Subtract the overlap when joining fraction lengths

4.NF.B.3 · take · grade 4

Archetype: Overlap Reduces the Total · step in a 4-type progression

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 $17\dfrac{2}{5}$ km, the distance from B to D is $19\dfrac{4}{5}$ km, and the distance from D to E is $4\dfrac{3}{5}$ km. Also, the distance from B to C (the overlapping section) is $6\dfrac{1}{5}$ 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. AE = AC + CD + DE, where CD is the part of BD past C. Equivalently, AE = AC + BD + DE - BC, because adding AC and BD counts the overlap BC twice.

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: 2/5 + 4/5 + 3/5 - 1/5 = (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 8/5 as 1 3/5.

#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 = 11 1/5 + 6 1/5 + 13 3/5 + 4 3/5 = 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 fifths and regrouping 8/5 into 1 3/5.

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