Mass

To estimate a mass without weighing, keep a few reference masses in mind (approximate values):

\(30\) mg	a grain of rice
\(1\) g	a paper clip
\(1\) kg	a full one-litre bottle of water
\(1\) t	a small car

To estimate a mass:

1. Choose the reference suited to the object.
2. Estimate how many reference objects together would have the same mass.
3. Give the estimate with a suitable unit.

A mass such as \(3.4~\mathrm{kg}\) is a whole number of kilograms and a part of a kilogram: \(3\) kilograms and \(4\) tenths of a kilogram. Since \(1~\mathrm{kg}=1\,000~\mathrm{g}\):

• a tenth of a kilogram is \(100~\mathrm{g}\): the \(1\,000~\mathrm{g}\) of one kilogram are cut into \(10\) equal parts;
• a hundredth of a kilogram is \(10~\mathrm{g}\): the \(1\,000~\mathrm{g}\) are cut into \(100\) equal parts.

• Towards the smaller unit: split the mass into its whole units and its tenths (or hundredths), then change each part. $$\begin{aligned}[t] 3.4~\mathrm{kg} &= 3~\mathrm{kg} + 4 \text{ tenths of a kilogram} &&\text{(3 units and 4 tenths)}\\ &= 3\,000~\mathrm{g} + 4 \times 100~\mathrm{g} &&\text{(1 kg = 1 000 g; a tenth = 100 g)}\\ &= 3\,000~\mathrm{g} + 400~\mathrm{g} &&\text{(multiplication)}\\ &= 3\,400~\mathrm{g} \end{aligned}$$ $$\begin{aligned}[t] 1.25~\mathrm{kg} &= 1~\mathrm{kg} + 2 \text{ tenths} + 5 \text{ hundredths of a kilogram} &&\text{(the digits after the point)}\\ &= 1\,000~\mathrm{g} + 200~\mathrm{g} + 50~\mathrm{g} &&\text{(a tenth = 100 g; a hundredth = 10 g)}\\ &= 1\,250~\mathrm{g} \end{aligned}$$
• Towards the bigger unit: group the whole units first, then write what is left as tenths (or hundredths). $$\begin{aligned}[t] 2\,700~\mathrm{g} &= 2\,000~\mathrm{g} + 700~\mathrm{g} &&\text{(the thousands first)}\\ &= 2~\mathrm{kg} + 7 \text{ tenths of a kilogram} &&\text{(1 000 g = 1 kg; 700 g is 7 tenths)}\\ &= 2.7~\mathrm{kg} \end{aligned}$$ $$\begin{aligned}[t] 2\,750~\mathrm{g} &= 2\,000~\mathrm{g} + 700~\mathrm{g} + 50~\mathrm{g} &&\text{(thousands, hundreds, tens)}\\ &= 2~\mathrm{kg} + 7 \text{ tenths} + 5 \text{ hundredths of a kilogram} &&\text{(100 g is a tenth; 10 g is a hundredth)}\\ &= 2.75~\mathrm{kg} \end{aligned}$$ Here the part left is always a whole number of tenths or hundredths of the bigger unit.
• The same with the other units: \(1~\mathrm{t}=1\,000~\mathrm{kg}\), so a tenth of a tonne is \(100~\mathrm{kg}\): \(2.3~\mathrm{t} = 2\,000~\mathrm{kg} + 300~\mathrm{kg} = 2\,300~\mathrm{kg}\). And \(1~\mathrm{g}=1\,000~\mathrm{mg}\), so a tenth of a gram is \(100~\mathrm{mg}\): \(0.4~\mathrm{g} = 4 \times 100~\mathrm{mg} = 400~\mathrm{mg}\).
• Check: \(3.4~\mathrm{kg}\) lies between \(3~\mathrm{kg}\) and \(4~\mathrm{kg}\), so the answer lies between \(3\,000~\mathrm{g}\) and \(4\,000~\mathrm{g}\).

• To convert from a bigger unit to a smaller one (kg \(\to\) g), you multiply by \(1\,000\).
  Example: \(3 \,\text{kg} = 3 \times 1\,000 \,\text{g} = 3\,000 \,\text{g}\).
• To convert from a smaller unit to a bigger one (g \(\to\) kg), you divide by \(1\,000\).
  Example: \(2\,500 \div 1\,000 = 2.5\), so \(2\,500 \,\text{g} = 2.5 \,\text{kg}\).

Convert 2 kilograms to grams.

1. Draw the metric place value table. Each unit has its own column; the two empty columns are for other units.

   \(\mathrm{kg}\)	\(\quad\;\)	\(\quad\;\)	\(\;\mathrm{g}\;\)

2. Place the ones digit of your number in the starting unit's column. Here, place the 2 in the kg column.

   \(\mathrm{kg}\)	\(\quad\;\)	\(\quad\;\)	\(\;\mathrm{g}\;\)
   2

3. Fill with zeros in each column until you reach the column of the target unit. We want to convert to grams (g).

   \(\mathrm{kg}\)	\(\quad\;\)	\(\quad\;\)	\(\;\mathrm{g}\;\)
   2	0	0	0

4. Read the number to find the converted value. \(2 \,\text{kg} = 2\,000 \,\text{g}\).