# Units and equations

## In a nutshell

Units are used to measure quantities. Equations are mathematical expressions that show the relationship between different variables.

## SI units

SI units is an abbreviation for International System of Units. SI units are standard units of measurement used throughout the world. Some SI units are highlighted in the table below.

#### Quantity | #### SI unit | #### Symbol |

$temperature$ | $kelvin$ | $K$ |

$time$ | $second$ | $s$ |

$mass$ | $kilogram$ | $kg$ |

$length$ | $metre$ | $m$ |

## Metric prefixes

To convert between different units of measurement, you need to know the difference in magnitude between metric prefixes.

#### Metric prefix | #### symbol | #### magnitude |

$mega$ | $M$ | $\times 10^6$ |

$kilo$ | $k$ | $\times 10^3$ |

$deci$ | $d$ | $\times10^{-1}$ |

$centi$ | $c$ | $\times10^{-2}$ |

$milli$ | $m$ | $\times10^{-3}$ |

$micro$ | $\mu$ | $\times 10^{-6}$ |

## Converting larger numbers

You will need to be able to convert a large number to a suitable prefix. As the number in front of the prefix is smaller than the original number, you will need to divide.

#### procedure

1. | Work out how many times bigger one unit is compared to the other. e.g a mega is $1000$ times bigger than kilo. |

2. | Divide the original number by this value. |

Alternatively, you could convert values into standard form, and use your knowledge of magnitudes to convert into prefixes. Prefixes are used to replace powers of ten.

##### Example

*Convert $8000 \ g$ into $kg$.*

*Method 1:*

*There are $1000 \, g$ ($\times 10^3 \ g$) in $1 \ kg$. Therefore to convert from grams to kilograms, you need to divide by $1000$*.

*$\dfrac{8000}{1000} = 8$*

*$8000 \ g$ is equal to $\underline{8 \ kg}$.*

**Method 2:**

*Alternatively, convert $8000$ into standard form. *

*$8000 = 8 \times 10^3$*

*You know that $\times 10^3$ is the same as kilo ($k$) so $8000 \ g$ is equal to $\underline{8 \ kg}$.*

## Converting smaller numbers

You will need to be able to convert a small number to a suitable prefix. As the number in front of the prefix is bigger than the original number, you will need to multiply.

#### Procedure

1. | Work out how many times bigger one unit is compared to the other. e.g a kilo is $1000$ times smaller than mega. |

2. | Multiply the original number by this value. |

##### Example

*Convert $0.03 \ m$ into $cm$.*

**Method 1:**

*There are $100 \ cm$ ($\times 10^{-2} \ cm$) in $1 \ m$. Therefore to convert from metres to centimetres, you will need to multiply by $100$.*

*$0.03 \times 100 = 3$*

*$0.03 \ m$ is equal to $\underline{3 \ cm}$.*

**Method 2:**

*Alternatively, convert $0.03$ into standard form.*

*$0.03 = 3 \times 10^{-2}$*

*You know that $\times 10^{-2}$ is the same as centi ($c$) so $0.03 \ m$ is equal to $\underline{3 \ cm}$. *

## Equations

Equations are mathematical expressions that show the relationship between different variables.

Equations can be rearranged to answer certain questions.

##### Example

*A man drives at a speed $50~miles~per~hour$ to his Grandma's house. His Grandma's house is $18~miles$ away from his. How long will it take, in minutes, to arrive to his Grandma's house?*

*You will need to use the following equation:*

$speed = \dfrac{distance}{time}$

*Rearrange the equation to make time the subject. You need to multiply by time and divide by speed.*

$time = \dfrac{distance}{speed}$

*Insert values into the equation:*

*$time = \dfrac{18}{50} = 0.36 \ hours$*

*Convert units to get the final answer in minutes:*

$0.36 \times 60 = 21.6$

*It will take $\underline{21.6 \ minutes}$ to get to Grandma's house. *