Solve problems involving length
Theory
Length problems often mix units, such as metres and centimetres. The key step is to change everything to the same unit first, then add, subtract, multiply or divide. This Year 5 maths page shows how to do that, how to cut a length into equal pieces and find what is left, and how to use perimeter for fences, posts and bunting.
Use \(1\text{ cm} = 10\text{ mm}\), \(1\text{ m} = 100\text{ cm}\) and \(1\text{ km} = 1000\text{ m}\) to change units. \(2.6\text{ km} = 2600\text{ m}\) and \(4\text{ m} = 400\text{ cm}\).
For equal pieces, divide the total length by the length of one piece. The remainder is the leftover, in the same unit.
If everything must be covered (bunting, fencing packs), round up to the next whole pack.
Posts placed every few metres around a closed shape: number of posts = perimeter \(\div\) gap.
Convert first, then calculate:
How to solve a length problem
- Read the problem and decide the unit for the answer.
- Convert every length to that unit.
- Choose the operation (or operations) and calculate.
- Answer with the unit, and check it is sensible.
| \(1450 + 760 + 1900\) | \(=\) | \(4110\text{ m}\) |
\(4110\) m (about \(4.1\) km).
| \(4\text{ m}\) | \(=\) | \(400\text{ cm}\) |
| \(400 - 135\) | \(=\) | \(265\text{ cm}\) |
\(265\) cm.
| \(2600 - 1450\) | \(=\) | \(1150\text{ m}\) |
\(1150\) m longer.
(i) How many whole pieces can be cut?
(ii) How much rope is left over?
| \(5\text{ m}\) | \(=\) | \(500\text{ cm}\) |
| \(500 \div 60\) | \(=\) | \(8 \text{ r } 20\) |
(i) \(8\) pieces
(ii) \(20\) cm
| \(120 + 85 + 150\) | \(=\) | \(355\text{ cm}\) |
| \(600 - 355\) | \(=\) | \(245\text{ cm}\) |
\(245\) cm.
| \(2 \times (36 + 24)\) | \(=\) | \(120\text{ m}\) |
| \(120 \div 4\) | \(=\) | \(30\) |
\(30\) posts (the last gap ends back at the first post).
| \(2 \times (15 + 9)\) | \(=\) | \(48\text{ m}\) |
| \(48 \div 10\) | \(=\) | \(4 \text{ r } 8\) |
\(4\) packs are not enough, so \(5\) packs are needed.
The working is in centimetres, so the answer is \(370\) cm, which is \(3.7\) m, not \(370\) m.
Common pitfalls
Frequently asked questions
Why convert units first?
You can only add or subtract lengths in the same unit. \(4\) m and \(135\) cm become \(400\) cm and \(135\) cm.
How do you find how many pieces can be cut?
Divide the total by the piece length: \(500 \div 60 = 8 \text{ r } 20\), so \(8\) pieces with \(20\) cm left.
When should you round up?
When everything must be covered, such as buying enough packs of bunting.
How many posts go around a rectangle?
Perimeter divided by the gap between posts: \(120 \div 4 = 30\).
How do you change kilometres to metres?
Multiply by \(1000\): \(2.6\) km \(= 2600\) m.