Find a missing side, then sum sides as fractions

4.NF.B.34.MD.A.3 · take · grade 4

Archetype: Perimeter by Tracing Every Side · step in a 11-type progression

The width of a rectangle is $3\dfrac{5}{15}$ cm, and its height is $1\dfrac{11}{15}$ cm longer than the width. Find the sum of the lengths of the four sides of this rectangle, in cm.

Show solution

Understand

A rectangle has width 3 5/15 cm. Its height is 1 11/15 cm longer than the width. I must find the perimeter (the sum of all four side lengths) in cm.

Givens

Width = 3 5/15 cm.
Height = width + 1 11/15 cm.
A rectangle has two widths and two heights.

Unknowns

The perimeter (sum of the four sides) of the rectangle.

Constraints

Fraction parts share denominator 15; perimeter = 2 × (width + height).

Plan

#7 Identify Subproblems · also uses: #1 Draw a Diagram

First find the height by adding to the width, then apply the rectangle perimeter formula P = 2 × (width + height). Two clean subproblems, both with denominator 15.

Execute

#7 Identify Subproblems 4.NF.B.3

Height = width + 1 11/15 = 3 5/15 + 1 11/15. Whole: 3+1 = 4. Fractions: 5/15 + 11/15 = 16/15 = 1 1/15. So height = 4 + 1 1/15 = 5 1/15 cm.

3\tfrac{5}{15}+1\tfrac{11}{15}=4+\dfrac{16}{15}=5\tfrac{1}{15}

Same-denominator add, then regroup 16/15 as one whole and 1/15.

#7 Identify Subproblems 4.NF.B.3

width + height = 3 5/15 + 5 1/15. Whole: 3+5 = 8. Fractions: 5/15 + 1/15 = 6/15. So the sum is 8 6/15 cm.

3\tfrac{5}{15}+5\tfrac{1}{15}=8\tfrac{6}{15}

Adding one width and one height gives half the perimeter.

#7 Identify Subproblems 4.MD.A.3

Perimeter = 2 × (width + height) = 2 × 8 6/15. Whole: 2×8 = 16. Fraction: 2 × 6/15 = 12/15 = 4/5. So perimeter = 16 12/15 = 16 4/5 cm.

2\times 8\tfrac{6}{15}=16\tfrac{12}{15}=16\tfrac{4}{5}

A rectangle has two equal widths and two equal heights, so doubling the half-perimeter gives the whole.

Answer: 16 4/5 cm

Review

Width ≈ 3.33 cm and height ≈ 5.07 cm, so perimeter ≈ 2 × 8.4 ≈ 16.8 cm, matching 16 4/5 = 16.8 cm. Units stay in cm and 12/15 simplifies neatly to 4/5.

Add all four sides separately (tool 1): 3 5/15 + 3 5/15 + 5 1/15 + 5 1/15 = 16 12/15 = 16 4/5 cm, the same result.

Standards · min grade 4

4.NF.B.3 Understand a fraction with numerator greater than one as sum of unit fractions — Adding the mixed-number side lengths with denominator 15 and regrouping improper fractions.
4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems — Using perimeter = 2 × (width + height) to total the four sides.

💡 This only needs Grade 4 fraction adding plus the rectangle perimeter formula — two widths plus two heights!