Ratios and proportion

1. A ratio compares two quantities by dividing one by the other, so it tells you how many times larger one quantity is than the other.
2. A coach that travels $90km$ while a cyclist travels $30km$ has covered $3$ times the distance, a ratio of $3:1$.
3. Two quantities are directly proportional when doubling one doubles the other, tripling one triples the other, and so on.
4. For a vehicle moving at a constant speed, the distance travelled is directly proportional to the time taken.
5. Proportional reasoning uses this pattern to scale a known value up or down without starting the calculation again.
6. One method is to find the value for one unit first and then multiply by the number of units needed.
7. A second method is to find the scale factor between the two situations and multiply the known value by it.

Converting to SI units

1. Physics equations such as $distance=speed\times time$ and $F=ma$ only give correct answers when every quantity is in SI units.
2. The SI unit of distance is the metre, $m$, the SI unit of time is the second, $s$, and the SI unit of mass is the kilogram, $kg$.
3. Road information is often given in non-SI units, such as kilometres, miles, hours, minutes and tonnes, so it must be converted before it is used.
4. A conversion multiplies by a ratio equal to one, such as $\frac{1000m}{1km}$, so the size of the quantity stays the same and only the unit changes.
5. The conversions you need most often on the road are these:
   1. $1km=1000m$
   2. $1mile\approx 1600m$
   3. $1minute=60s$ and $1hour=3600s$
   4. $1tonne=1000kg$ and $1g=0.001kg$
6. Converting to a smaller unit gives a larger number, so $2.5km$ becomes $2500m$.

Converting road speeds

1. To change $km/h$ into $m/s$, convert the kilometres into metres and the hours into seconds, which means multiplying by $1000$ and dividing by $3600$.
2. Multiplying by $\frac{1000}{3600}$ is the same as dividing by $3.6$, so $speed\; in\; m/s=\frac{speed\; in\; km/h}{3.6}$.
3. To change $m/s$ back into $km/h$, multiply by $3.6$.
4. To change miles per hour into $m/s$, multiply by $1600$ and divide by $3600$, which is close to multiplying by $0.45$.
5. Useful results to remember are $30$ mph $\approx 13m/s$ and $70$ mph $\approx 31m/s$.
6. A speed in $m/s$ is always a smaller number than the same speed in $km/h$ or mph, which gives a fast way to spot a conversion done the wrong way round.

Computing rates

1. A rate tells you how much a quantity changes in each unit of time, so speed is the rate of change of distance and acceleration is the rate of change of velocity.
2. A rate is found by dividing the change in a quantity by the time taken for that change.
3. The unit of a rate is the unit of the quantity divided by the unit of time, so a distance in $m$ divided by a time in $s$ gives $m/s$.
4. The average speed for a whole journey is the total distance divided by the total time, including any time spent stopped.
5. Every quantity in a rate calculation must be in matching units, so a speed in $km/h$ must be converted to $m/s$ before it is divided by a time in seconds.