Mixed-unit arithmetic and comparisons

Many real-life problems give you lengths in different units: km here, m there, even cm. Before you can add, subtract or compare them, you must bring everything to the same unit. After that it is just ordinary arithmetic with whole numbers.

The golden rule

Convert all values to the same unit first. Then add, subtract or compare.

The smartest choice is usually the smallest unit that appears in the problem. That way every other value is multiplied by an integer factor and stays a whole number.

Two units in one expression

Sometimes the same number is split between a larger and a smaller unit, like $80\text{dm}5\text{cm}$. The two parts mean the sum of the two lengths.

To convert it to a single unit (centimetres in this case):

Adding and subtracting in mixed units

Example: $7\text{km}-4000\text{m}-10000\text{cm}=?\text{m}$.

Convert everything to metres:

• $7\text{km}=7\times 1000=7000\text{m}$
• $4000\text{m}$ stays as $4000\text{m}$
• $10000\text{cm}=10000/100=100\text{m}$

Then operate on the integers:

Comparing two lengths in different units

Question: which is longer, $5\text{km}$ or $4500\text{m}$?

Convert one of them to the other unit:

• $5\text{km}=5000\text{m}$, and $5000\text{m}>4500\text{m}$. ✓

So $5\text{km}>4500\text{m}$.

A handy strategy: pick the smaller unit (here metres) and convert the other value to it. The numbers stay integers and the comparison becomes obvious.

Comparing two compound expressions

Question: is $4000\text{m}10000\text{cm}$ greater than, equal to, or less than $6000\text{m}1000\text{cm}$?

Convert both sides to centimetres (the smaller unit):

• $4000\text{m}10000\text{cm}=4000\times 100+10000=400000+10000=410000\text{cm}$
• $6000\text{m}1000\text{cm}=6000\times 100+1000=600000+1000=601000\text{cm}$

So $4000\text{m}10000\text{cm}<6000\text{m}1000\text{cm}$.

Common pitfalls

• Working in the wrong unit. If you choose metres for an expression that has centimetres, you may end up with non-integer values. The smallest unit is the safest.
• Ignoring the order of magnitude. $1000\text{cm}$ feels small, but it is $10\text{m}$ — a big number once converted.
• Mixing systems. Never compare metric to imperial without converting both to the same family.

Read more

• Length unit conversions – guide
• Metric system
• Converting larger to smaller
• Converting smaller to larger

Practise

• Convert two units to one
• Arithmetic with mixed units
• Compare lengths in different units
• Compare lengths in mixed units