← Overlap shrinks the total length · Overlap Reduces the Total

Overlap shrinks the total length · 10 practice problems

2.MD.B.53.NBT.A.2

Generated variants — 10

Freshly produced from the archetype’s parameters — problem, figure, and solution derived together.

Variant 1 easy answer: 305 cm

What is the length from $A$ to $D$ , in centimeters?

Four points $A$ , $B$ , $C$ , $D$ lie on a straight line in that order. The length from $A$ to $C$ is $144\ \text{cm}$ , the length from $B$ to $D$ is $256\ \text{cm}$ , and the overlapping part from $B$ to $C$ is $95\ \text{cm}$ .

Show solution

1 · Understandwhat's really being asked

Four points A, B, C, D sit on a line in that order. I know A-to-C (144 cm), B-to-D (256 cm), and the middle part B-to-C that both spans share (95 cm). I need the whole length from A to D.

Givens

Points A, B, C, D lie on one straight line in that order.
Length A to C is 144 cm.
Length B to D is 256 cm.
The overlapping part B to C is 95 cm.

Unknowns

The length from A to D, in centimeters.

Constraints

Lengths add along the line: AD = AB + BC + CD.
All measurements are in the same unit (cm).

2 · Planchoose the strategy

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

Adding span A-C and span B-D counts the overlap B-C twice, so AD = AC + BD - BC. A picture makes the double-counted middle visible.

3 · Execute3 carry out the plan

1See why the overlap is counted twice

#1 Draw a Diagram 2.MD.B.5

Span A-to-C covers AB and BC; span B-to-D covers BC and CD. Adding both spans counts the middle piece BC two times.

AC + BD = (AB + BC) + (BC + CD)

Drawing the two bars on one line shows the shared middle plainly.

2Add the two spans

#7 Identify Subproblems 3.NBT.A.2

First add the lengths A-to-C and B-to-D together.

144 + 256 = 400

Three-digit addition is the regrouping skill practiced in Grade 3.

3Subtract the overlap once

#7 Identify Subproblems 3.NBT.A.2

Because B-to-C was counted twice, take it away once to get the true length from A to D.

400 - 95 = 305

Removing the extra copy of the overlap is a single subtraction.

Answer: 305 cm

4 · Reviewdoes it hold up?

The answer is in cm. AD should be longer than either single span (144 and 256) but shorter than their full sum (400) because the bars overlap; 305 cm sits sensibly between them.

Another way: Work piece by piece: AB = 144 - 95 = 49, CD = 256 - 95 = 161, then AD = 49 + 95 + 161 = 305 cm, the same result.

Standardsmin grade 3

2.MD.B.5 Solve word problems involving lengths using same units — Modeling the overlapping length spans on a single line.
3.NBT.A.2 Fluently add and subtract within 1000 — Adding the two spans and subtracting the double-counted overlap.

💡Takeaway. When two lengths overlap, add them and subtract the shared middle once -- just Grade 3 add-and-subtract!

Variant 2 easy answer: 377 cm

What is the length from $A$ to $D$ , in centimeters?

Four points $A$ , $B$ , $C$ , $D$ lie on a straight line in that order. The length from $A$ to $C$ is $190\ \text{cm}$ , the length from $B$ to $D$ is $275\ \text{cm}$ , and the overlapping part from $B$ to $C$ is $88\ \text{cm}$ .

Show solution

1 · Understandwhat's really being asked

Four points A, B, C, D sit on a line in that order. I know A-to-C (190 cm), B-to-D (275 cm), and the middle part B-to-C that both spans share (88 cm). I need the whole length from A to D.

Givens

Points A, B, C, D lie on one straight line in that order.
Length A to C is 190 cm.
Length B to D is 275 cm.
The overlapping part B to C is 88 cm.

Unknowns

The length from A to D, in centimeters.

Constraints

Lengths add along the line: AD = AB + BC + CD.
All measurements are in the same unit (cm).

2 · Planchoose the strategy

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

Adding span A-C and span B-D counts the overlap B-C twice, so AD = AC + BD - BC. A picture makes the double-counted middle visible.

3 · Execute3 carry out the plan

1See why the overlap is counted twice

#1 Draw a Diagram 2.MD.B.5

Span A-to-C covers AB and BC; span B-to-D covers BC and CD. Adding both spans counts the middle piece BC two times.

AC + BD = (AB + BC) + (BC + CD)

Drawing the two bars on one line shows the shared middle plainly.

2Add the two spans

#7 Identify Subproblems 3.NBT.A.2

First add the lengths A-to-C and B-to-D together.

190 + 275 = 465

Three-digit addition is the regrouping skill practiced in Grade 3.

3Subtract the overlap once

#7 Identify Subproblems 3.NBT.A.2

Because B-to-C was counted twice, take it away once to get the true length from A to D.

465 - 88 = 377

Removing the extra copy of the overlap is a single subtraction.

Answer: 377 cm

4 · Reviewdoes it hold up?

The answer is in cm. AD should be longer than either single span (190 and 275) but shorter than their full sum (465) because the bars overlap; 377 cm sits sensibly between them.

Another way: Work piece by piece: AB = 190 - 88 = 102, CD = 275 - 88 = 187, then AD = 102 + 88 + 187 = 377 cm, the same result.

Standardsmin grade 3

2.MD.B.5 Solve word problems involving lengths using same units — Modeling the overlapping length spans on a single line.
3.NBT.A.2 Fluently add and subtract within 1000 — Adding the two spans and subtracting the double-counted overlap.

💡Takeaway. When two lengths overlap, add them and subtract the shared middle once -- just Grade 3 add-and-subtract!

Variant 3 easy answer: 424 cm

What is the length from $A$ to $D$ , in centimeters?

Four points $A$ , $B$ , $C$ , $D$ lie on a straight line in that order. The length from $A$ to $C$ is $208\ \text{cm}$ , the length from $B$ to $D$ is $333\ \text{cm}$ , and the overlapping part from $B$ to $C$ is $117\ \text{cm}$ .

Show solution

1 · Understandwhat's really being asked

Four points A, B, C, D sit on a line in that order. I know A-to-C (208 cm), B-to-D (333 cm), and the middle part B-to-C that both spans share (117 cm). I need the whole length from A to D.

Givens

Points A, B, C, D lie on one straight line in that order.
Length A to C is 208 cm.
Length B to D is 333 cm.
The overlapping part B to C is 117 cm.

Unknowns

The length from A to D, in centimeters.

Constraints

Lengths add along the line: AD = AB + BC + CD.
All measurements are in the same unit (cm).

2 · Planchoose the strategy

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

Adding span A-C and span B-D counts the overlap B-C twice, so AD = AC + BD - BC. A picture makes the double-counted middle visible.

3 · Execute3 carry out the plan

1See why the overlap is counted twice

#1 Draw a Diagram 2.MD.B.5

Span A-to-C covers AB and BC; span B-to-D covers BC and CD. Adding both spans counts the middle piece BC two times.

AC + BD = (AB + BC) + (BC + CD)

Drawing the two bars on one line shows the shared middle plainly.

2Add the two spans

#7 Identify Subproblems 3.NBT.A.2

First add the lengths A-to-C and B-to-D together.

208 + 333 = 541

Three-digit addition is the regrouping skill practiced in Grade 3.

3Subtract the overlap once

#7 Identify Subproblems 3.NBT.A.2

Because B-to-C was counted twice, take it away once to get the true length from A to D.

541 - 117 = 424

Removing the extra copy of the overlap is a single subtraction.

Answer: 424 cm

4 · Reviewdoes it hold up?

The answer is in cm. AD should be longer than either single span (208 and 333) but shorter than their full sum (541) because the bars overlap; 424 cm sits sensibly between them.

Another way: Work piece by piece: AB = 208 - 117 = 91, CD = 333 - 117 = 216, then AD = 91 + 117 + 216 = 424 cm, the same result.

Standardsmin grade 3

2.MD.B.5 Solve word problems involving lengths using same units — Modeling the overlapping length spans on a single line.
3.NBT.A.2 Fluently add and subtract within 1000 — Adding the two spans and subtracting the double-counted overlap.

💡Takeaway. When two lengths overlap, add them and subtract the shared middle once -- just Grade 3 add-and-subtract!

Variant 4 medium answer: 545 cm

What is the length from $A$ to $D$ , in centimeters?

Four points $A$ , $B$ , $C$ , $D$ lie on a straight line in that order. The length from $A$ to $C$ is $360\ \text{cm}$ , the length from $B$ to $D$ is $360\ \text{cm}$ , and the overlapping part from $B$ to $C$ is $175\ \text{cm}$ .

Show solution

1 · Understandwhat's really being asked

Four points A, B, C, D sit on a line in that order. I know A-to-C (360 cm), B-to-D (360 cm), and the middle part B-to-C that both spans share (175 cm). I need the whole length from A to D.

Givens

Points A, B, C, D lie on one straight line in that order.
Length A to C is 360 cm.
Length B to D is 360 cm.
The overlapping part B to C is 175 cm.

Unknowns

The length from A to D, in centimeters.

Constraints

Lengths add along the line: AD = AB + BC + CD.
All measurements are in the same unit (cm).

2 · Planchoose the strategy

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

Adding span A-C and span B-D counts the overlap B-C twice, so AD = AC + BD - BC. A picture makes the double-counted middle visible.

3 · Execute3 carry out the plan

1See why the overlap is counted twice

#1 Draw a Diagram 2.MD.B.5

Span A-to-C covers AB and BC; span B-to-D covers BC and CD. Adding both spans counts the middle piece BC two times.

AC + BD = (AB + BC) + (BC + CD)

Drawing the two bars on one line shows the shared middle plainly.

2Add the two spans

#7 Identify Subproblems 3.NBT.A.2

First add the lengths A-to-C and B-to-D together.

360 + 360 = 720

Three-digit addition is the regrouping skill practiced in Grade 3.

3Subtract the overlap once

#7 Identify Subproblems 3.NBT.A.2

Because B-to-C was counted twice, take it away once to get the true length from A to D.

720 - 175 = 545

Removing the extra copy of the overlap is a single subtraction.

Answer: 545 cm

4 · Reviewdoes it hold up?

The answer is in cm. AD should be longer than either single span (360 and 360) but shorter than their full sum (720) because the bars overlap; 545 cm sits sensibly between them.

Another way: Work piece by piece: AB = 360 - 175 = 185, CD = 360 - 175 = 185, then AD = 185 + 175 + 185 = 545 cm, the same result.

Standardsmin grade 3

2.MD.B.5 Solve word problems involving lengths using same units — Modeling the overlapping length spans on a single line.
3.NBT.A.2 Fluently add and subtract within 1000 — Adding the two spans and subtracting the double-counted overlap.

💡Takeaway. When two lengths overlap, add them and subtract the shared middle once -- just Grade 3 add-and-subtract!

Variant 5 medium answer: 515 cm

What is the length from $A$ to $D$ , in centimeters?

Four points $A$ , $B$ , $C$ , $D$ lie on a straight line in that order. The length from $A$ to $C$ is $255\ \text{cm}$ , the length from $B$ to $D$ is $380\ \text{cm}$ , and the overlapping part from $B$ to $C$ is $120\ \text{cm}$ .

Show solution

1 · Understandwhat's really being asked

Four points A, B, C, D sit on a line in that order. I know A-to-C (255 cm), B-to-D (380 cm), and the middle part B-to-C that both spans share (120 cm). I need the whole length from A to D.

Givens

Points A, B, C, D lie on one straight line in that order.
Length A to C is 255 cm.
Length B to D is 380 cm.
The overlapping part B to C is 120 cm.

Unknowns

The length from A to D, in centimeters.

Constraints

Lengths add along the line: AD = AB + BC + CD.
All measurements are in the same unit (cm).

2 · Planchoose the strategy

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

Adding span A-C and span B-D counts the overlap B-C twice, so AD = AC + BD - BC. A picture makes the double-counted middle visible.

3 · Execute3 carry out the plan

1See why the overlap is counted twice

#1 Draw a Diagram 2.MD.B.5

Span A-to-C covers AB and BC; span B-to-D covers BC and CD. Adding both spans counts the middle piece BC two times.

AC + BD = (AB + BC) + (BC + CD)

Drawing the two bars on one line shows the shared middle plainly.

2Add the two spans

#7 Identify Subproblems 3.NBT.A.2

First add the lengths A-to-C and B-to-D together.

255 + 380 = 635

Three-digit addition is the regrouping skill practiced in Grade 3.

3Subtract the overlap once

#7 Identify Subproblems 3.NBT.A.2

Because B-to-C was counted twice, take it away once to get the true length from A to D.

635 - 120 = 515

Removing the extra copy of the overlap is a single subtraction.

Answer: 515 cm

4 · Reviewdoes it hold up?

The answer is in cm. AD should be longer than either single span (255 and 380) but shorter than their full sum (635) because the bars overlap; 515 cm sits sensibly between them.

Another way: Work piece by piece: AB = 255 - 120 = 135, CD = 380 - 120 = 260, then AD = 135 + 120 + 260 = 515 cm, the same result.

Standardsmin grade 3

2.MD.B.5 Solve word problems involving lengths using same units — Modeling the overlapping length spans on a single line.
3.NBT.A.2 Fluently add and subtract within 1000 — Adding the two spans and subtracting the double-counted overlap.

💡Takeaway. When two lengths overlap, add them and subtract the shared middle once -- just Grade 3 add-and-subtract!

Variant 6 medium answer: 536 cm

What is the length from $A$ to $D$ , in centimeters?

Four points $A$ , $B$ , $C$ , $D$ lie on a straight line in that order. The length from $A$ to $C$ is $288\ \text{cm}$ , the length from $B$ to $D$ is $417\ \text{cm}$ , and the overlapping part from $B$ to $C$ is $169\ \text{cm}$ .

Show solution

1 · Understandwhat's really being asked

Four points A, B, C, D sit on a line in that order. I know A-to-C (288 cm), B-to-D (417 cm), and the middle part B-to-C that both spans share (169 cm). I need the whole length from A to D.

Givens

Points A, B, C, D lie on one straight line in that order.
Length A to C is 288 cm.
Length B to D is 417 cm.
The overlapping part B to C is 169 cm.

Unknowns

The length from A to D, in centimeters.

Constraints

Lengths add along the line: AD = AB + BC + CD.
All measurements are in the same unit (cm).

2 · Planchoose the strategy

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

Adding span A-C and span B-D counts the overlap B-C twice, so AD = AC + BD - BC. A picture makes the double-counted middle visible.

3 · Execute3 carry out the plan

1See why the overlap is counted twice

#1 Draw a Diagram 2.MD.B.5

Span A-to-C covers AB and BC; span B-to-D covers BC and CD. Adding both spans counts the middle piece BC two times.

AC + BD = (AB + BC) + (BC + CD)

Drawing the two bars on one line shows the shared middle plainly.

2Add the two spans

#7 Identify Subproblems 3.NBT.A.2

First add the lengths A-to-C and B-to-D together.

288 + 417 = 705

Three-digit addition is the regrouping skill practiced in Grade 3.

3Subtract the overlap once

#7 Identify Subproblems 3.NBT.A.2

Because B-to-C was counted twice, take it away once to get the true length from A to D.

705 - 169 = 536

Removing the extra copy of the overlap is a single subtraction.

Answer: 536 cm

4 · Reviewdoes it hold up?

The answer is in cm. AD should be longer than either single span (288 and 417) but shorter than their full sum (705) because the bars overlap; 536 cm sits sensibly between them.

Another way: Work piece by piece: AB = 288 - 169 = 119, CD = 417 - 169 = 248, then AD = 119 + 169 + 248 = 536 cm, the same result.

Standardsmin grade 3

2.MD.B.5 Solve word problems involving lengths using same units — Modeling the overlapping length spans on a single line.
3.NBT.A.2 Fluently add and subtract within 1000 — Adding the two spans and subtracting the double-counted overlap.

💡Takeaway. When two lengths overlap, add them and subtract the shared middle once -- just Grade 3 add-and-subtract!

Variant 7 medium answer: 600 cm

What is the length from $A$ to $D$ , in centimeters?

Four points $A$ , $B$ , $C$ , $D$ lie on a straight line in that order. The length from $A$ to $C$ is $300\ \text{cm}$ , the length from $B$ to $D$ is $450\ \text{cm}$ , and the overlapping part from $B$ to $C$ is $150\ \text{cm}$ .

Show solution

1 · Understandwhat's really being asked

Four points A, B, C, D sit on a line in that order. I know A-to-C (300 cm), B-to-D (450 cm), and the middle part B-to-C that both spans share (150 cm). I need the whole length from A to D.

Givens

Points A, B, C, D lie on one straight line in that order.
Length A to C is 300 cm.
Length B to D is 450 cm.
The overlapping part B to C is 150 cm.

Unknowns

The length from A to D, in centimeters.

Constraints

Lengths add along the line: AD = AB + BC + CD.
All measurements are in the same unit (cm).

2 · Planchoose the strategy

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

Adding span A-C and span B-D counts the overlap B-C twice, so AD = AC + BD - BC. A picture makes the double-counted middle visible.

3 · Execute3 carry out the plan

1See why the overlap is counted twice

#1 Draw a Diagram 2.MD.B.5

Span A-to-C covers AB and BC; span B-to-D covers BC and CD. Adding both spans counts the middle piece BC two times.

AC + BD = (AB + BC) + (BC + CD)

Drawing the two bars on one line shows the shared middle plainly.

2Add the two spans

#7 Identify Subproblems 3.NBT.A.2

First add the lengths A-to-C and B-to-D together.

300 + 450 = 750

Three-digit addition is the regrouping skill practiced in Grade 3.

3Subtract the overlap once

#7 Identify Subproblems 3.NBT.A.2

Because B-to-C was counted twice, take it away once to get the true length from A to D.

750 - 150 = 600

Removing the extra copy of the overlap is a single subtraction.

Answer: 600 cm

4 · Reviewdoes it hold up?

The answer is in cm. AD should be longer than either single span (300 and 450) but shorter than their full sum (750) because the bars overlap; 600 cm sits sensibly between them.

Another way: Work piece by piece: AB = 300 - 150 = 150, CD = 450 - 150 = 300, then AD = 150 + 150 + 300 = 600 cm, the same result.

Standardsmin grade 3

2.MD.B.5 Solve word problems involving lengths using same units — Modeling the overlapping length spans on a single line.
3.NBT.A.2 Fluently add and subtract within 1000 — Adding the two spans and subtracting the double-counted overlap.

💡Takeaway. When two lengths overlap, add them and subtract the shared middle once -- just Grade 3 add-and-subtract!

Variant 8 hard answer: 790 cm

What is the length from $A$ to $D$ , in centimeters?

Four points $A$ , $B$ , $C$ , $D$ lie on a straight line in that order. The length from $A$ to $C$ is $501\ \text{cm}$ , the length from $B$ to $D$ is $488\ \text{cm}$ , and the overlapping part from $B$ to $C$ is $199\ \text{cm}$ .

Show solution

1 · Understandwhat's really being asked

Four points A, B, C, D sit on a line in that order. I know A-to-C (501 cm), B-to-D (488 cm), and the middle part B-to-C that both spans share (199 cm). I need the whole length from A to D.

Givens

Points A, B, C, D lie on one straight line in that order.
Length A to C is 501 cm.
Length B to D is 488 cm.
The overlapping part B to C is 199 cm.

Unknowns

The length from A to D, in centimeters.

Constraints

Lengths add along the line: AD = AB + BC + CD.
All measurements are in the same unit (cm).

2 · Planchoose the strategy

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

Adding span A-C and span B-D counts the overlap B-C twice, so AD = AC + BD - BC. A picture makes the double-counted middle visible.

3 · Execute3 carry out the plan

1See why the overlap is counted twice

#1 Draw a Diagram 2.MD.B.5

Span A-to-C covers AB and BC; span B-to-D covers BC and CD. Adding both spans counts the middle piece BC two times.

AC + BD = (AB + BC) + (BC + CD)

Drawing the two bars on one line shows the shared middle plainly.

2Add the two spans

#7 Identify Subproblems 3.NBT.A.2

First add the lengths A-to-C and B-to-D together.

501 + 488 = 989

Three-digit addition is the regrouping skill practiced in Grade 3.

3Subtract the overlap once

#7 Identify Subproblems 3.NBT.A.2

Because B-to-C was counted twice, take it away once to get the true length from A to D.

989 - 199 = 790

Removing the extra copy of the overlap is a single subtraction.

Answer: 790 cm

4 · Reviewdoes it hold up?

The answer is in cm. AD should be longer than either single span (501 and 488) but shorter than their full sum (989) because the bars overlap; 790 cm sits sensibly between them.

Another way: Work piece by piece: AB = 501 - 199 = 302, CD = 488 - 199 = 289, then AD = 302 + 199 + 289 = 790 cm, the same result.

Standardsmin grade 3

2.MD.B.5 Solve word problems involving lengths using same units — Modeling the overlapping length spans on a single line.
3.NBT.A.2 Fluently add and subtract within 1000 — Adding the two spans and subtracting the double-counted overlap.

💡Takeaway. When two lengths overlap, add them and subtract the shared middle once -- just Grade 3 add-and-subtract!

Variant 9 hard answer: 701 cm

What is the length from $A$ to $D$ , in centimeters?

Four points $A$ , $B$ , $C$ , $D$ lie on a straight line in that order. The length from $A$ to $C$ is $412\ \text{cm}$ , the length from $B$ to $D$ is $533\ \text{cm}$ , and the overlapping part from $B$ to $C$ is $244\ \text{cm}$ .

Show solution

1 · Understandwhat's really being asked

Four points A, B, C, D sit on a line in that order. I know A-to-C (412 cm), B-to-D (533 cm), and the middle part B-to-C that both spans share (244 cm). I need the whole length from A to D.

Givens

Points A, B, C, D lie on one straight line in that order.
Length A to C is 412 cm.
Length B to D is 533 cm.
The overlapping part B to C is 244 cm.

Unknowns

The length from A to D, in centimeters.

Constraints

Lengths add along the line: AD = AB + BC + CD.
All measurements are in the same unit (cm).

2 · Planchoose the strategy

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

Adding span A-C and span B-D counts the overlap B-C twice, so AD = AC + BD - BC. A picture makes the double-counted middle visible.

3 · Execute3 carry out the plan

1See why the overlap is counted twice

#1 Draw a Diagram 2.MD.B.5

Span A-to-C covers AB and BC; span B-to-D covers BC and CD. Adding both spans counts the middle piece BC two times.

AC + BD = (AB + BC) + (BC + CD)

Drawing the two bars on one line shows the shared middle plainly.

2Add the two spans

#7 Identify Subproblems 3.NBT.A.2

First add the lengths A-to-C and B-to-D together.

412 + 533 = 945

Three-digit addition is the regrouping skill practiced in Grade 3.

3Subtract the overlap once

#7 Identify Subproblems 3.NBT.A.2

Because B-to-C was counted twice, take it away once to get the true length from A to D.

945 - 244 = 701

Removing the extra copy of the overlap is a single subtraction.

Answer: 701 cm

4 · Reviewdoes it hold up?

The answer is in cm. AD should be longer than either single span (412 and 533) but shorter than their full sum (945) because the bars overlap; 701 cm sits sensibly between them.

Another way: Work piece by piece: AB = 412 - 244 = 168, CD = 533 - 244 = 289, then AD = 168 + 244 + 289 = 701 cm, the same result.

Standardsmin grade 3

2.MD.B.5 Solve word problems involving lengths using same units — Modeling the overlapping length spans on a single line.
3.NBT.A.2 Fluently add and subtract within 1000 — Adding the two spans and subtracting the double-counted overlap.

💡Takeaway. When two lengths overlap, add them and subtract the shared middle once -- just Grade 3 add-and-subtract!

Variant 10 hard answer: 859 cm

What is the length from $A$ to $D$ , in centimeters?

Four points $A$ , $B$ , $C$ , $D$ lie on a straight line in that order. The length from $A$ to $C$ is $623\ \text{cm}$ , the length from $B$ to $D$ is $547\ \text{cm}$ , and the overlapping part from $B$ to $C$ is $311\ \text{cm}$ .

Show solution

1 · Understandwhat's really being asked

Four points A, B, C, D sit on a line in that order. I know A-to-C (623 cm), B-to-D (547 cm), and the middle part B-to-C that both spans share (311 cm). I need the whole length from A to D.

Givens

Points A, B, C, D lie on one straight line in that order.
Length A to C is 623 cm.
Length B to D is 547 cm.
The overlapping part B to C is 311 cm.

Unknowns

The length from A to D, in centimeters.

Constraints

Lengths add along the line: AD = AB + BC + CD.
All measurements are in the same unit (cm).

2 · Planchoose the strategy

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

Adding span A-C and span B-D counts the overlap B-C twice, so AD = AC + BD - BC. A picture makes the double-counted middle visible.

3 · Execute3 carry out the plan

1See why the overlap is counted twice

#1 Draw a Diagram 2.MD.B.5

Span A-to-C covers AB and BC; span B-to-D covers BC and CD. Adding both spans counts the middle piece BC two times.

AC + BD = (AB + BC) + (BC + CD)

Drawing the two bars on one line shows the shared middle plainly.

2Add the two spans

#7 Identify Subproblems 3.NBT.A.2

First add the lengths A-to-C and B-to-D together.

623 + 547 = 1170

Three-digit addition is the regrouping skill practiced in Grade 3.

3Subtract the overlap once

#7 Identify Subproblems 3.NBT.A.2

Because B-to-C was counted twice, take it away once to get the true length from A to D.

1170 - 311 = 859

Removing the extra copy of the overlap is a single subtraction.

Answer: 859 cm

4 · Reviewdoes it hold up?

The answer is in cm. AD should be longer than either single span (623 and 547) but shorter than their full sum (1170) because the bars overlap; 859 cm sits sensibly between them.

Another way: Work piece by piece: AB = 623 - 311 = 312, CD = 547 - 311 = 236, then AD = 312 + 311 + 236 = 859 cm, the same result.

Standardsmin grade 3

2.MD.B.5 Solve word problems involving lengths using same units — Modeling the overlapping length spans on a single line.
3.NBT.A.2 Fluently add and subtract within 1000 — Adding the two spans and subtracting the double-counted overlap.

💡Takeaway. When two lengths overlap, add them and subtract the shared middle once -- just Grade 3 add-and-subtract!