by another route: center-to-fire-station = A to D minus A to C = 2600 - 1200 = 1400 m, and bank-to-center = 500 m, so bank-to-fire-station = 500 + 1400 = 1900 m = 1 km 900 m. It also must be less than the full 2 km 600 m, and 1 km 900 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 (1400 m) and C-to-B (500 m), reaching the same 1900 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!

Variant 4 answer: 2 km 500 m

If the distance from Gyuri's house to the fire station is $3 km 800 m$ , 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 $1 km 900 m$ , and the distance from the bank to the community center is $600 m$ .

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 900 m, the bank-to-community-center distance is 600 m, and the house-to-fire-station distance is 3 km 800 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 900 m = 1900 m.
Bank to community center (B to C) = 600 m.
House to fire station (A to D) = 3 km 800 m = 3800 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 900 m = 1900 m, B to C = 600 m, A to D = 3 km 800 m = 3800 m.

1 \text{ km } 900 \text{ m} = 1900 \text{ m}, \quad 3 \text{ km } 800 \text{ m} = 3800 \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 1900 m and B to C is 600 m, the part from the house to the bank is the difference.

1900 - 600 = 1300 \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 = 3800 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.

3800 - 1300 = 2500 \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 2500 m using 1000 m = 1 km: 2500 = 2000 + 500, so 2500 m = 2 km 500 m.

2500 \text{ m} = 2 \text{ km } 500 \text{ m}

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

Answer: 2 km 500 m

Review

Check by another route: center-to-fire-station = A to D minus A to C = 3800 - 1900 = 1900 m, and bank-to-center = 600 m, so bank-to-fire-station = 600 + 1900 = 2500 m = 2 km 500 m. It also must be less than the full 3 km 800 m, and 2 km 500 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 (1900 m) and C-to-B (600 m), reaching the same 2500 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!

Variant 5 answer: 2 km 950 m

If the distance from Gyuri's house to the fire station is $4 km 100 m$ , 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 $2 km 100 m$ , and the distance from the bank to the community center is $950 m$ .

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 2 km 100 m, the bank-to-community-center distance is 950 m, and the house-to-fire-station distance is 4 km 100 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) = 2 km 100 m = 2100 m.
Bank to community center (B to C) = 950 m.
House to fire station (A to D) = 4 km 100 m = 4100 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 = 2 km 100 m = 2100 m, B to C = 950 m, A to D = 4 km 100 m = 4100 m.

2 \text{ km } 100 \text{ m} = 2100 \text{ m}, \quad 4 \text{ km } 100 \text{ m} = 4100 \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 2100 m and B to C is 950 m, the part from the house to the bank is the difference.

2100 - 950 = 1150 \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 = 4100 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.

4100 - 1150 = 2950 \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 2950 m using 1000 m = 1 km: 2950 = 2000 + 950, so 2950 m = 2 km 950 m.

2950 \text{ m} = 2 \text{ km } 950 \text{ m}

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

Answer: 2 km 950 m

Review

Check by another route: center-to-fire-station = A to D minus A to C = 4100 - 2100 = 2000 m, and bank-to-center = 950 m, so bank-to-fire-station = 950 + 2000 = 2950 m = 2 km 950 m. It also must be less than the full 4 km 100 m, and 2 km 950 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 (2000 m) and C-to-B (950 m), reaching the same 2950 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!

Variant 6 answer: 2 km 20 m

If the distance from Gyuri's house to the fire station is $2 km 900 m$ , 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 $1 km 620 m$ , and the distance from the bank to the community center is $740 m$ .

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 620 m, the bank-to-community-center distance is 740 m, and the house-to-fire-station distance is 2 km 900 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 620 m = 1620 m.
Bank to community center (B to C) = 740 m.
House to fire station (A to D) = 2 km 900 m = 2900 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 620 m = 1620 m, B to C = 740 m, A to D = 2 km 900 m = 2900 m.

1 \text{ km } 620 \text{ m} = 1620 \text{ m}, \quad 2 \text{ km } 900 \text{ m} = 2900 \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 1620 m and B to C is 740 m, the part from the house to the bank is the difference.

1620 - 740 = 880 \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 = 2900 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.

2900 - 880 = 2020 \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 2020 m using 1000 m = 1 km: 2020 = 2000 + 20, so 2020 m = 2 km 20 m.

2020 \text{ m} = 2 \text{ km } 20 \text{ m}

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

Answer: 2 km 20 m

Review

Check by another route: center-to-fire-station = A to D minus A to C = 2900 - 1620 = 1280 m, and bank-to-center = 740 m, so bank-to-fire-station = 740 + 1280 = 2020 m = 2 km 20 m. It also must be less than the full 2 km 900 m, and 2 km 20 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 (1280 m) and C-to-B (740 m), reaching the same 2020 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!

Variant 7 answer: 2 km 550 m

If the distance from Gyuri's house to the fire station is $3 km 500 m$ , 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 $1 km 750 m$ , and the distance from the bank to the community center is $800 m$ .

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 750 m, the bank-to-community-center distance is 800 m, and the house-to-fire-station distance is 3 km 500 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 750 m = 1750 m.
Bank to community center (B to C) = 800 m.
House to fire station (A to D) = 3 km 500 m = 3500 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 750 m = 1750 m, B to C = 800 m, A to D = 3 km 500 m = 3500 m.

1 \text{ km } 750 \text{ m} = 1750 \text{ m}, \quad 3 \text{ km } 500 \text{ m} = 3500 \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 1750 m and B to C is 800 m, the part from the house to the bank is the difference.

1750 - 800 = 950 \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 = 3500 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.

3500 - 950 = 2550 \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 2550 m using 1000 m = 1 km: 2550 = 2000 + 550, so 2550 m = 2 km 550 m.

2550 \text{ m} = 2 \text{ km } 550 \text{ m}

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

Answer: 2 km 550 m

Review

Check by another route: center-to-fire-station = A to D minus A to C = 3500 - 1750 = 1750 m, and bank-to-center = 800 m, so bank-to-fire-station = 800 + 1750 = 2550 m = 2 km 550 m. It also must be less than the full 3 km 500 m, and 2 km 550 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 (1750 m) and C-to-B (800 m), reaching the same 2550 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!

Variant 8 answer: 1 km 550 m

If the distance from Gyuri's house to the fire station is $2 km 750 m$ , 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 $1 km 550 m$ , and the distance from the bank to the community center is $350 m$ .

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 550 m, the bank-to-community-center distance is 350 m, and the house-to-fire-station distance is 2 km 750 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 550 m = 1550 m.
Bank to community center (B to C) = 350 m.
House to fire station (A to D) = 2 km 750 m = 2750 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 550 m = 1550 m, B to C = 350 m, A to D = 2 km 750 m = 2750 m.

1 \text{ km } 550 \text{ m} = 1550 \text{ m}, \quad 2 \text{ km } 750 \text{ m} = 2750 \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 1550 m and B to C is 350 m, the part from the house to the bank is the difference.

1550 - 350 = 1200 \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 = 2750 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.

2750 - 1200 = 1550 \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 1550 m using 1000 m = 1 km: 1550 = 1000 + 550, so 1550 m = 1 km 550 m.

1550 \text{ m} = 1 \text{ km } 550 \text{ m}

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

Answer: 1 km 550 m

Review

Check by another route: center-to-fire-station = A to D minus A to C = 2750 - 1550 = 1200 m, and bank-to-center = 350 m, so bank-to-fire-station = 350 + 1200 = 1550 m = 1 km 550 m. It also must be less than the full 2 km 750 m, and 1 km 550 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 (1200 m) and C-to-B (350 m), reaching the same 1550 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!

Variant 9 answer: 2 km 390 m

If the distance from Gyuri's house to the fire station is $3 km 200 m$ , 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 $1 km 480 m$ , and the distance from the bank to the community center is $670 m$ .

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!

Variant 10 answer: 3 km 300 m

If the distance from Gyuri's house to the fire station is $4 km 600 m$ , 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 $2 km 400 m$ , and the distance from the bank to the community center is $1 km 100 m$ .

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 2 km 400 m, the bank-to-community-center distance is 1 km 100 m, and the house-to-fire-station distance is 4 km 600 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) = 2 km 400 m = 2400 m.
Bank to community center (B to C) = 1 km 100 m.
House to fire station (A to D) = 4 km 600 m = 4600 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 = 2 km 400 m = 2400 m, B to C = 1100 m, A to D = 4 km 600 m = 4600 m.

2 \text{ km } 400 \text{ m} = 2400 \text{ m}, \quad 4 \text{ km } 600 \text{ m} = 4600 \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 2400 m and B to C is 1100 m, the part from the house to the bank is the difference.

2400 - 1100 = 1300 \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 = 4600 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.

4600 - 1300 = 3300 \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 3300 m using 1000 m = 1 km: 3300 = 3000 + 300, so 3300 m = 3 km 300 m.

3300 \text{ m} = 3 \text{ km } 300 \text{ m}

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

Answer: 3 km 300 m

Review

Check by another route: center-to-fire-station = A to D minus A to C = 4600 - 2400 = 2200 m, and bank-to-center = 1100 m, so bank-to-fire-station = 1100 + 2200 = 3300 m = 3 km 300 m. It also must be less than the full 4 km 600 m, and 3 km 300 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 (2200 m) and C-to-B (1100 m), reaching the same 3300 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!

Total length is the sum of its parts · 10 practice problems

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