← 4-2 · Subtract the overlap when joining fraction lengths · Overlap Reduces the Total

Subtract the overlap when joining fraction lengths · 10 practice problems

4.OA.A.34.NF.B.3

Generated variants — 10

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

Variant 1 answer: 34 6/8 km

Use the figure to find the distance from point A to point E, in km.

Five points A, B, C, D, and E lie on one straight line in that order. The distance from A to C is $14\tfrac{3}{8}$ km, the distance from B to D is $18\tfrac{5}{8}$ km, and the distance from D to E is $6\tfrac{7}{8}$ km. Also, the distance from B to C (the overlapping section) is $5\tfrac{1}{8}$ km.

Show solution

Understand

Five points A, B, C, D, E sit on a line in that order. I'm given the spans AC, BD, DE, and the overlapping span BC. Using the number-line figure I must find the total distance from A to E.

Givens

Points lie in order A, B, C, D, E on one line.
AC = 14 3/8 km.
BD = 18 5/8 km.
DE = 6 7/8 km.
BC = 5 1/8 km is the overlap shared by AC and BD.

Unknowns

The distance from A to E, in km.

Constraints

AC and BD overlap exactly on the segment BC, since B is between A and C and C is between B and D.

Plan

#1 Draw a Diagram · also uses: #7 Identify Subproblems#16 Count the Complement

The figure shows A-B-C-D-E with AC and BD sharing the overlap BC. Adding AC + BD double-counts BC, so AE = AC + BD + DE - BC. The diagram makes the overlap easy to subtract.

Execute

#1 Draw a Diagram 4.OA.A.3

From the figure: AC stretches from A to C, BD from B to D, and these two arcs overlap on segment BC. Adding AC and BD counts the overlap BC twice, so AE = AC + BD + DE - BC.

AE = AC + BD + DE - BC

Drawing the line shows the two arcs share the middle piece BC, so it must be removed once.

#7 Identify Subproblems 4.OA.A.3

Combine the whole numbers: 14 + 18 + 6 - 5 = 33.

14+18+6-5=33

Handling the whole parts first keeps the fraction work small.

#7 Identify Subproblems 4.NF.B.3

All fractions share denominator 8: (3+5+7-1)/8 = 14/8 = 1 6/8.

\dfrac{3}{8}+\dfrac{5}{8}+\dfrac{7}{8}-\dfrac{1}{8}=\dfrac{14}{8}=1\tfrac{6}{8}

Like denominators let you add and subtract numerators, then regroup.

#16 Count the Complement 4.NF.B.3

Add the whole-part total and the fraction-part total: 33 + 1 6/8 = 34 6/8.

33+1\tfrac{6}{8}=34\tfrac{6}{8}

Putting the pieces back together gives the full A-to-E distance.

Answer: 34 6/8 km

Review

Cross-check with segment pieces: AB = AC - BC = 9 2/8, CD = BD - BC = 13 4/8, so AE = AB + BC + CD + DE = 34 6/8 km. Same answer, and 34 6/8 is sensibly larger than each given span.

Work piece by piece (tool 7): find AB and CD by subtracting the overlap, then add AB + BC + CD + DE directly instead of using the overlap-subtraction formula.

Standards · min grade 4

4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Setting up AE = AC + BD + DE - BC and combining the whole-number parts.
4.NF.B.3 Understand a fraction with numerator greater than one as sum of unit fractions — Adding and subtracting the like-denominator 8ths and regrouping.

💡 This only needs Grade 4 fraction add/subtract — draw the line and remember the overlap gets counted twice, so subtract it once!

Variant 2 answer: 23 km

Use the figure to find the distance from point A to point E, in km.

Five points A, B, C, D, and E lie on one straight line in that order. The distance from A to C is $10\tfrac{1}{4}$ km, the distance from B to D is $13\tfrac{3}{4}$ km, and the distance from D to E is $2\tfrac{2}{4}$ km. Also, the distance from B to C (the overlapping section) is $3\tfrac{2}{4}$ km.

Show solution

Understand

Five points A, B, C, D, E sit on a line in that order. I'm given the spans AC, BD, DE, and the overlapping span BC. Using the number-line figure I must find the total distance from A to E.

Givens

Points lie in order A, B, C, D, E on one line.
AC = 10 1/4 km.
BD = 13 3/4 km.
DE = 2 2/4 km.
BC = 3 2/4 km is the overlap shared by AC and BD.

Unknowns

The distance from A to E, in km.

Constraints

AC and BD overlap exactly on the segment BC, since B is between A and C and C is between B and D.

Plan

#1 Draw a Diagram · also uses: #7 Identify Subproblems#16 Count the Complement

The figure shows A-B-C-D-E with AC and BD sharing the overlap BC. Adding AC + BD double-counts BC, so AE = AC + BD + DE - BC. The diagram makes the overlap easy to subtract.

Execute

#1 Draw a Diagram 4.OA.A.3

From the figure: AC stretches from A to C, BD from B to D, and these two arcs overlap on segment BC. Adding AC and BD counts the overlap BC twice, so AE = AC + BD + DE - BC.

AE = AC + BD + DE - BC

Drawing the line shows the two arcs share the middle piece BC, so it must be removed once.

#7 Identify Subproblems 4.OA.A.3

Combine the whole numbers: 10 + 13 + 2 - 3 = 22.

10+13+2-3=22

Handling the whole parts first keeps the fraction work small.

#7 Identify Subproblems 4.NF.B.3

All fractions share denominator 4: (1+3+2-2)/4 = 4/4 = 1.

\dfrac{1}{4}+\dfrac{3}{4}+\dfrac{2}{4}-\dfrac{2}{4}=\dfrac{4}{4}=1

Like denominators let you add and subtract numerators, then regroup.

#16 Count the Complement 4.NF.B.3

Add the whole-part total and the fraction-part total: 22 + 1 = 23.

22+1=23

Putting the pieces back together gives the full A-to-E distance.

Answer: 23 km

Review

Cross-check with segment pieces: AB = AC - BC = 6 3/4, CD = BD - BC = 10 1/4, so AE = AB + BC + CD + DE = 23 km. Same answer, and 23 is sensibly larger than each given span.

Work piece by piece (tool 7): find AB and CD by subtracting the overlap, then add AB + BC + CD + DE directly instead of using the overlap-subtraction formula.

Standards · min grade 4

4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Setting up AE = AC + BD + DE - BC and combining the whole-number parts.
4.NF.B.3 Understand a fraction with numerator greater than one as sum of unit fractions — Adding and subtracting the like-denominator 4ths and regrouping.

💡 This only needs Grade 4 fraction add/subtract — draw the line and remember the overlap gets counted twice, so subtract it once!

Variant 3 answer: 26 4/5 km

Use the figure to find the distance from point A to point E, in km.

Five points A, B, C, D, and E lie on one straight line in that order. The distance from A to C is $12\tfrac{1}{5}$ km, the distance from B to D is $15\tfrac{3}{5}$ km, and the distance from D to E is $3\tfrac{2}{5}$ km. Also, the distance from B to C (the overlapping section) is $4\tfrac{2}{5}$ km.

Show solution

Understand

Five points A, B, C, D, E sit on a line in that order. I'm given the spans AC, BD, DE, and the overlapping span BC. Using the number-line figure I must find the total distance from A to E.

Givens

Points lie in order A, B, C, D, E on one line.
AC = 12 1/5 km.
BD = 15 3/5 km.
DE = 3 2/5 km.
BC = 4 2/5 km is the overlap shared by AC and BD.

Unknowns

The distance from A to E, in km.

Constraints

AC and BD overlap exactly on the segment BC, since B is between A and C and C is between B and D.

Plan

#1 Draw a Diagram · also uses: #7 Identify Subproblems#16 Count the Complement

The figure shows A-B-C-D-E with AC and BD sharing the overlap BC. Adding AC + BD double-counts BC, so AE = AC + BD + DE - BC. The diagram makes the overlap easy to subtract.

Execute

#1 Draw a Diagram 4.OA.A.3

From the figure: AC stretches from A to C, BD from B to D, and these two arcs overlap on segment BC. Adding AC and BD counts the overlap BC twice, so AE = AC + BD + DE - BC.

AE = AC + BD + DE - BC

Drawing the line shows the two arcs share the middle piece BC, so it must be removed once.

#7 Identify Subproblems 4.OA.A.3

Combine the whole numbers: 12 + 15 + 3 - 4 = 26.

12+15+3-4=26

Handling the whole parts first keeps the fraction work small.

#7 Identify Subproblems 4.NF.B.3

All fractions share denominator 5: (1+3+2-2)/5 = 4/5 = 4/5.

\dfrac{1}{5}+\dfrac{3}{5}+\dfrac{2}{5}-\dfrac{2}{5}=\dfrac{4}{5}=\dfrac{4}{5}

Like denominators let you add and subtract numerators, then regroup.

#16 Count the Complement 4.NF.B.3

Add the whole-part total and the fraction-part total: 26 + 4/5 = 26 4/5.

26+\dfrac{4}{5}=26\tfrac{4}{5}

Putting the pieces back together gives the full A-to-E distance.

Answer: 26 4/5 km

Review

Cross-check with segment pieces: AB = AC - BC = 7 4/5, CD = BD - BC = 11 1/5, so AE = AB + BC + CD + DE = 26 4/5 km. Same answer, and 26 4/5 is sensibly larger than each given span.

Work piece by piece (tool 7): find AB and CD by subtracting the overlap, then add AB + BC + CD + DE directly instead of using the overlap-subtraction formula.

Standards · min grade 4

4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Setting up AE = AC + BD + DE - BC and combining the whole-number parts.
4.NF.B.3 Understand a fraction with numerator greater than one as sum of unit fractions — Adding and subtracting the like-denominator 5ths and regrouping.

💡 This only needs Grade 4 fraction add/subtract — draw the line and remember the overlap gets counted twice, so subtract it once!

Variant 4 answer: 19 4/5 km

Use the figure to find the distance from point A to point E, in km.

Five points A, B, C, D, and E lie on one straight line in that order. The distance from A to C is $8\tfrac{4}{5}$ km, the distance from B to D is $11\tfrac{2}{5}$ km, and the distance from D to E is $3\tfrac{1}{5}$ km. Also, the distance from B to C (the overlapping section) is $3\tfrac{3}{5}$ km.

Show solution

Understand

Five points A, B, C, D, E sit on a line in that order. I'm given the spans AC, BD, DE, and the overlapping span BC. Using the number-line figure I must find the total distance from A to E.

Givens

Points lie in order A, B, C, D, E on one line.
AC = 8 4/5 km.
BD = 11 2/5 km.
DE = 3 1/5 km.
BC = 3 3/5 km is the overlap shared by AC and BD.

Unknowns

The distance from A to E, in km.

Constraints

AC and BD overlap exactly on the segment BC, since B is between A and C and C is between B and D.

Plan

#1 Draw a Diagram · also uses: #7 Identify Subproblems#16 Count the Complement

The figure shows A-B-C-D-E with AC and BD sharing the overlap BC. Adding AC + BD double-counts BC, so AE = AC + BD + DE - BC. The diagram makes the overlap easy to subtract.

Execute

#1 Draw a Diagram 4.OA.A.3

From the figure: AC stretches from A to C, BD from B to D, and these two arcs overlap on segment BC. Adding AC and BD counts the overlap BC twice, so AE = AC + BD + DE - BC.

AE = AC + BD + DE - BC

Drawing the line shows the two arcs share the middle piece BC, so it must be removed once.

#7 Identify Subproblems 4.OA.A.3

Combine the whole numbers: 8 + 11 + 3 - 3 = 19.

8+11+3-3=19

Handling the whole parts first keeps the fraction work small.

#7 Identify Subproblems 4.NF.B.3

All fractions share denominator 5: (4+2+1-3)/5 = 4/5 = 4/5.

\dfrac{4}{5}+\dfrac{2}{5}+\dfrac{1}{5}-\dfrac{3}{5}=\dfrac{4}{5}=\dfrac{4}{5}

Like denominators let you add and subtract numerators, then regroup.

#16 Count the Complement 4.NF.B.3

Add the whole-part total and the fraction-part total: 19 + 4/5 = 19 4/5.

19+\dfrac{4}{5}=19\tfrac{4}{5}

Putting the pieces back together gives the full A-to-E distance.

Answer: 19 4/5 km

Review

Cross-check with segment pieces: AB = AC - BC = 5 1/5, CD = BD - BC = 7 4/5, so AE = AB + BC + CD + DE = 19 4/5 km. Same answer, and 19 4/5 is sensibly larger than each given span.

Work piece by piece (tool 7): find AB and CD by subtracting the overlap, then add AB + BC + CD + DE directly instead of using the overlap-subtraction formula.

Standards · min grade 4

4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Setting up AE = AC + BD + DE - BC and combining the whole-number parts.
4.NF.B.3 Understand a fraction with numerator greater than one as sum of unit fractions — Adding and subtracting the like-denominator 5ths and regrouping.

💡 This only needs Grade 4 fraction add/subtract — draw the line and remember the overlap gets counted twice, so subtract it once!

Variant 5 answer: 30 5/7 km

Use the figure to find the distance from point A to point E, in km.

Five points A, B, C, D, and E lie on one straight line in that order. The distance from A to C is $13\tfrac{6}{7}$ km, the distance from B to D is $17\tfrac{4}{7}$ km, and the distance from D to E is $4\tfrac{5}{7}$ km. Also, the distance from B to C (the overlapping section) is $5\tfrac{3}{7}$ km.

Show solution

Understand

Five points A, B, C, D, E sit on a line in that order. I'm given the spans AC, BD, DE, and the overlapping span BC. Using the number-line figure I must find the total distance from A to E.

Givens

Points lie in order A, B, C, D, E on one line.
AC = 13 6/7 km.
BD = 17 4/7 km.
DE = 4 5/7 km.
BC = 5 3/7 km is the overlap shared by AC and BD.

Unknowns

The distance from A to E, in km.

Constraints

AC and BD overlap exactly on the segment BC, since B is between A and C and C is between B and D.

Plan

#1 Draw a Diagram · also uses: #7 Identify Subproblems#16 Count the Complement

The figure shows A-B-C-D-E with AC and BD sharing the overlap BC. Adding AC + BD double-counts BC, so AE = AC + BD + DE - BC. The diagram makes the overlap easy to subtract.

Execute

#1 Draw a Diagram 4.OA.A.3

From the figure: AC stretches from A to C, BD from B to D, and these two arcs overlap on segment BC. Adding AC and BD counts the overlap BC twice, so AE = AC + BD + DE - BC.

AE = AC + BD + DE - BC

Drawing the line shows the two arcs share the middle piece BC, so it must be removed once.

#7 Identify Subproblems 4.OA.A.3

Combine the whole numbers: 13 + 17 + 4 - 5 = 29.

13+17+4-5=29

Handling the whole parts first keeps the fraction work small.

#7 Identify Subproblems 4.NF.B.3

All fractions share denominator 7: (6+4+5-3)/7 = 12/7 = 1 5/7.

\dfrac{6}{7}+\dfrac{4}{7}+\dfrac{5}{7}-\dfrac{3}{7}=\dfrac{12}{7}=1\tfrac{5}{7}

Like denominators let you add and subtract numerators, then regroup.

#16 Count the Complement 4.NF.B.3

Add the whole-part total and the fraction-part total: 29 + 1 5/7 = 30 5/7.

29+1\tfrac{5}{7}=30\tfrac{5}{7}

Putting the pieces back together gives the full A-to-E distance.

Answer: 30 5/7 km

Review

Cross-check with segment pieces: AB = AC - BC = 8 3/7, CD = BD - BC = 12 1/7, so AE = AB + BC + CD + DE = 30 5/7 km. Same answer, and 30 5/7 is sensibly larger than each given span.

Work piece by piece (tool 7): find AB and CD by subtracting the overlap, then add AB + BC + CD + DE directly instead of using the overlap-subtraction formula.

Standards · min grade 4

4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Setting up AE = AC + BD + DE - BC and combining the whole-number parts.
4.NF.B.3 Understand a fraction with numerator greater than one as sum of unit fractions — Adding and subtracting the like-denominator 7ths and regrouping.

💡 This only needs Grade 4 fraction add/subtract — draw the line and remember the overlap gets counted twice, so subtract it once!

Variant 6 answer: 51 8/12 km

Use the figure to find the distance from point A to point E, in km.

Five points A, B, C, D, and E lie on one straight line in that order. The distance from A to C is $25\tfrac{5}{12}$ km, the distance from B to D is $28\tfrac{11}{12}$ km, and the distance from D to E is $7\tfrac{6}{12}$ km. Also, the distance from B to C (the overlapping section) is $10\tfrac{2}{12}$ km.

Show solution

Understand

Five points A, B, C, D, E sit on a line in that order. I'm given the spans AC, BD, DE, and the overlapping span BC. Using the number-line figure I must find the total distance from A to E.

Givens

Points lie in order A, B, C, D, E on one line.
AC = 25 5/12 km.
BD = 28 11/12 km.
DE = 7 6/12 km.
BC = 10 2/12 km is the overlap shared by AC and BD.

Unknowns

The distance from A to E, in km.

Constraints

AC and BD overlap exactly on the segment BC, since B is between A and C and C is between B and D.

Plan

#1 Draw a Diagram · also uses: #7 Identify Subproblems#16 Count the Complement

The figure shows A-B-C-D-E with AC and BD sharing the overlap BC. Adding AC + BD double-counts BC, so AE = AC + BD + DE - BC. The diagram makes the overlap easy to subtract.

Execute

#1 Draw a Diagram 4.OA.A.3

From the figure: AC stretches from A to C, BD from B to D, and these two arcs overlap on segment BC. Adding AC and BD counts the overlap BC twice, so AE = AC + BD + DE - BC.

AE = AC + BD + DE - BC

Drawing the line shows the two arcs share the middle piece BC, so it must be removed once.

#7 Identify Subproblems 4.OA.A.3

Combine the whole numbers: 25 + 28 + 7 - 10 = 50.

25+28+7-10=50

Handling the whole parts first keeps the fraction work small.

#7 Identify Subproblems 4.NF.B.3

All fractions share denominator 12: (5+11+6-2)/12 = 20/12 = 1 8/12.

\dfrac{5}{12}+\dfrac{11}{12}+\dfrac{6}{12}-\dfrac{2}{12}=\dfrac{20}{12}=1\tfrac{8}{12}

Like denominators let you add and subtract numerators, then regroup.

#16 Count the Complement 4.NF.B.3

Add the whole-part total and the fraction-part total: 50 + 1 8/12 = 51 8/12.

50+1\tfrac{8}{12}=51\tfrac{8}{12}

Putting the pieces back together gives the full A-to-E distance.

Answer: 51 8/12 km

Review

Cross-check with segment pieces: AB = AC - BC = 15 3/12, CD = BD - BC = 18 9/12, so AE = AB + BC + CD + DE = 51 8/12 km. Same answer, and 51 8/12 is sensibly larger than each given span.

Work piece by piece (tool 7): find AB and CD by subtracting the overlap, then add AB + BC + CD + DE directly instead of using the overlap-subtraction formula.

Standards · min grade 4

4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Setting up AE = AC + BD + DE - BC and combining the whole-number parts.
4.NF.B.3 Understand a fraction with numerator greater than one as sum of unit fractions — Adding and subtracting the like-denominator 12ths and regrouping.

💡 This only needs Grade 4 fraction add/subtract — draw the line and remember the overlap gets counted twice, so subtract it once!

Variant 7 answer: 35 3/5 km

Use the figure to find the distance from point A to point E, in km.

Five points A, B, C, D, and E lie on one straight line in that order. The distance from A to C is $17\tfrac{2}{5}$ km, the distance from B to D is $19\tfrac{4}{5}$ km, and the distance from D to E is $4\tfrac{3}{5}$ km. Also, the distance from B to C (the overlapping section) is $6\tfrac{1}{5}$ km.

Show solution

Understand

Five points A, B, C, D, E sit on a line in that order. I'm given the spans AC, BD, DE, and the overlapping span BC. Using the number-line figure I must find the total distance from A to E.

Givens

Points lie in order A, B, C, D, E on one line.
AC = 17 2/5 km.
BD = 19 4/5 km.
DE = 4 3/5 km.
BC = 6 1/5 km is the overlap shared by AC and BD.

Unknowns

The distance from A to E, in km.

Constraints

AC and BD overlap exactly on the segment BC, since B is between A and C and C is between B and D.

Plan

#1 Draw a Diagram · also uses: #7 Identify Subproblems#16 Count the Complement

The figure shows A-B-C-D-E with AC and BD sharing the overlap BC. Adding AC + BD double-counts BC, so AE = AC + BD + DE - BC. The diagram makes the overlap easy to subtract.

Execute

#1 Draw a Diagram 4.OA.A.3

From the figure: AC stretches from A to C, BD from B to D, and these two arcs overlap on segment BC. Adding AC and BD counts the overlap BC twice, so AE = AC + BD + DE - BC.

AE = AC + BD + DE - BC

Drawing the line shows the two arcs share the middle piece BC, so it must be removed once.

#7 Identify Subproblems 4.OA.A.3

Combine the whole numbers: 17 + 19 + 4 - 6 = 34.

17+19+4-6=34

Handling the whole parts first keeps the fraction work small.

#7 Identify Subproblems 4.NF.B.3

All fractions share denominator 5: (2+4+3-1)/5 = 8/5 = 1 3/5.

\dfrac{2}{5}+\dfrac{4}{5}+\dfrac{3}{5}-\dfrac{1}{5}=\dfrac{8}{5}=1\tfrac{3}{5}

Like denominators let you add and subtract numerators, then regroup.

#16 Count the Complement 4.NF.B.3

Add the whole-part total and the fraction-part total: 34 + 1 3/5 = 35 3/5.

34+1\tfrac{3}{5}=35\tfrac{3}{5}

Putting the pieces back together gives the full A-to-E distance.

Answer: 35 3/5 km

Review

Cross-check with segment pieces: AB = AC - BC = 11 1/5, CD = BD - BC = 13 3/5, so AE = AB + BC + CD + DE = 35 3/5 km. Same answer, and 35 3/5 is sensibly larger than each given span.

Work piece by piece (tool 7): find AB and CD by subtracting the overlap, then add AB + BC + CD + DE directly instead of using the overlap-subtraction formula.

Standards · min grade 4

4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Setting up AE = AC + BD + DE - BC and combining the whole-number parts.
4.NF.B.3 Understand a fraction with numerator greater than one as sum of unit fractions — Adding and subtracting the like-denominator 5ths and regrouping.

💡 This only needs Grade 4 fraction add/subtract — draw the line and remember the overlap gets counted twice, so subtract it once!

Variant 8 answer: 45 5/10 km

Use the figure to find the distance from point A to point E, in km.

Five points A, B, C, D, and E lie on one straight line in that order. The distance from A to C is $21\tfrac{7}{10}$ km, the distance from B to D is $24\tfrac{9}{10}$ km, and the distance from D to E is $8\tfrac{3}{10}$ km. Also, the distance from B to C (the overlapping section) is $9\tfrac{4}{10}$ km.

Show solution

Understand

Five points A, B, C, D, E sit on a line in that order. I'm given the spans AC, BD, DE, and the overlapping span BC. Using the number-line figure I must find the total distance from A to E.

Givens

Points lie in order A, B, C, D, E on one line.
AC = 21 7/10 km.
BD = 24 9/10 km.
DE = 8 3/10 km.
BC = 9 4/10 km is the overlap shared by AC and BD.

Unknowns

The distance from A to E, in km.

Constraints

AC and BD overlap exactly on the segment BC, since B is between A and C and C is between B and D.

Plan

#1 Draw a Diagram · also uses: #7 Identify Subproblems#16 Count the Complement

The figure shows A-B-C-D-E with AC and BD sharing the overlap BC. Adding AC + BD double-counts BC, so AE = AC + BD + DE - BC. The diagram makes the overlap easy to subtract.

Execute

#1 Draw a Diagram 4.OA.A.3

From the figure: AC stretches from A to C, BD from B to D, and these two arcs overlap on segment BC. Adding AC and BD counts the overlap BC twice, so AE = AC + BD + DE - BC.

AE = AC + BD + DE - BC

Drawing the line shows the two arcs share the middle piece BC, so it must be removed once.

#7 Identify Subproblems 4.OA.A.3

Combine the whole numbers: 21 + 24 + 8 - 9 = 44.

21+24+8-9=44

Handling the whole parts first keeps the fraction work small.

#7 Identify Subproblems 4.NF.B.3

All fractions share denominator 10: (7+9+3-4)/10 = 15/10 = 1 5/10.

\dfrac{7}{10}+\dfrac{9}{10}+\dfrac{3}{10}-\dfrac{4}{10}=\dfrac{15}{10}=1\tfrac{5}{10}

Like denominators let you add and subtract numerators, then regroup.

#16 Count the Complement 4.NF.B.3

Add the whole-part total and the fraction-part total: 44 + 1 5/10 = 45 5/10.

44+1\tfrac{5}{10}=45\tfrac{5}{10}

Putting the pieces back together gives the full A-to-E distance.

Answer: 45 5/10 km

Review

Cross-check with segment pieces: AB = AC - BC = 12 3/10, CD = BD - BC = 15 5/10, so AE = AB + BC + CD + DE = 45 5/10 km. Same answer, and 45 5/10 is sensibly larger than each given span.

Work piece by piece (tool 7): find AB and CD by subtracting the overlap, then add AB + BC + CD + DE directly instead of using the overlap-subtraction formula.

Standards · min grade 4

4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Setting up AE = AC + BD + DE - BC and combining the whole-number parts.
4.NF.B.3 Understand a fraction with numerator greater than one as sum of unit fractions — Adding and subtracting the like-denominator 10ths and regrouping.

💡 This only needs Grade 4 fraction add/subtract — draw the line and remember the overlap gets counted twice, so subtract it once!

Variant 9 answer: 36 1/3 km

Use the figure to find the distance from point A to point E, in km.

Five points A, B, C, D, and E lie on one straight line in that order. The distance from A to C is $16\tfrac{2}{3}$ km, the distance from B to D is $20\tfrac{1}{3}$ km, and the distance from D to E is $5\tfrac{2}{3}$ km. Also, the distance from B to C (the overlapping section) is $6\tfrac{1}{3}$ km.

Show solution

Understand

Five points A, B, C, D, E sit on a line in that order. I'm given the spans AC, BD, DE, and the overlapping span BC. Using the number-line figure I must find the total distance from A to E.

Givens

Points lie in order A, B, C, D, E on one line.
AC = 16 2/3 km.
BD = 20 1/3 km.
DE = 5 2/3 km.
BC = 6 1/3 km is the overlap shared by AC and BD.

Unknowns

The distance from A to E, in km.

Constraints

AC and BD overlap exactly on the segment BC, since B is between A and C and C is between B and D.

Plan

#1 Draw a Diagram · also uses: #7 Identify Subproblems#16 Count the Complement

The figure shows A-B-C-D-E with AC and BD sharing the overlap BC. Adding AC + BD double-counts BC, so AE = AC + BD + DE - BC. The diagram makes the overlap easy to subtract.

Execute

#1 Draw a Diagram 4.OA.A.3

From the figure: AC stretches from A to C, BD from B to D, and these two arcs overlap on segment BC. Adding AC and BD counts the overlap BC twice, so AE = AC + BD + DE - BC.

AE = AC + BD + DE - BC

Drawing the line shows the two arcs share the middle piece BC, so it must be removed once.

#7 Identify Subproblems 4.OA.A.3

Combine the whole numbers: 16 + 20 + 5 - 6 = 35.

16+20+5-6=35

Handling the whole parts first keeps the fraction work small.

#7 Identify Subproblems 4.NF.B.3

All fractions share denominator 3: (2+1+2-1)/3 = 4/3 = 1 1/3.

\dfrac{2}{3}+\dfrac{1}{3}+\dfrac{2}{3}-\dfrac{1}{3}=\dfrac{4}{3}=1\tfrac{1}{3}

Like denominators let you add and subtract numerators, then regroup.

#16 Count the Complement 4.NF.B.3

Add the whole-part total and the fraction-part total: 35 + 1 1/3 = 36 1/3.

35+1\tfrac{1}{3}=36\tfrac{1}{3}

Putting the pieces back together gives the full A-to-E distance.

Answer: 36 1/3 km

Review

Cross-check with segment pieces: AB = AC - BC = 10 1/3, CD = BD - BC = 14, so AE = AB + BC + CD + DE = 36 1/3 km. Same answer, and 36 1/3 is sensibly larger than each given span.

Work piece by piece (tool 7): find AB and CD by subtracting the overlap, then add AB + BC + CD + DE directly instead of using the overlap-subtraction formula.

Standards · min grade 4

4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Setting up AE = AC + BD + DE - BC and combining the whole-number parts.
4.NF.B.3 Understand a fraction with numerator greater than one as sum of unit fractions — Adding and subtracting the like-denominator 3ths and regrouping.

💡 This only needs Grade 4 fraction add/subtract — draw the line and remember the overlap gets counted twice, so subtract it once!

Variant 10 answer: 41 2/6 km

Use the figure to find the distance from point A to point E, in km.

Five points A, B, C, D, and E lie on one straight line in that order. The distance from A to C is $20\tfrac{5}{6}$ km, the distance from B to D is $22\tfrac{1}{6}$ km, and the distance from D to E is $5\tfrac{4}{6}$ km. Also, the distance from B to C (the overlapping section) is $7\tfrac{2}{6}$ km.

Show solution

Understand

Five points A, B, C, D, E sit on a line in that order. I'm given the spans AC, BD, DE, and the overlapping span BC. Using the number-line figure I must find the total distance from A to E.

Givens

Points lie in order A, B, C, D, E on one line.
AC = 20 5/6 km.
BD = 22 1/6 km.
DE = 5 4/6 km.
BC = 7 2/6 km is the overlap shared by AC and BD.

Unknowns

The distance from A to E, in km.

Constraints

AC and BD overlap exactly on the segment BC, since B is between A and C and C is between B and D.

Plan

#1 Draw a Diagram · also uses: #7 Identify Subproblems#16 Count the Complement

The figure shows A-B-C-D-E with AC and BD sharing the overlap BC. Adding AC + BD double-counts BC, so AE = AC + BD + DE - BC. The diagram makes the overlap easy to subtract.

Execute

#1 Draw a Diagram 4.OA.A.3

From the figure: AC stretches from A to C, BD from B to D, and these two arcs overlap on segment BC. Adding AC and BD counts the overlap BC twice, so AE = AC + BD + DE - BC.

AE = AC + BD + DE - BC

Drawing the line shows the two arcs share the middle piece BC, so it must be removed once.

#7 Identify Subproblems 4.OA.A.3

Combine the whole numbers: 20 + 22 + 5 - 7 = 40.

20+22+5-7=40

Handling the whole parts first keeps the fraction work small.

#7 Identify Subproblems 4.NF.B.3

All fractions share denominator 6: (5+1+4-2)/6 = 8/6 = 1 2/6.

\dfrac{5}{6}+\dfrac{1}{6}+\dfrac{4}{6}-\dfrac{2}{6}=\dfrac{8}{6}=1\tfrac{2}{6}

Like denominators let you add and subtract numerators, then regroup.

#16 Count the Complement 4.NF.B.3

Add the whole-part total and the fraction-part total: 40 + 1 2/6 = 41 2/6.

40+1\tfrac{2}{6}=41\tfrac{2}{6}

Putting the pieces back together gives the full A-to-E distance.

Answer: 41 2/6 km

Review

Cross-check with segment pieces: AB = AC - BC = 13 3/6, CD = BD - BC = 14 5/6, so AE = AB + BC + CD + DE = 41 2/6 km. Same answer, and 41 2/6 is sensibly larger than each given span.

Work piece by piece (tool 7): find AB and CD by subtracting the overlap, then add AB + BC + CD + DE directly instead of using the overlap-subtraction formula.

Standards · min grade 4

4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Setting up AE = AC + BD + DE - BC and combining the whole-number parts.
4.NF.B.3 Understand a fraction with numerator greater than one as sum of unit fractions — Adding and subtracting the like-denominator 6ths and regrouping.

💡 This only needs Grade 4 fraction add/subtract — draw the line and remember the overlap gets counted twice, so subtract it once!