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← 2-2 · Subtract the overlap from total length · Overlap Reduces the Total

Subtract the overlap from total length · 8 practice problems

2.MD.B.52.MD.A.4

Generated variants — 8

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

Variant 1 answer: 6 m 50 cm

Find the length from A to D in meters and centimeters.

On a single straight line the four points A, B, C, D are marked in order from left to right. The length from A to C is 5 m 30 cm5\ \mathrm{m}\ 30\ \mathrm{cm}, the length from B to D is 4 m 80 cm4\ \mathrm{m}\ 80\ \mathrm{cm}, and the overlapping part from B to C is 3 m 60 cm3\ \mathrm{m}\ 60\ \mathrm{cm}.

A B C D 5 m 30 cm 4 m 80 cm 3 m 60 cm
Show solution

Understand

Four points A, B, C, D lie in order on a line. The span A to C is 5 m 30 cm, the span B to D is 4 m 80 cm, and the overlapping middle span B to C is 3 m 60 cm. Find the full length from A to D in meters and centimeters.

Givens
  • A, B, C, D are in order from left to right on one straight line.
  • A to C = 5 m 30 cm.
  • B to D = 4 m 80 cm.
  • Overlap B to C = 3 m 60 cm.
Unknowns
  • The length from A to D in meters and centimeters.
Constraints
  • 1 m = 100 cm.
  • Adding A-C and B-D counts the middle overlap B-C twice, so it must be subtracted once.

Plan

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

The picture shows two overlapping spans whose overlap is B-C. Adding the two spans double-counts the overlap, so A-D = (A-C) + (B-D) - (B-C). Work in centimeters, then convert back.

Execute

#1 Draw a Diagram 2.MD.B.5
Write each length in centimeters so they can be combined.
530 cm, 480 cm, 360 cm530\,\text{cm},\ 480\,\text{cm},\ 360\,\text{cm}
Same-unit lengths add and subtract cleanly.
#7 Identify Subproblems 2.MD.A.4
Add A-C and B-D; this counts the middle part B-C twice.
530+480=1010 cm530 + 480 = 1010 \text{ cm}
Combining the two measured pieces gives a total that includes the overlap region twice.
#7 Identify Subproblems 2.MD.B.5
Remove one copy of the overlap B-C (360 cm) to get the true length A-D.
1010−360=650 cm=6 m 50 cm1010 - 360 = 650 \text{ cm} = 6\,\text{m}\ 50\,\text{cm}
Taking out the part counted twice leaves the full stretch from A to D exactly once.
Answer: 6 m 50 cm

Review

A-D should be longer than either single span (5 m 30 cm or 4 m 80 cm) but less than their sum (10 m 10 cm); 6 m 50 cm fits between, so it is reasonable.

Subproblems by segments (tool 7): A-B = (A-C) - (B-C) = 530 - 360 = 170 cm; then A-D = A-B + B-D = 170 + 480 = 650 cm = 6 m 50 cm.

Standards · min grade 2

  • 2.MD.B.5 Solve word problems involving lengths using same units — Adding and subtracting same-unit lengths to find A to D.
  • 2.MD.A.4 Measure to determine how much longer one object is than another — Reasoning about the overlapping spans and the doubly counted middle part.
💡 When two lengths overlap, add them and take the overlap back out once — Grade 2 length adding and subtracting!
Variant 2 answer: 4 m 5 cm

Find the length from A to D in meters and centimeters.

On a single straight line the four points A, B, C, D are marked in order from left to right. The length from A to C is 2 m 90 cm2\ \mathrm{m}\ 90\ \mathrm{cm}, the length from B to D is 3 m 10 cm3\ \mathrm{m}\ 10\ \mathrm{cm}, and the overlapping part from B to C is 1 m 95 cm1\ \mathrm{m}\ 95\ \mathrm{cm}.

A B C D 2 m 90 cm 3 m 10 cm 1 m 95 cm
Show solution

Understand

Four points A, B, C, D lie in order on a line. The span A to C is 2 m 90 cm, the span B to D is 3 m 10 cm, and the overlapping middle span B to C is 1 m 95 cm. Find the full length from A to D in meters and centimeters.

Givens
  • A, B, C, D are in order from left to right on one straight line.
  • A to C = 2 m 90 cm.
  • B to D = 3 m 10 cm.
  • Overlap B to C = 1 m 95 cm.
Unknowns
  • The length from A to D in meters and centimeters.
Constraints
  • 1 m = 100 cm.
  • Adding A-C and B-D counts the middle overlap B-C twice, so it must be subtracted once.

Plan

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

The picture shows two overlapping spans whose overlap is B-C. Adding the two spans double-counts the overlap, so A-D = (A-C) + (B-D) - (B-C). Work in centimeters, then convert back.

Execute

#1 Draw a Diagram 2.MD.B.5
Write each length in centimeters so they can be combined.
290 cm, 310 cm, 195 cm290\,\text{cm},\ 310\,\text{cm},\ 195\,\text{cm}
Same-unit lengths add and subtract cleanly.
#7 Identify Subproblems 2.MD.A.4
Add A-C and B-D; this counts the middle part B-C twice.
290+310=600 cm290 + 310 = 600 \text{ cm}
Combining the two measured pieces gives a total that includes the overlap region twice.
#7 Identify Subproblems 2.MD.B.5
Remove one copy of the overlap B-C (195 cm) to get the true length A-D.
600−195=405 cm=4 m 5 cm600 - 195 = 405 \text{ cm} = 4\,\text{m}\ 5\,\text{cm}
Taking out the part counted twice leaves the full stretch from A to D exactly once.
Answer: 4 m 5 cm

Review

A-D should be longer than either single span (2 m 90 cm or 3 m 10 cm) but less than their sum (6 m 0 cm); 4 m 5 cm fits between, so it is reasonable.

Subproblems by segments (tool 7): A-B = (A-C) - (B-C) = 290 - 195 = 95 cm; then A-D = A-B + B-D = 95 + 310 = 405 cm = 4 m 5 cm.

Standards · min grade 2

  • 2.MD.B.5 Solve word problems involving lengths using same units — Adding and subtracting same-unit lengths to find A to D.
  • 2.MD.A.4 Measure to determine how much longer one object is than another — Reasoning about the overlapping spans and the doubly counted middle part.
💡 When two lengths overlap, add them and take the overlap back out once — Grade 2 length adding and subtracting!
Variant 3 answer: 2 m 65 cm

Find the length from A to D in meters and centimeters.

On a single straight line the four points A, B, C, D are marked in order from left to right. The length from A to C is 1 m 80 cm1\ \mathrm{m}\ 80\ \mathrm{cm}, the length from B to D is 1 m 95 cm1\ \mathrm{m}\ 95\ \mathrm{cm}, and the overlapping part from B to C is 1 m 10 cm1\ \mathrm{m}\ 10\ \mathrm{cm}.

A B C D 1 m 80 cm 1 m 95 cm 1 m 10 cm
Show solution

Understand

Four points A, B, C, D lie in order on a line. The span A to C is 1 m 80 cm, the span B to D is 1 m 95 cm, and the overlapping middle span B to C is 1 m 10 cm. Find the full length from A to D in meters and centimeters.

Givens
  • A, B, C, D are in order from left to right on one straight line.
  • A to C = 1 m 80 cm.
  • B to D = 1 m 95 cm.
  • Overlap B to C = 1 m 10 cm.
Unknowns
  • The length from A to D in meters and centimeters.
Constraints
  • 1 m = 100 cm.
  • Adding A-C and B-D counts the middle overlap B-C twice, so it must be subtracted once.

Plan

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

The picture shows two overlapping spans whose overlap is B-C. Adding the two spans double-counts the overlap, so A-D = (A-C) + (B-D) - (B-C). Work in centimeters, then convert back.

Execute

#1 Draw a Diagram 2.MD.B.5
Write each length in centimeters so they can be combined.
180 cm, 195 cm, 110 cm180\,\text{cm},\ 195\,\text{cm},\ 110\,\text{cm}
Same-unit lengths add and subtract cleanly.
#7 Identify Subproblems 2.MD.A.4
Add A-C and B-D; this counts the middle part B-C twice.
180+195=375 cm180 + 195 = 375 \text{ cm}
Combining the two measured pieces gives a total that includes the overlap region twice.
#7 Identify Subproblems 2.MD.B.5
Remove one copy of the overlap B-C (110 cm) to get the true length A-D.
375−110=265 cm=2 m 65 cm375 - 110 = 265 \text{ cm} = 2\,\text{m}\ 65\,\text{cm}
Taking out the part counted twice leaves the full stretch from A to D exactly once.
Answer: 2 m 65 cm

Review

A-D should be longer than either single span (1 m 80 cm or 1 m 95 cm) but less than their sum (3 m 75 cm); 2 m 65 cm fits between, so it is reasonable.

Subproblems by segments (tool 7): A-B = (A-C) - (B-C) = 180 - 110 = 70 cm; then A-D = A-B + B-D = 70 + 195 = 265 cm = 2 m 65 cm.

Standards · min grade 2

  • 2.MD.B.5 Solve word problems involving lengths using same units — Adding and subtracting same-unit lengths to find A to D.
  • 2.MD.A.4 Measure to determine how much longer one object is than another — Reasoning about the overlapping spans and the doubly counted middle part.
💡 When two lengths overlap, add them and take the overlap back out once — Grade 2 length adding and subtracting!
Variant 4 answer: 3 m 55 cm

Find the length from A to D in meters and centimeters.

On a single straight line the four points A, B, C, D are marked in order from left to right. The length from A to C is 2 m 15 cm2\ \mathrm{m}\ 15\ \mathrm{cm}, the length from B to D is 2 m 45 cm2\ \mathrm{m}\ 45\ \mathrm{cm}, and the overlapping part from B to C is 1 m 5 cm1\ \mathrm{m}\ 5\ \mathrm{cm}.

A B C D 2 m 15 cm 2 m 45 cm 1 m 5 cm
Show solution

Understand

Four points A, B, C, D lie in order on a line. The span A to C is 2 m 15 cm, the span B to D is 2 m 45 cm, and the overlapping middle span B to C is 1 m 5 cm. Find the full length from A to D in meters and centimeters.

Givens
  • A, B, C, D are in order from left to right on one straight line.
  • A to C = 2 m 15 cm.
  • B to D = 2 m 45 cm.
  • Overlap B to C = 1 m 5 cm.
Unknowns
  • The length from A to D in meters and centimeters.
Constraints
  • 1 m = 100 cm.
  • Adding A-C and B-D counts the middle overlap B-C twice, so it must be subtracted once.

Plan

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

The picture shows two overlapping spans whose overlap is B-C. Adding the two spans double-counts the overlap, so A-D = (A-C) + (B-D) - (B-C). Work in centimeters, then convert back.

Execute

#1 Draw a Diagram 2.MD.B.5
Write each length in centimeters so they can be combined.
215 cm, 245 cm, 105 cm215\,\text{cm},\ 245\,\text{cm},\ 105\,\text{cm}
Same-unit lengths add and subtract cleanly.
#7 Identify Subproblems 2.MD.A.4
Add A-C and B-D; this counts the middle part B-C twice.
215+245=460 cm215 + 245 = 460 \text{ cm}
Combining the two measured pieces gives a total that includes the overlap region twice.
#7 Identify Subproblems 2.MD.B.5
Remove one copy of the overlap B-C (105 cm) to get the true length A-D.
460−105=355 cm=3 m 55 cm460 - 105 = 355 \text{ cm} = 3\,\text{m}\ 55\,\text{cm}
Taking out the part counted twice leaves the full stretch from A to D exactly once.
Answer: 3 m 55 cm

Review

A-D should be longer than either single span (2 m 15 cm or 2 m 45 cm) but less than their sum (4 m 60 cm); 3 m 55 cm fits between, so it is reasonable.

Subproblems by segments (tool 7): A-B = (A-C) - (B-C) = 215 - 105 = 110 cm; then A-D = A-B + B-D = 110 + 245 = 355 cm = 3 m 55 cm.

Standards · min grade 2

  • 2.MD.B.5 Solve word problems involving lengths using same units — Adding and subtracting same-unit lengths to find A to D.
  • 2.MD.A.4 Measure to determine how much longer one object is than another — Reasoning about the overlapping spans and the doubly counted middle part.
💡 When two lengths overlap, add them and take the overlap back out once — Grade 2 length adding and subtracting!
Variant 5 answer: 4 m 60 cm

Find the length from A to D in meters and centimeters.

On a single straight line the four points A, B, C, D are marked in order from left to right. The length from A to C is 3 m 20 cm3\ \mathrm{m}\ 20\ \mathrm{cm}, the length from B to D is 2 m 70 cm2\ \mathrm{m}\ 70\ \mathrm{cm}, and the overlapping part from B to C is 1 m 30 cm1\ \mathrm{m}\ 30\ \mathrm{cm}.

A B C D 3 m 20 cm 2 m 70 cm 1 m 30 cm
Show solution

Understand

Four points A, B, C, D lie in order on a line. The span A to C is 3 m 20 cm, the span B to D is 2 m 70 cm, and the overlapping middle span B to C is 1 m 30 cm. Find the full length from A to D in meters and centimeters.

Givens
  • A, B, C, D are in order from left to right on one straight line.
  • A to C = 3 m 20 cm.
  • B to D = 2 m 70 cm.
  • Overlap B to C = 1 m 30 cm.
Unknowns
  • The length from A to D in meters and centimeters.
Constraints
  • 1 m = 100 cm.
  • Adding A-C and B-D counts the middle overlap B-C twice, so it must be subtracted once.

Plan

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

The picture shows two overlapping spans whose overlap is B-C. Adding the two spans double-counts the overlap, so A-D = (A-C) + (B-D) - (B-C). Work in centimeters, then convert back.

Execute

#1 Draw a Diagram 2.MD.B.5
Write each length in centimeters so they can be combined.
320 cm, 270 cm, 130 cm320\,\text{cm},\ 270\,\text{cm},\ 130\,\text{cm}
Same-unit lengths add and subtract cleanly.
#7 Identify Subproblems 2.MD.A.4
Add A-C and B-D; this counts the middle part B-C twice.
320+270=590 cm320 + 270 = 590 \text{ cm}
Combining the two measured pieces gives a total that includes the overlap region twice.
#7 Identify Subproblems 2.MD.B.5
Remove one copy of the overlap B-C (130 cm) to get the true length A-D.
590−130=460 cm=4 m 60 cm590 - 130 = 460 \text{ cm} = 4\,\text{m}\ 60\,\text{cm}
Taking out the part counted twice leaves the full stretch from A to D exactly once.
Answer: 4 m 60 cm

Review

A-D should be longer than either single span (3 m 20 cm or 2 m 70 cm) but less than their sum (5 m 90 cm); 4 m 60 cm fits between, so it is reasonable.

Subproblems by segments (tool 7): A-B = (A-C) - (B-C) = 320 - 130 = 190 cm; then A-D = A-B + B-D = 190 + 270 = 460 cm = 4 m 60 cm.

Standards · min grade 2

  • 2.MD.B.5 Solve word problems involving lengths using same units — Adding and subtracting same-unit lengths to find A to D.
  • 2.MD.A.4 Measure to determine how much longer one object is than another — Reasoning about the overlapping spans and the doubly counted middle part.
💡 When two lengths overlap, add them and take the overlap back out once — Grade 2 length adding and subtracting!
Variant 6 answer: 5 m 75 cm

Find the length from A to D in meters and centimeters.

On a single straight line the four points A, B, C, D are marked in order from left to right. The length from A to C is 4 m 60 cm4\ \mathrm{m}\ 60\ \mathrm{cm}, the length from B to D is 3 m 40 cm3\ \mathrm{m}\ 40\ \mathrm{cm}, and the overlapping part from B to C is 2 m 25 cm2\ \mathrm{m}\ 25\ \mathrm{cm}.

A B C D 4 m 60 cm 3 m 40 cm 2 m 25 cm
Show solution

Understand

Four points A, B, C, D lie in order on a line. The span A to C is 4 m 60 cm, the span B to D is 3 m 40 cm, and the overlapping middle span B to C is 2 m 25 cm. Find the full length from A to D in meters and centimeters.

Givens
  • A, B, C, D are in order from left to right on one straight line.
  • A to C = 4 m 60 cm.
  • B to D = 3 m 40 cm.
  • Overlap B to C = 2 m 25 cm.
Unknowns
  • The length from A to D in meters and centimeters.
Constraints
  • 1 m = 100 cm.
  • Adding A-C and B-D counts the middle overlap B-C twice, so it must be subtracted once.

Plan

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

The picture shows two overlapping spans whose overlap is B-C. Adding the two spans double-counts the overlap, so A-D = (A-C) + (B-D) - (B-C). Work in centimeters, then convert back.

Execute

#1 Draw a Diagram 2.MD.B.5
Write each length in centimeters so they can be combined.
460 cm, 340 cm, 225 cm460\,\text{cm},\ 340\,\text{cm},\ 225\,\text{cm}
Same-unit lengths add and subtract cleanly.
#7 Identify Subproblems 2.MD.A.4
Add A-C and B-D; this counts the middle part B-C twice.
460+340=800 cm460 + 340 = 800 \text{ cm}
Combining the two measured pieces gives a total that includes the overlap region twice.
#7 Identify Subproblems 2.MD.B.5
Remove one copy of the overlap B-C (225 cm) to get the true length A-D.
800−225=575 cm=5 m 75 cm800 - 225 = 575 \text{ cm} = 5\,\text{m}\ 75\,\text{cm}
Taking out the part counted twice leaves the full stretch from A to D exactly once.
Answer: 5 m 75 cm

Review

A-D should be longer than either single span (4 m 60 cm or 3 m 40 cm) but less than their sum (8 m 0 cm); 5 m 75 cm fits between, so it is reasonable.

Subproblems by segments (tool 7): A-B = (A-C) - (B-C) = 460 - 225 = 235 cm; then A-D = A-B + B-D = 235 + 340 = 575 cm = 5 m 75 cm.

Standards · min grade 2

  • 2.MD.B.5 Solve word problems involving lengths using same units — Adding and subtracting same-unit lengths to find A to D.
  • 2.MD.A.4 Measure to determine how much longer one object is than another — Reasoning about the overlapping spans and the doubly counted middle part.
💡 When two lengths overlap, add them and take the overlap back out once — Grade 2 length adding and subtracting!
Variant 7 answer: 3 m 94 cm

Find the length from A to D in meters and centimeters.

On a single straight line the four points A, B, C, D are marked in order from left to right. The length from A to C is 2 m 55 cm2\ \mathrm{m}\ 55\ \mathrm{cm}, the length from B to D is 2 m 89 cm2\ \mathrm{m}\ 89\ \mathrm{cm}, and the overlapping part from B to C is 1 m 50 cm1\ \mathrm{m}\ 50\ \mathrm{cm}.

A B C D 2 m 55 cm 2 m 89 cm 1 m 50 cm
Show solution

Understand

Four points A, B, C, D lie in order on a line. The span A to C is 2 m 55 cm, the span B to D is 2 m 89 cm, and the overlapping middle span B to C is 1 m 50 cm. Find the full length from A to D in meters and centimeters.

Givens
  • A, B, C, D are in order from left to right on one straight line.
  • A to C = 2 m 55 cm.
  • B to D = 2 m 89 cm.
  • Overlap B to C = 1 m 50 cm.
Unknowns
  • The length from A to D in meters and centimeters.
Constraints
  • 1 m = 100 cm.
  • Adding A-C and B-D counts the middle overlap B-C twice, so it must be subtracted once.

Plan

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

The picture shows two overlapping spans whose overlap is B-C. Adding the two spans double-counts the overlap, so A-D = (A-C) + (B-D) - (B-C). Work in centimeters, then convert back.

Execute

#1 Draw a Diagram 2.MD.B.5
Write each length in centimeters so they can be combined.
255 cm, 289 cm, 150 cm255\,\text{cm},\ 289\,\text{cm},\ 150\,\text{cm}
Same-unit lengths add and subtract cleanly.
#7 Identify Subproblems 2.MD.A.4
Add A-C and B-D; this counts the middle part B-C twice.
255+289=544 cm255 + 289 = 544 \text{ cm}
Combining the two measured pieces gives a total that includes the overlap region twice.
#7 Identify Subproblems 2.MD.B.5
Remove one copy of the overlap B-C (150 cm) to get the true length A-D.
544−150=394 cm=3 m 94 cm544 - 150 = 394 \text{ cm} = 3\,\text{m}\ 94\,\text{cm}
Taking out the part counted twice leaves the full stretch from A to D exactly once.
Answer: 3 m 94 cm

Review

A-D should be longer than either single span (2 m 55 cm or 2 m 89 cm) but less than their sum (5 m 44 cm); 3 m 94 cm fits between, so it is reasonable.

Subproblems by segments (tool 7): A-B = (A-C) - (B-C) = 255 - 150 = 105 cm; then A-D = A-B + B-D = 105 + 289 = 394 cm = 3 m 94 cm.

Standards · min grade 2

  • 2.MD.B.5 Solve word problems involving lengths using same units — Adding and subtracting same-unit lengths to find A to D.
  • 2.MD.A.4 Measure to determine how much longer one object is than another — Reasoning about the overlapping spans and the doubly counted middle part.
💡 When two lengths overlap, add them and take the overlap back out once — Grade 2 length adding and subtracting!
Variant 8 answer: 4 m 50 cm

Find the length from A to D in meters and centimeters.

On a single straight line the four points A, B, C, D are marked in order from left to right. The length from A to C is 3 m 75 cm3\ \mathrm{m}\ 75\ \mathrm{cm}, the length from B to D is 3 m 25 cm3\ \mathrm{m}\ 25\ \mathrm{cm}, and the overlapping part from B to C is 2 m 50 cm2\ \mathrm{m}\ 50\ \mathrm{cm}.

A B C D 3 m 75 cm 3 m 25 cm 2 m 50 cm
Show solution

Understand

Four points A, B, C, D lie in order on a line. The span A to C is 3 m 75 cm, the span B to D is 3 m 25 cm, and the overlapping middle span B to C is 2 m 50 cm. Find the full length from A to D in meters and centimeters.

Givens
  • A, B, C, D are in order from left to right on one straight line.
  • A to C = 3 m 75 cm.
  • B to D = 3 m 25 cm.
  • Overlap B to C = 2 m 50 cm.
Unknowns
  • The length from A to D in meters and centimeters.
Constraints
  • 1 m = 100 cm.
  • Adding A-C and B-D counts the middle overlap B-C twice, so it must be subtracted once.

Plan

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

The picture shows two overlapping spans whose overlap is B-C. Adding the two spans double-counts the overlap, so A-D = (A-C) + (B-D) - (B-C). Work in centimeters, then convert back.

Execute

#1 Draw a Diagram 2.MD.B.5
Write each length in centimeters so they can be combined.
375 cm, 325 cm, 250 cm375\,\text{cm},\ 325\,\text{cm},\ 250\,\text{cm}
Same-unit lengths add and subtract cleanly.
#7 Identify Subproblems 2.MD.A.4
Add A-C and B-D; this counts the middle part B-C twice.
375+325=700 cm375 + 325 = 700 \text{ cm}
Combining the two measured pieces gives a total that includes the overlap region twice.
#7 Identify Subproblems 2.MD.B.5
Remove one copy of the overlap B-C (250 cm) to get the true length A-D.
700−250=450 cm=4 m 50 cm700 - 250 = 450 \text{ cm} = 4\,\text{m}\ 50\,\text{cm}
Taking out the part counted twice leaves the full stretch from A to D exactly once.
Answer: 4 m 50 cm

Review

A-D should be longer than either single span (3 m 75 cm or 3 m 25 cm) but less than their sum (7 m 0 cm); 4 m 50 cm fits between, so it is reasonable.

Subproblems by segments (tool 7): A-B = (A-C) - (B-C) = 375 - 250 = 125 cm; then A-D = A-B + B-D = 125 + 325 = 450 cm = 4 m 50 cm.

Standards · min grade 2

  • 2.MD.B.5 Solve word problems involving lengths using same units — Adding and subtracting same-unit lengths to find A to D.
  • 2.MD.A.4 Measure to determine how much longer one object is than another — Reasoning about the overlapping spans and the doubly counted middle part.
💡 When two lengths overlap, add them and take the overlap back out once — Grade 2 length adding and subtracting!