← 3-1 · 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 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

Understand

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).

Plan

#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.

Execute

#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.

#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.

#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

Review

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.

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

Standards · min 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.

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

Variant 2 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

Understand

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).

Plan

#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.

Execute

#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.

#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.

#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

Review

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.

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

Standards · min 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.

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

Variant 3 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

Understand

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).

Plan

#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.

Execute

#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.

#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.

#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

Review

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.

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

Standards · min 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.

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

Variant 4 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

Understand

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).

Plan

#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.

Execute

#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.

#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.

#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

Review

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.

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

Standards · min 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.

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

Variant 5 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

Understand

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).

Plan

#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.

Execute

#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.

#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.

#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

Review

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.

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

Standards · min 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.

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

Variant 6 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

Understand

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).

Plan

#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.

Execute

#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.

#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.

#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

Review

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.

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

Standards · min 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.

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

Variant 7 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

Understand

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).

Plan

#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.

Execute

#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.

#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.

#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

Review

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.

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

Standards · min 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.

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

Variant 8 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

Understand

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).

Plan

#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.

Execute

#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.

#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.

#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

Review

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.

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

Standards · min 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.

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

Variant 9 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

Understand

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).

Plan

#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.

Execute

#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.

#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.

#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

Review

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.

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

Standards · min 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.

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

Variant 10 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

Understand

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).

Plan

#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.

Execute

#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.

#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.

#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

Review

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.

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

Standards · min 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.

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