Find a rectangle side via one unknown

3.MD.D.83.OA.A.4 · adapt · grade 3

Archetype: Find Two Unknowns from Sum and Difference · step in a 8-type progression

Liam used a piece of wire $88$ inches long, using all of it, to bend a rectangle. The long side of the rectangle is $3$ times the length of the short side. How many inches long is the long side?

Show solution

Understand

An 88-inch wire is bent into a rectangle whose long side is 3 times the short side, using all the wire. We need the length of the long side.

Givens

The wire is 88 inches long and all of it is used.
It forms a rectangle (the wire is the perimeter).
The long side is 3 times the short side.

Unknowns

The length of the long side in inches.

Constraints

The perimeter equals 88 inches (all wire used).
Long side = 3 times short side.

Plan

#1 Draw a Diagram · also uses: #6 Guess and Check

Sketch the rectangle and count the short-side units around the perimeter: a short, a long (3 shorts), and again, giving 8 equal short-side pieces; divide the wire among them.

Execute

#1 Draw a Diagram 3.MD.D.8

Going around: short + long + short + long. Each long is 3 shorts, so the perimeter is 1 + 3 + 1 + 3 = 8 short-side lengths.

1 + 3 + 1 + 3 = 8 \text{ short sides}

Measuring the whole border in short-side units turns the wire into 8 equal pieces.

#6 Guess and Check 3.OA.C.7

The 88-inch wire is divided into 8 equal short-side pieces.

88 \div 8 = 11 \text{ inches}

Sharing the total length equally among the 8 units gives one short side.

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

The long side is 3 times the short side, so multiply 11 by 3.

11 \times 3 = 33 \text{ inches}

The long side is three short pieces joined, so triple the short side.

Answer: 33 inches

Review

Check: short 11, long 33, perimeter = 2 times (11 + 33) = 2 times 44 = 88 inches, matching the wire exactly. The long side being longer than the short fits the picture.

Convert to algebra (tool 13): with short side s, perimeter 2(s + 3s) = 8s = 88, so s = 11 and long side = 3s = 33.

Standards · min grade 3

3.MD.D.8 Solve real-world problems involving perimeters of polygons — Expressing the 88-inch perimeter as a sum of short-side units.
3.OA.C.7 Fluently multiply and divide within 100 — Dividing 88 by 8 to find the short side.
3.OA.A.3 Solve multiplication and division word problems within 100 — Tripling the short side to get the long side.

💡 This only needs Grade 3 perimeter sense: the border is 8 short sides, so divide, then triple for the long side!