Overlap shrinks the total length

2.MD.B.53.OA.D.8 · adapt · grade 3

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

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 the span A-to-C (288 cm), the span 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 all the way 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

The figure shows two spans that overlap in the middle. 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, and breaking the line into the pieces AB, BC, CD turns it into a tidy addition and subtraction.

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. If I add the two spans together, the middle piece BC gets included in both, so it is counted two times.

AC + BD = (AB + BC) + (BC + CD)

Drawing the two bars on the same line shows the shared middle plainly, the way a Grade 2 length-line picture does.

#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 exactly the regrouping skill practiced in Grade 3.

#7 Identify Subproblems 3.NBT.A.2

Because the middle part B-to-C was counted twice, take it away one time to get the true length from A to D.

705 - 169 = 536

Removing the extra copy of the overlap is a single three-digit subtraction.

Answer: 536 cm

Review

The answer is in cm, as asked. 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 417 and 705.

Work piece by piece (Tool 9, easier related problem): AB = AC - BC = 288 - 169 = 119, CD = BD - BC = 417 - 169 = 248, then AD = AB + BC + CD = 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 and reasoning about how the pieces add up.
3.NBT.A.2 Fluently add and subtract within 1000 — Adding the two spans (288 + 417) and subtracting the double-counted overlap (705 - 169).

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