Large & small units
Large & small units
- Unit 1: Angles & Constructions
- Unit 2: Shapes & Symmetry
- Unit 3: Position & Transformation
- Unit 4: Area, Volume & Symmetry
🎯 In this topic you will
- Know and recognise very small or very large units of length, capacity, and mass.
🧠 Key Words
- prefix
- tonne
Show Definitions
- prefix: A group of letters placed before a unit (such as kilo– or milli–) that changes its size by a power of ten.
- tonne: A unit of mass equal to 1000 kilograms (also called a metric ton).
📏 Units of Length (recap)
You already know some length units: kilometre (km), metre (m), centimetre (cm), and millimetre (mm).
- $1000\ \text{m}=1\ \text{km}$
- $100\ \text{cm}=1\ \text{m}$
- $10\ \text{mm}=1\ \text{cm}$
🥤 Units of Capacity (recap)
Main units are litre (L) and millilitre (mL).
- $1000\ \text{mL}=1\ \text{L}$
⚖️ Units of Mass (recap)
Main units are kilogram (kg) and gram (g).
- $1000\ \text{g}=1\ \text{kg}$
🔤 Metric Prefixes
Very large or very small measurements use prefixes before a unit. The table shows common prefixes and their size as decimals and powers of ten.
| Prefix | Letter | Multiply by | Power of ten |
|---|---|---|---|
| tera | T | $1\,000\,000\,000\,000$ | $10^{12}$ |
| giga | G | $1\,000\,000\,000$ | $10^{9}$ |
| mega | M | $1\,000\,000$ | $10^{6}$ |
| kilo | k | $1000$ | $10^{3}$ |
| hecto | h | $100$ | $10^{2}$ |
| centi | c | $0.01$ | $10^{-2}$ |
| milli | m | $0.001$ | $10^{-3}$ |
| micro | μ | $0.000001$ | $10^{-6}$ |
| nano | n | $0.000000001$ | $10^{-9}$ |
✅ Prefixes you’ve met before
- kilo–: $1\ \text{kg}=1000\ \text{g}$, $1\ \text{km}=1000\ \text{m}$
- centi–: $1\ \text{cm}=0.01\ \text{m}$
- milli–: $1\ \text{mL}=0.001\ \text{L}$, $1\ \text{mm}=0.001\ \text{m}$
🟠 Special mass unit: tonne
Another unit of mass with a special name is the tonne.
$1\ \text{tonne (t)}=1000\ \text{kilograms (kg)}$
❓ EXERCISES
💡 Tip
Look at Worked example 7.3 to help you.
1. Copy and complete these descriptions. Use all the words and letters in the box.
| $length$ | $one\ thousandth$ | $mg$ | $mass$ | $one\ billionth$ | $nm$ | $metres$ | $one\ thousand$ | $g$ | $m$ | $one\ billion$ | $grams$ |
|---|
a. A milligram is a very small measure of …….
It is represented by the letters ……
$1$ milligram $= 0.001$ …… which is the same as
$1\ mg = 1 \times 10^{-3}$ ……
You can also say that there are …….. milligrams in a gram or that $1$ milligram is …….. of a gram.
👀 Show answer
It is represented by the letters mg.
$1$ milligram $= 0.001\ g$ which is the same as $1\ mg = 1 \times 10^{-3}\ g$.
You can also say that there are $1000$ milligrams in a gram or that $1$ milligram is $\tfrac{1}{1000}$ of a gram.
b. A nanometre is a very small measure of …….
It is represented by the letters ……
$1$ nanometre $= 0.000\ 000\ 001$ …… which is the same as
$1\ nm = 1 \times 10^{-9}$ ……
You can also say that there are …….. nanometres in a metre or that $1$ nanometre is …….. of a metre.
👀 Show answer
It is represented by the letters nm.
$1$ nanometre $= 0.000\ 000\ 001\ m$ which is the same as $1\ nm = 1 \times 10^{-9}\ m$.
You can also say that there are $1\ 000\ 000\ 000$ nanometres in a metre or that $1$ nanometre is $\tfrac{1}{1\ 000\ 000\ 000}$ of a metre.
❓ EXERCISES
2. Copy and complete these descriptions. Use all the words, letters and numbers in the box.
| kL | one thousandth | capacity | large | Gm | one billion |
|---|---|---|---|---|---|
| litres | $9$ | one billionth | kilolitre | metres | $3$ |
a. A kilolitre is a very large measure of …….
It is represented by the letters ……
$1$ ……. = $1000$ litres which is the same as $1$ kL = $1 \times 10^{\;…}$ L.
You can also say that there are one thousand ……. in a kilolitre or that $1$ litre is ……. of a kilolitre.
👀 Show answer
It is represented by the letters kL.
$1$kilolitre$=1000$ litres, which is the same as $1$ kL $= 1\times10^{3}$ L.
There are one thousand litres in a kilolitre, and $1$ litre is one thousandth of a kilolitre.
b. A gigametre is a very ……. measure of length.
It is represented by the letters ……
$1$ gigametre $=$$1\,000\,000\,000$ ……. which is the same as
$1$ Gm $= 1 \times 10^{\;…}$ metres.
You can also say that there are ……. metres in a gigametre or $1$ metre is ……. of a gigametre.
👀 Show answer
It is represented by the letters Gm.
$1$ gigametre $=$$1\,000\,000\,000$metres, which is the same as $1$ Gm $= 1\times10^{9}$ metres.
There are one billion metres in a gigametre, and $1$ metre is one billionth of a gigametre.
3. a. Write these lengths in order of size, starting with the smallest.
$8$ centimetres $8$ gigametres $8$ micrometres $8$ millimetres $8$ metres $8$ kilometres
👀 Show answer
1)$8$ micrometres → $8\ \mu\text{m}$
2)$8$ millimetres → $8\ \text{mm}$
3)$8$ centimetres → $8\ \text{cm}$
4)$8$ metres → $8\ \text{m}$
5)$8$ kilometres → $8\ \text{km}$
6)$8$ gigametres → $8\ \text{Gm}$
b. Underneath each of the lengths in part a, write the length using the correct letters for the units, not words. For example, underneath $8$ millimetres you write $8$ mm.
👀 Show answer
$8$ gigametres → $8\ \text{Gm}$
$8$ micrometres → $8\ \mu\text{m}$
$8$ millimetres → $8\ \text{mm}$
$8$ metres → $8\ \text{m}$
$8$ kilometres → $8\ \text{km}$
🧠 Think like a Mathematician
Investigation: Marcus and Arun make these conjectures:
- Marcus: “I think one tonne is the same as one megagram, so $1\,t = 1\,Mg$.”
- Arun: “I think 100 millilitres is the same as one centilitre, so $100\,mL = 1\,cL$.”
Tasks:
- Decide if Marcus’ and Arun’s conjectures are correct. Show your working to justify your decisions.
- Write down your reasoning clearly.
- Make a new conjecture that is different from Marcus and Arun’s. For example: “I think there are 1000 nanograms in a microgram.” Then check if it is true or false by calculation.
Follow-up Questions:
👀 show answer
- a) Marcus is correct. By definition, $1\,\text{tonne} = 1000\,kg = 1\,Mg$. Arun is incorrect. $1\,cL = 0.01\,L = 10\,mL$, not 100 mL.
- b) A tonne is exactly the same as a megagram since both equal 1000 kilograms. But 100 millilitres equals 0.1 litres, which is 10 centilitres. So Arun’s claim is wrong by a factor of 10.
- c) Example conjecture: “There are 1000 micrograms in a milligram.” Calculation: $1\,mg = 0.001\,g$ and $1\,\mu g = 0.000001\,g$. Dividing gives $\tfrac{0.001}{0.000001} = 1000$. The conjecture is true.
❓ EXERCISES
5. Copy and complete these conversions.
a. $2.5 \, \text{Mm to m} \quad \Rightarrow \quad 1 \, \text{Mm} = 1000000 \, \text{m}$, so $2.5 \, \text{Mm} = 2.5 \times 1000000 = \ldots \, \text{m}$
👀 Show answer
b. $0.75 \, \text{GL to L} \quad \Rightarrow \quad 1 \, \text{GL} = 1000000000 \, \text{L}$, so $0.75 \, \text{GL} = 0.75 \times 1000000000 = \ldots \, \text{L}$
👀 Show answer
c. $13.2 \, \text{hg to g} \quad \Rightarrow \quad 1 \, \text{hg} = \ldots \, \text{g}$, so $13.2 \, \text{hg} = 13.2 \times \ldots = \ldots \, \text{g}$
👀 Show answer
6. This is how Hania converts $225000000$ nanograms into grams.
Use Hania’s method to copy and complete these conversions.
a. $364 \, \text{cL to L} \quad \Rightarrow \quad 100 \, \text{cL} = 1 \, \text{L}$, so $364 \, \text{cL} = 364 \div 100 = \ldots \, \text{L}$
👀 Show answer
b. $12000 \, \text{mg to g} \quad \Rightarrow \quad 1000 \, \text{mg} = 1 \, \text{g}$, so $12000 \, \text{mg} = 12000 \div 1000 = \ldots \, \text{g}$
👀 Show answer
c. $620000 \, \mu \text{m to m} \quad \Rightarrow \quad 1000000 \, \mu \text{m} = 1 \, \text{m}$, so $620000 \, \mu \text{m} = 620000 \div \ldots = \ldots \, \text{m}$
👀 Show answer
7. The table shows the approximate distances from Earth to some other planets. Copy and complete the table.
| From Earth to: | Distance in m | Distance in … |
|---|---|---|
| Mars | $78340000000 \, \text{m}$ | $78.34 \, \text{Gm}$ |
| Jupiter | $628700000000 \, \text{m}$ | $628.7 \, \text{Gm}$ |
| Saturn | $1280000000000 \, \text{m}$ | $1.28 \, \text{Tm}$ |
| Uranus | $2724000000000 \, \text{m}$ | $2.724 \, \text{Tm}$ |
| Neptune | $4350000000000 \, \text{m}$ | $4.35 \, \text{Tm}$ |
👀 Show answer
❓ EXERCISES
8. The yellow cards show the approximate mass, in grams, of some very small objects.
The blue cards show the masses of the objects measured in milligrams, micrograms or nanograms.
Match each yellow card with the correct blue card.
👀 Show answer
Conversions (using $1\,\text{g}=1000\,\text{mg}=10^6\,\mu\text{g}=10^9\,\text{ng}$):
- A$0.0006\,\text{g}=0.6\,\text{mg}=600\,\mu\text{g}$ → v$600\,\mu\text{g}$
- B$0.0003\,\text{g}=0.3\,\text{mg}=300\,\mu\text{g}$ → iv$300\,\mu\text{g}$
- C$0.000\,000\,6\,\text{g}=600\,\text{ng}$ → i$600\,\text{ng}$
- D$0.03\,\text{g}=30\,\text{mg}$ → iii$30\,\text{mg}$
- E$0.000\,06\,\text{g}=0.06\,\text{mg}=60\,\mu\text{g}$ → ii$60\,\mu\text{g}$
🧠 Think like a Mathematician
Investigation: Work out whether Sofia’s statement about the length of a light year is correct, and calculate related values.
Information given:
- A light year is the distance light travels in one year.
- Light travels at a speed of approximately $3.0 \times 10^{8}$ m/s.
- You can use the formula: $\text{distance} = \text{speed} \times \text{time}$
- There are:
- 60 seconds in 1 minute
- 60 minutes in 1 hour
- 24 hours in 1 day
- 365.25 days in 1 year
Tasks:
- Sofia says a light year is approximately $9.47 \times 10^{15}$ m. Check if this is correct.
- Use the more accurate speed of light: 299,792,458 m/s. Calculate a more precise value for one light year.
- The abbreviation ly is used for light years. Express $9.460730472580800 \times 10^{15}$ m as 3 significant figures.
- The distance to Barnard’s Star is about 6 ly. Work out the distance in metres.
- Reflect on your answers and how they compare with Sofia’s estimate.
👀 Show Answer
- a: Yes. Using $3.0 \times 10^{8}$ m/s × (365.25 × 24 × 3600 s) ≈ $9.47 \times 10^{15}$ m. Sofia is correct.
- b: Using 299,792,458 m/s × 31,557,600 s ≈ $9.4607 \times 10^{15}$ m.
- c:$9.460730472580800 \times 10^{15}$ m to 3 significant figures is $9.46 \times 10^{15}$ m.
- d: Distance to Barnard’s Star = 6 × $9.46 \times 10^{15}$ ≈ $5.68 \times 10^{16}$ m.
- e: Sofia’s value is a good approximation; the exact figure is slightly smaller, but still extremely close.
❓ EXERCISES
| Unit of memory | Number of bytes as a power of $2$ | Number of bytes as a number |
|---|---|---|
| $1\ \text{KB}$ | $2^{10}$ | $1024$ |
| $1\ \text{MB}$ | $2^{20}$ | $1\,048\,576$ |
| $1\ \text{GB}$ | $2^{30}$ | $1\,073\,741\,824$ |
| $1\ \text{TB}$ | $2^{40}$ | $1\,099\,511\,627\,776$ |
10.
a. A shop sells these items. Write the items in order of memory size, from the smallest to the largest.
👀 Show answer
D (Camera, $512\ \text{MB}$) → B (Phone, $64\ \text{GB}$) → C (Tablet, $128\ \text{GB}$) → A (Laptop, $1\ \text{TB}$).
b. Ela buys a computer with $2\ \text{GB}$ of RAM memory. How many bytes is $2\ \text{GB}$?
👀 Show answer
$2\ \text{GB} = 2 \times 1\,073\,741\,824 = 2\,147\,483\,648\ \text{bytes}$.
c. Anoop buys a $32\ \text{GB}$ memory card for his camera.
It is possible to store approximately $340$ photographs on $1\ \text{GB}$ of memory.
Approximately how many photographs can Anoop store on his memory card?
👀 Show answer
$32\ \text{GB} \approx 32 \times 340 = 10\,880$ photographs.
d. Doroata buys an external hard drive for her computer which has $8\ \text{TB}$ of memory.
It is possible to store approximately $233$ films on $1\ \text{TB}$ of memory.
Approximately how many films can Doroata store on her external hard drive?
👀 Show answer
$8\ \text{TB} \approx 8 \times 233 = 1864$ films.
❓ EXERCISES
11. Magnar wants to buy a new computer. He looks at three different models, A, B and C. He looks at the speed, in nanoseconds, at which each computer can access the memory.
Magnar thinks Model C is the fastest. Is he correct? Explain your answer.
👀 Show answer
No. A smaller time in nanoseconds means memory can be accessed faster.
- Model A: $40\,\text{ns}$
- Model B: $10\,\text{ns}$ ← fastest
- Model C: $60\,\text{ns}$ ← slowest
Therefore, Model B is the fastest and Model C is the slowest.