Practice · Subtract the overlap when joining fraction lengths · Sensim Math

Variant 1 easy answer:

19\frac{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

1 · Understandwhat's really being asked

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 $\frac{4}{5}$ km.
BD = 11 $\frac{2}{5}$ km.
DE = 3 $\frac{1}{5}$ km.
BC = 3 $\frac{3}{5}$ 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.

2 · Planchoose the strategy

#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.

3 · Execute4 carry out the plan

1Read the figure and set up the spans

#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.

2Add the whole-number parts

#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.

3Add the fraction parts

#7 Identify Subproblems 4.NF.B.3

All fractions share denominator 5:

\frac{4+2+1-3}{5}

=

\frac{4}{5}

=

\frac{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.

4Combine for AE

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

Add the whole-part total and the fraction-part total: 19 +

\frac{4}{5}

= 19

\frac{4}{5}

.

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

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

Answer:

19\frac{4}{5}

km

4 · Reviewdoes it hold up?

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

Another way: 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.

Standardsmin 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.

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

Variant 2 easy 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

1 · Understandwhat's really being asked

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 $\frac{1}{4}$ km.
BD = 13 $\frac{3}{4}$ km.
DE = 2 $\frac{2}{4}$ km.
BC = 3 $\frac{2}{4}$ 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.

2 · Planchoose the strategy

#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.

3 · Execute4 carry out the plan

1Read the figure and set up the spans

#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.

2Add the whole-number parts

#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.

3Add the fraction parts

#7 Identify Subproblems 4.NF.B.3

All fractions share denominator 4:

\frac{1+3+2-2}{4}

=

\frac{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.

4Combine for AE

#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

4 · Reviewdoes it hold up?

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

Another way: 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.

Standardsmin 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.

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

Variant 3 easy answer:

26\frac{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

1 · Understandwhat's really being asked

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 $\frac{1}{5}$ km.
BD = 15 $\frac{3}{5}$ km.
DE = 3 $\frac{2}{5}$ km.
BC = 4 $\frac{2}{5}$ 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.

2 · Planchoose the strategy

#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.

3 · Execute4 carry out the plan

1Read the figure and set up the spans

#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.

2Add the whole-number parts

#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.

3Add the fraction parts

#7 Identify Subproblems 4.NF.B.3

All fractions share denominator 5:

\frac{1+3+2-2}{5}

=

\frac{4}{5}

=

\frac{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.

4Combine for AE

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

Add the whole-part total and the fraction-part total: 26 +

\frac{4}{5}

= 26

\frac{4}{5}

.

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

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

Answer:

26\frac{4}{5}

km

4 · Reviewdoes it hold up?

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

Another way: 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.

Standardsmin 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.

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

Variant 4 medium answer:

30\frac{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

1 · Understandwhat's really being asked

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 $\frac{6}{7}$ km.
BD = 17 $\frac{4}{7}$ km.
DE = 4 $\frac{5}{7}$ km.
BC = 5 $\frac{3}{7}$ 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.

2 · Planchoose the strategy

#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.

3 · Execute4 carry out the plan

1Read the figure and set up the spans

#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.

2Add the whole-number parts

#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.

3Add the fraction parts

#7 Identify Subproblems 4.NF.B.3

All fractions share denominator 7:

\frac{6+4+5-3}{7}

=

\frac{12}{7}

= 1

\frac{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.

4Combine for AE

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

Add the whole-part total and the fraction-part total: 29 + 1

\frac{5}{7}

= 30

\frac{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\frac{5}{7}

km

4 · Reviewdoes it hold up?

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

Another way: 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.

Standardsmin 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.

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

Variant 5 medium answer:

34\frac{3}{4}

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

1 · Understandwhat's really being asked

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 $\frac{3}{8}$ km.
BD = 18 $\frac{5}{8}$ km.
DE = 6 $\frac{7}{8}$ km.
BC = 5 $\frac{1}{8}$ 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.

2 · Planchoose the strategy

#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.

3 · Execute4 carry out the plan

1Read the figure and set up the spans

#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.

2Add the whole-number parts

#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.

3Add the fraction parts

#7 Identify Subproblems 4.NF.B.3

All fractions share denominator 8:

\frac{3+5+7-1}{8}

=

\frac{14}{8}

= 1

\frac{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.

4Combine for AE

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

Add the whole-part total and the fraction-part total: 33 + 1

\frac{6}{8}

= 34

\frac{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\frac{3}{4}

km

4 · Reviewdoes it hold up?

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

Another way: 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.

Standardsmin 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.

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

Variant 6 medium answer:

35\frac{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

1 · Understandwhat's really being asked

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 $\frac{2}{5}$ km.
BD = 19 $\frac{4}{5}$ km.
DE = 4 $\frac{3}{5}$ km.
BC = 6 $\frac{1}{5}$ 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.

2 · Planchoose the strategy

#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.

3 · Execute4 carry out the plan

1Read the figure and set up the spans

#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.

2Add the whole-number parts

#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.

3Add the fraction parts

#7 Identify Subproblems 4.NF.B.3

All fractions share denominator 5:

\frac{2+4+3-1}{5}

=

\frac{8}{5}

= 1

\frac{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.

4Combine for AE

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

Add the whole-part total and the fraction-part total: 34 + 1

\frac{3}{5}

= 35

\frac{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\frac{3}{5}

km

4 · Reviewdoes it hold up?

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

Another way: 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.

Standardsmin 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.

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

Variant 7 medium answer:

36\frac{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

1 · Understandwhat's really being asked

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 $\frac{2}{3}$ km.
BD = 20 $\frac{1}{3}$ km.
DE = 5 $\frac{2}{3}$ km.
BC = 6 $\frac{1}{3}$ 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.

2 · Planchoose the strategy

#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.

3 · Execute4 carry out the plan

1Read the figure and set up the spans

#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.

2Add the whole-number parts

#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.

3Add the fraction parts

#7 Identify Subproblems 4.NF.B.3

All fractions share denominator 3:

\frac{2+1+2-1}{3}

=

\frac{4}{3}

= 1

\frac{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.

4Combine for AE

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

Add the whole-part total and the fraction-part total: 35 + 1

\frac{1}{3}

= 36

\frac{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\frac{1}{3}

km

4 · Reviewdoes it hold up?

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

Another way: 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.

Standardsmin 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.

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

Variant 8 hard answer:

41\frac{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 $20\tfrac{5}{6}$ 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

1 · Understandwhat's really being asked

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 $\frac{5}{6}$ km.
BD = 22 $\frac{1}{6}$ km.
DE = 5 $\frac{4}{6}$ km.
BC = 7 $\frac{2}{6}$ 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.

2 · Planchoose the strategy

#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.

3 · Execute4 carry out the plan

1Read the figure and set up the spans

#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.

2Add the whole-number parts

#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.

3Add the fraction parts

#7 Identify Subproblems 4.NF.B.3

All fractions share denominator 6:

\frac{5+1+4-2}{6}

=

\frac{8}{6}

= 1

\frac{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.

4Combine for AE

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

Add the whole-part total and the fraction-part total: 40 + 1

\frac{2}{6}

= 41

\frac{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\frac{1}{3}

km

4 · Reviewdoes it hold up?

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

Another way: 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.

Standardsmin 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.

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

Variant 9 hard answer:

45\frac{1}{2}

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

1 · Understandwhat's really being asked

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 $\frac{7}{10}$ km.
BD = 24 $\frac{9}{10}$ km.
DE = 8 $\frac{3}{10}$ km.
BC = 9 $\frac{4}{10}$ 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.

2 · Planchoose the strategy

#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.

3 · Execute4 carry out the plan

1Read the figure and set up the spans

#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.

2Add the whole-number parts

#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.

3Add the fraction parts

#7 Identify Subproblems 4.NF.B.3

All fractions share denominator 10:

\frac{7+9+3-4}{10}

=

\frac{15}{10}

= 1

\frac{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.

4Combine for AE

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

Add the whole-part total and the fraction-part total: 44 + 1

\frac{5}{10}

= 45

\frac{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\frac{1}{2}

km

4 · Reviewdoes it hold up?

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

Another way: 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.

Standardsmin 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.

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

Variant 10 hard answer:

51\frac{2}{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 $25\tfrac{5}{12}$ 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

1 · Understandwhat's really being asked

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 $\frac{5}{12}$ km.
BD = 28 $\frac{11}{12}$ km.
DE = 7 $\frac{6}{12}$ km.
BC = 10 $\frac{2}{12}$ 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.

2 · Planchoose the strategy

#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.

3 · Execute4 carry out the plan

1Read the figure and set up the spans

#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.

2Add the whole-number parts

#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.

3Add the fraction parts

#7 Identify Subproblems 4.NF.B.3

All fractions share denominator 12:

\frac{5+11+6-2}{12}

=

\frac{20}{12}

= 1

\frac{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.

4Combine for AE

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

Add the whole-part total and the fraction-part total: 50 + 1

\frac{8}{12}

= 51

\frac{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\frac{2}{3}

km

4 · Reviewdoes it hold up?

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

Another way: 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.

Standardsmin 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.

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

Subtract the overlap when joining fraction lengths · 10 practice problems

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