FN13, FR1, FG12, HN13, HR1, HG14 - Units and conversions

Metric measures

If $2.4\,\mathrm m=240\,\mathrm{cm}$, does $2.4\,\mathrm m^2=240\,\mathrm{cm}^2$ as well? We will return to that question after finding what a unit conversion actually counts.

Units tell you the size of the pieces you are counting. One metre contains $100$ centimetres, so the same length needs a number $100$ times as large when you count centimetres. The length has not grown: the unit is smaller.

Choose the factor from the units, rather than move the decimal point by a guessed number of places.

Smaller units give a larger numerical value; larger units give a smaller numerical value. This is a sense check, not a substitute for knowing the factor. For instance, converting kilometres to metres and kilograms to grams uses $1000$; converting metres to centimetres uses $100$.

Keep the same numerical value, but change the units. This time the factor is $1000$, not $100$.

In the reverse direction, divide by the factor. Thus $1450\,\mathrm{ml}=1.45\,\mathrm L$, because $1450\div1000=1.45$. Money works the same way: $275$ pence is £$2.75$, because $275\div100=2.75$; £$4.08$ is $408$ pence. The zero in a money amount keeps the pence in the correct place.

Convert to a common unit before combining measures. A ribbon of $1.2\,\mathrm m$ and a ribbon of $35\,\mathrm{cm}$ have total length $120+35=155\,\mathrm{cm}$, not $36.2$ of an unspecified unit.

Use the factor and keep all lengths in one unit.

Time

Time conversions use groups of $60$, rather than the groups of $10$, $100$ and $1000$ used in metric measures.

Use $60$ seconds $=1$ minute, $60$ minutes $=1$ hour and $24$ hours $=1$ day. There are $7$ days in a week, but months do not all have the same number of days.

The 24-hour clock runs from 00:00 at midnight through 12:00 at noon to 23:59 just before the next midnight. For example, 16:20 means 4:20 pm. On a 12-hour clock, 12 am means midnight and 12 pm means noon. A time such as 18:45 has $18$ hours and $45$ minutes, not $18.45$ hours.

For elapsed time, count forwards through convenient hour boundaries. Specify the next day when the finish time is after midnight.

Now move the same length of journey across midnight. The clock restarts at 00:00, but elapsed time keeps increasing.

Decimal hours need a conversion too. In $1.35$ hours, the decimal part means $0.35$ of an hour:

So $1.35$ hours means $1$ hour $21$ minutes, not $1$ hour $35$ minutes. In the reverse direction, $2$ hours $18$ minutes is $2+18\div60=2.3$ hours. To change $2.5$ minutes to seconds, calculate $2.5\times60=150$ seconds.

Handle the day boundary and a decimal duration separately.

Area and volume units

An area counts unit squares; a volume counts unit cubes. A length conversion therefore acts in two directions for area and three directions for volume.

Because $1\,\mathrm{cm}=10\,\mathrm{mm}$, a square centimetre contains $10$ rows of $10$ square millimetres: $100\,\mathrm{mm}^2$. A cubic centimetre contains $10$ layers, each with $100$ cubic millimetres: $1000\,\mathrm{mm}^3$.

[DIAGRAM: asset_name: Squared and cubed unit factors; asset_slug: squared-and-cubed-units; caption: Ten smaller units on each edge give one hundred squares or one thousand cubes.; description: A square with sides one centimetre is divided into ten rows of ten one-millimetre squares, showing one square centimetre equals one hundred square millimetres. A cube with edges one centimetre has ten unit lengths on each of its three edges, giving ten layers of one hundred one-millimetre cubes and one thousand cubic millimetres.]

The same reasoning applies to metres and centimetres. The length factor is $100$, so the area factor is $100\times100=10\,000$, and the volume factor is $100\times100\times100=1\,000\,000$.

In general, square the length factor for area and cube it for volume. Divide by the corresponding factor when converting in the reverse direction.

Change the dimension from area to volume while keeping the same starting numerical value. A third direction adds another factor of $100$.

Volume and capacity are linked by $1\,\mathrm{cm}^3=1\,\mathrm{ml}$ and $1000\,\mathrm{cm}^3=1\,\mathrm L$. Therefore $1\,\mathrm m^3=1000\,\mathrm L$. For example, $2500\,\mathrm{cm}^3$ is $2500\,\mathrm{ml}=2.5\,\mathrm L$. Do not confuse a cubic metre with a litre.

This answers the opening question: $2.4\,\mathrm m$ is $240\,\mathrm{cm}$, but $2.4\,\mathrm m^2$ is $24\,000\,\mathrm{cm}^2$. The numerical conversion factor depends on what the units measure.

Convert a fresh volume using two different unit relationships.

Quick Recap

• A conversion changes the numerical value and unit together while preserving the quantity.
• Convert to a common unit before combining measures.
• Time uses $60$ minutes per hour. Count forwards across hour and midnight boundaries.
• Square a length conversion factor for area and cube it for volume.

The opening area needs a factor of $10\,000$, even though the corresponding length conversion needs only $100$.