Problem
Find a missing side, then sum sides as fractions
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.
FractionsMeasurement & data
Your answer
cm
Watch out Height is not simply 1 11/15 cm — that's how much longer than the width it is; you must add it to the width first.
How to solve
Strategy Identify Subproblems — 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.
1STEP 1
Find the height
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 5/15+1 11/15=4+16/15=5 1/15
Same-denominator add, then regroup 16/15 as one whole and 1/15.
4.NF.B.3Identify Subproblems2STEP 2
Add width and height
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 5/15+5 1/15=8 6/15
Adding one width and one height gives half the perimeter.
4.NF.B.3Identify Subproblems3STEP 3
Double to get the perimeter
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.
2×8 6/15=16 12/15=16 4/5
A rectangle has two equal widths and two equal heights, so doubling the half-perimeter gives the whole.
4.MD.A.3Identify SubproblemsDoubling the sum of one width and one height gives the rectangle's whole perimeter.
Why?
The perimeter is the whole trip around the rectangle, which is its four sides put back together with no gaps or overlaps.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
Why?
The four sides are two equal widths and two equal heights, so their total, width plus width plus height plus height, is the same as two of one width and one height together.
Why?
Adding the same width twice is two equal groups of that width, and adding the same height twice is two equal groups of that height.
🧱Multiplication as equal groupsa times b means a equal groups of b — it is just repeated adding of the same amount.
Why?
Two widths plus two heights is the same as two of the width-and-height pair, so the shared count of two can be taken outside the sum.
🧱Distributive propertya groups of (b and c) is a groups of b plus a groups of c — you can split the group.
Answer
16 4/5 cm
2 × 8 6/15 = 16 4/5
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.
Takeaway
This only needs Grade 4 fraction adding plus the rectangle perimeter formula — two widths plus two heights!
- Find the height
- Add width and height
- Double to get the perimeter
Where next?
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