Solve length problems
Theory
Solving length problems is a Year 4 measurement skill: reading a word problem, changing every length to the same metric unit (millimetres, centimetres, metres or kilometres), then adding, subtracting, multiplying or dividing to find the answer. This page shows how to use bar models, work out a missing side, and find a simple perimeter. Curriculum: VC2M4M02 / AC9M4M01, AC9M4M02.
A length problem is a word problem about how long, how far, or how much longer one thing is than another. Length is measured in millimetres (mm), centimetres (cm), metres (m) and kilometres (km).
You can only add or subtract lengths when they are in the same unit. So the first job is often to convert, using \(10\text{ mm} = 1\text{ cm}\), \(100\text{ cm} = 1\text{ m}\) and \(1000\text{ m} = 1\text{ km}\). Then you add to join lengths, subtract to cut off or compare, and multiply or divide for equal pieces.
Perimeter is the distance all the way around the edge of a shape. To find it, add the length of every side. A bar model is a drawing that shows the parts and the whole, which makes it clear whether to add or subtract.
Length problems do not need formulas, but these metric conversions are worth remembering. Bigger units are made of many smaller ones.
How to solve a length problem
- Match the units. Change every length to one unit, such as centimetres, using \(100\text{ cm} = 1\text{ m}\) and \(10\text{ mm} = 1\text{ cm}\).
- Draw a bar model. Show the parts you know and mark what you are looking for, so you can see whether to add or subtract.
- Work it out and rename. Add, subtract, multiply or divide, then write the answer in the unit the question asks for, such as metres and centimetres.
Which operation?
- Add when lengths are joined end to end, or for a whole journey. Going there and back means doubling.
- Subtract to find how much is left after cutting, or how much longer one length is.
- Multiply when the same length repeats, like steps or equal pieces in a row.
- Divide to find how many equal pieces fit into a length, or how many fence panels go around a perimeter.
| \(\text{units}\) | \(\text{both cm}\) | |
| \(24 + 38\) | \(=\) | \(62\) |
Together they are \(62\) cm long.
| \(1\text{ m } 30\text{ cm}\) | \(=\) | \(130\text{ cm}\) |
| \(130 - 85\) | \(=\) | \(45\) |
The blue rope is \(45\) cm longer.
| \(8 + 8\) | \(=\) | \(16\) |
| \(24 - 16\) | \(=\) | \(8\) |
| \(8 \div 2\) | \(=\) | \(4\) |
Each shorter side is \(4\) m.
| \(70 \times 4\) | \(=\) | \(280\text{ cm}\) |
| \(4\text{ m}\) | \(=\) | \(400\text{ cm}\) |
| \(400 - 280\) | \(=\) | \(120\text{ cm}\) |
| \(120\text{ cm}\) | \(=\) | \(1\text{ m } 20\text{ cm}\) |
There is \(1\) m \(20\) cm of tape left.
(i) How far is home to the shop?
(ii) He walks home the same way. How far does he walk altogether?
| \(\text{(i) } 430 + 290\) | \(=\) | \(720\text{ m}\) |
| \(\text{(ii) } 720 + 720\) | \(=\) | \(1440\text{ m}\) |
| \(1440\text{ m}\) | \(=\) | \(1\text{ km }440\text{ m}\) |
Going back the same way doubles the distance. \(1440\) m is more than \(1000\) m, so it is \(1\) km \(440\) m.
| \(1\text{ m }60\text{ cm}\) | \(=\) | \(160\text{ cm}\) |
| \(160 \div 40\) | \(=\) | \(4\) |
| \(\text{check: } 4 \times 40\) | \(=\) | \(160\) |
Equal pieces from a length means divide. There are \(4\) pieces.
| \(6 \times 60\) | \(=\) | \(360\text{ cm}\) |
| \(360\text{ cm}\) | \(=\) | \(3\text{ m }60\text{ cm}\) |
Equal lengths repeated means multiply. She walks about \(3\) m \(60\) cm.
| \(14\text{ cm}\) | \(=\) | \(140\text{ mm}\) |
| \(140 - 45\) | \(=\) | \(95\text{ mm}\) |
Change the centimetres to millimetres first, then subtract. The pencil is \(95\) mm longer.
| \(80 + 80\) | \(=\) | \(160\text{ cm}\) |
| \(300 - 160\) | \(=\) | \(140\text{ cm}\) |
| \(140\) | \(>\) | \(80\) |
Yes. \(140\) cm is left, which is more than \(80\) cm, so one more shelf can be cut with \(60\) cm to spare.
| \(\text{perimeter}\) | \(=\) | \(4 \times 9 = 36\text{ m}\) |
| \(36 \div 3\) | \(=\) | \(12\text{ panels}\) |
Find the perimeter first, then see how many panel lengths fit into it: \(12\) panels.
| \(1350 + 780\) | \(=\) | \(2130\) |
| \(2130 + 2150\) | \(=\) | \(4280\) |
| \(\text{total}\) | \(=\) | \(4280\text{ m}\) |
Add the legs one at a time. She rides \(4280\) m, which is \(4\) km \(280\) m.
| \(520 + 465\) | \(=\) | \(985\text{ km}\) |
| \(1340 - 985\) | \(=\) | \(355\text{ km}\) |
Add the parts already driven, then take them from the whole trip. \(355\) km is left.
Common pitfalls
Frequently asked questions
How do you solve a length word problem?
First change every length to the same unit, such as centimetres. Then decide whether to add, subtract, multiply or divide. A bar model helps you see the parts and the whole before you calculate.
How do you change centimetres to metres and metres to centimetres?
There are \(100\) centimetres in a metre. To change metres to centimetres, multiply by \(100\), so \(3\) m is \(300\) cm. To change centimetres to metres, group into hundreds, so \(250\) cm is \(2\) m \(50\) cm.
What is the difference between length and perimeter?
Length measures one side or one distance. Perimeter is the total distance all the way around the edge of a shape, so you add the lengths of every side.
How many millimetres are in a centimetre?
There are \(10\) millimetres in \(1\) centimetre. So \(7\) cm is \(70\) mm, and \(40\) mm is \(4\) cm.
How do you find a missing side when you know the perimeter?
Add the sides you already know, then subtract from the perimeter. For a rectangle, take away the two long sides, then share what is left between the two short sides by dividing by \(2\).
What unit should I use to measure length?
Use millimetres for very small lengths, centimetres for small objects, metres for rooms and gardens, and kilometres for long distances like a road or a run.