Problem
Perimeter is the sum of all side lengths
A rectangle has a perimeter of 10 2/15 cm. Its width (the horizontal side) is 2 2/5 cm. We need to find how long the height (the vertical side) is.
FractionsMeasurement & data
Your answer
How to solve
Strategy Draw a Diagram — Sketching the rectangle and labelling its four sides makes it clear that the perimeter is the sum of two widths and two heights. From there we work backwards: we already know the whole perimeter and the width, so we peel those amounts away to uncover the two heights, then split into one height. This keeps the reasoning visual and fraction-arithmetic based rather than reaching for algebra.
1STEP 1
Draw and label the rectangle
Label the two horizontal sides as the width and the two vertical ones as the unknown height.
P = w + w + h + h = 10 2/15 cm
Seeing all four sides on a picture makes 'perimeter is the sum of all the sides' concrete, which is exactly the grade-3 idea of perimeter.
3.MD.D.8Draw A Diagram2STEP 2
Add the two widths
The two horizontal sides are each 2 2/5 cm, so together they take up double that amount. Convert to an improper fraction and double it.
2 × 2 2/5 = 2 × 12/5 = 24/5 = 4 4/5 cm
Doubling a fraction is just adding it to itself; a fifth grader can scale 12/5 up to 24/5.
5.NF.A.1Work Backwards3STEP 3
Subtract the widths from the perimeter to leave the two heights
Taking the two widths off the perimeter leaves the two heights together: 16/3 cm.
10 2/15 - 4 4/5 = 152/15 - 72/15 = 80/15 = 16/3 cm
Working backwards from the total, removing what we know (the widths) leaves what we want (the two heights) — subtraction undoes the adding-on of the widths.
5.NF.A.2Draw A DiagramTaking the two widths off the perimeter leaves the two heights.
Why?
The perimeter is the whole way round the rectangle, made of the two widths and the two heights with no edge missed and none walked twice.
🧱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?
Removing the two widths undoes the part they contributed to that total, so what is left is what the heights contributed.
🧱Inverse operationsSubtraction undoes addition and division undoes multiplication — they reverse each other.
4STEP 4
Split the two heights into one height
The leftover 16/3 cm is two equal heights, so one height is half of it.
16/3 ÷ 2 = 8/3 = 2 2/3 cm
Two equal pieces add to 16/3, so each piece is half — splitting a fraction into two equal parts is sharing it evenly.
5.NF.A.1Work BackwardsAnswer
cm
Check by rebuilding the perimeter: 2 × 2 2/5 + 2 × 2 2/3 = 24/5 + 16/3 = 72/15 + 80/15 = 152/15 = 10 2/15 cm, which matches the given perimeter. The height 2 2/3 cm is a little larger than the width 2 2/5 cm, fitting a slightly taller (portrait) rectangle, and the units stay in centimeters throughout.
Takeaway
A rectangle's perimeter is all four sides added up, so take away the two sides you know and split what's left to find the missing side.
- Draw and label the rectangle
- Add the two widths
- Subtract the widths from the perimeter to leave the two heights
- Split the two heights into one height
Where next?
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