Ratio and proportion

1. A ratio compares two quantities by showing how many times bigger one is than the other, and it can be written with a colon, such as $3:1$, or as a fraction, such as $\frac{3}{1}$.
2. A ratio of two quantities measured in the same unit has no unit, because the units cancel when one is divided by the other.
3. Two quantities are directly proportional when they change by the same factor, so doubling one doubles the other and halving one halves the other.
4. At a steady speed, the distance travelled is directly proportional to the time taken, because the object covers the same distance in every second.
5. For example, a coach moving steadily along a motorway covers $24km$ in $20min$, so it covers $48km$ in $40min$ and $72km$ in $60min$.
6. Proportional reasoning means finding the scale factor between two values of one quantity and applying the same factor to the linked quantity.
7. In the coach example the time is multiplied by $3$ in going from $20min$ to $60min$, so the distance is also multiplied by $3$, from $24km$ to $72km$.
8. Two quantities are inversely proportional when multiplying one of them by a factor divides the other by the same factor.
9. For a fixed distance, the time taken is inversely proportional to the speed, so doubling the speed halves the journey time.
10. For example, a cyclist who takes $30min$ to ride to school at a steady speed would take $15min$ at twice that speed and $60min$ at half that speed.
11. A graph of two directly proportional quantities is a straight line through the origin, which is how a steady journey appears when distance is plotted against time.

Converting to SI units

1. SI units are the standard units used in science, and the SI units for motion are the metre for distance, the second for time and the metre per second for speed.
2. Many everyday measurements use other units, such as kilometres, centimetres, minutes, hours and kilometres per hour, so they are converted into SI units before they go into an equation.
3. A conversion factor is the number of smaller units in one larger unit, such as the $1000m$ in $1km$.
4. To convert from a larger unit to a smaller unit, multiply by the conversion factor, because each large unit contains many small ones.
5. To convert from a smaller unit to a larger unit, divide by the conversion factor.
6. The length conversions used in motion are $1km=1000m$, $1m=100cm$ and $1m=1000mm$.
7. The time conversions used in motion are $1min=60s$ and $1h=60min=3600s$.
   1. So $3.5km$ is $3.5\times 1000=3500m$, and $45cm$ is $\frac{45}{100}=0.45m$.
   2. Similarly, $12min$ is $12\times 60=720s$, and $2.5h$ is $2.5\times 3600=9000s$.
8. A speed has a distance unit on top and a time unit underneath, so converting a speed means converting both parts.
9. To change $km/h$ into $m/s$, multiply by $1000$ to turn kilometres into metres and divide by $3600$ to turn hours into seconds.
10. Multiplying by $1000$ and dividing by $3600$ is the same as dividing by $3.6$, so a speed in $km/h$ divided by $3.6$ gives the speed in $m/s$.
11. To change $m/s$ into $km/h$, multiply by $3.6$ instead.
12. The value in $km/h$ is always the larger number, because one kilometre per hour is a much smaller speed than one metre per second.
13. Very large or very small values can be written in standard form, so $3600s$ can be written as $3.6\times 10^{3}s$ and $0.0045m$ as $4.5\times 10^{-3}m$.

Calculating rates

1. A rate tells you how quickly one quantity changes, usually given as the change in each second, minute or hour.
2. A rate is found by dividing the change in a quantity by the time over which the change happens, $rate=\frac{change\; in\; quantity}{time\; taken}$.
3. Speed is a rate, because it is the distance travelled in each unit of time.
4. The unit of a rate is built from the units used, so metres divided by seconds gives $m/s$ and kilometres divided by hours gives $km/h$.
5. Other rates met in motion include the number of laps a runner completes per hour and the number of steps a walker takes per minute.
6. Before a rate is calculated, every quantity is converted into the units the answer needs, so a distance in kilometres and a time in minutes are changed into metres and seconds for an answer in $m/s$.
7. Two rates can only be compared when they are in the same units, so a runner at $5.0m/s$ and a cyclist at $20km/h$ are compared by converting one of them first.
8. Converting the cyclist's speed gives $\frac{20}{3.6}=5.6m/s$, so the cyclist is faster than the runner.