Rates, speed, distance and timeIB MYP Maths Extended: Revision notes

Speed is the distance travelled per unit of time. $\text{speed}=\frac{\text{distance}}{\text{time}},\quad \text{distance}=\text{speed}\times\text{time},\quad \text{time}=\frac{\text{distance}}{\text{speed}}.$ Example: $45$ km in $2.5$ hours is $45\div2.5=18$ km/h. At $18$ km/h, $63$ km takes $63\div18=3.5$ hours. A rate compares two quantities with different units, so speed is a rate (distance per time).

Keep the units consistent. Convert minutes to hours by dividing by $60$: $30$ min $=0.5$ h. Convert a decimal of an hour back by multiplying by $60$: $0.75$ h $=45$ min. So $2$ h $30$ min is $2.5$ h, not $2.3$ h. To change km/h to m/s, multiply by $1000$ (km to m) and divide by $3600$ (hours to seconds), which is dividing by $3.6$: $18$ km/h $=5$ m/s. To change m/s to km/h, multiply by $3.6$. For mass, $1$ kg $=1000$ g.

$\text{average speed}=\frac{\text{total distance}}{\text{total time}}.$ Find the time for each stage with $\frac{\text{distance}}{\text{speed}}$, add the times (including any rest stops), then divide the total distance by the total time. Example: $60$ km at $80$ km/h takes $0.75$ h; a rest of $0.5$ h; $90$ km at $60$ km/h takes $1.5$ h. Total distance $=150$ km, total time $=2.75$ h, so the average speed is $\frac{150}{2.75}=54.5$ km/h. This is not the mean of $80$ and $60$.

Density is mass per unit volume: $\text{density}=\frac{\text{mass}}{\text{volume}},\quad \text{mass}=\text{density}\times\text{volume},\quad \text{volume}=\frac{\text{mass}}{\text{density}}.$ A block of mass $540$ g and volume $200$ cm$^3$ has density $\frac{540}{200}=2.7$ g/cm$^3$. A $350$ cm$^3$ block of the same metal has mass $2.7\times350=945$ g. For $1.35$ kg, first change to $1350$ g, then volume $=\frac{1350}{2.7}=500$ cm$^3$. Other rates: price per unit (KES $180$ per litre), fuel use ($1$ litre per $30$ km) and cost per charge.

Rates let you compare options. For a $24$ km delivery: a motorbike at $48$ km/h takes $0.5$ h and uses $\frac{24}{30}=0.8$ litres, which costs $0.8\times180=\text{KES }144$. An electric bicycle at $20$ km/h takes $1.2$ h ($1$ h $12$ min) and costs $\frac{24}{40}\times12=\text{KES }7.20$. Work out each quantity separately, show the calculation, and give the unit.

A cheaper vehicle is not always better. If a courier is paid KES $250$ per delivery, the motorbike makes $16$ deliveries in $8$ hours and the electric bicycle $6$ (since $8\div1.2=6.67$ and only complete deliveries count). Profit $=$ earnings $-$ cost: motorbike $4000-2304=1696$, electric bicycle $1500-43.20=1456.80$. The motorbike earns more, but the electric bicycle is far cheaper to run. A conclusion should refer to the numbers and then to the context.