Speed, distance and time
The average speed of a vehicle is related to the total distance travelled and the total time taken by the formula:
$\text{average speed}=\frac{\text{total distance travelled}}{\text{total time taken}}$average speed=total distance travelledtotal time taken
$S=\frac{D}{T}$S=DT
We can rearrange the formula for speed to make either $D$D or $T$T the subject. For example, rearranging to make $T$T the subject:
| $S$S | $=$= | $\frac{D}{T}$DT |
|
|
| $S\times T$S×T | $=$= | $\frac{D}{T}\times T$DT×T |
multiply both sides by $T$T |
|
| $ST$ST | $=$= | $D$D |
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| $T$T | $=$= | $\frac{D}{S}$DS |
divide both sides by $S$S |
Similarly we can rearrange to make $D$D the subject and the formulas are below.
| Unknown time | Unknown distance |
|---|---|
| $T=\frac{D}{S}$T=DS | $D=S\times T$D=S×T |
Worked example
Example 1
Jim drives his car from Sydney to Canberra, a distance of $279$279 km. The journey takes him $3$3 hours and $45$45 minutes, including lunch and fuel stops along the way.
(a) What is Jim's average speed for the journey, correct to the nearest km/h?
Think: Whenever we use a formula, we must keep units consistent. In this case, speed is expressed in kilometres per hour and distance is in kilometres, so time must be expressed in hours.
Our first step is to convert the time of $3$3 hours $45$45 minutes into hours. Because there are $60$60 minutes in an hour, $45$45 minutes is equivalent to $\frac{45}{60}$4560 or $0.75$0.75 hours. Therefore $3$3 hours $45$45 minutes is the same as $3.75$3.75 hours.
Do: Substituting into the speed-distance-time formula:
| $S$S | $=$= | $\frac{D}{T}$DT |
|
| $=$= | $\frac{279}{3.75}$2793.75 |
substitute $D=279$D=279 and $T=3.75$T=3.75 |
|
| $=$= | $74.4$74.4 |
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|
| $=$= | $74$74 km/h (nearest km/h) |
|
(b) How far could Jim travel at this same average speed in $40$40 minutes. Round your answer to the nearest kilometre.
Think: A time of $40$40 minutes is equivalent to $\frac{40}{60}$4060 hours, which simplifies to $\frac{2}{3}$23 of an hour. In this case it is easier to leave the time as a fraction because the decimal is recurring. It is also easier to substitute our values into the formula when $D$D is the subject.
Do:
| $D$D | $=$= | $ST$ST |
|
| $=$= | $74\times\frac{2}{3}$74×23 |
substitute $S=74$S=74 and $T=\frac{2}{3}$T=23 |
|
| $=$= | $\frac{148}{3}$1483 |
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| $=$= | $49.\overline{3}$49.3 |
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| $=$= | $49$49 km (nearest km) |
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(c) If Jim travelled at this same average speed, how long would it take him to drive from Sydney to Melbourne, a distance of $875$875 km. Give your answer in hours and minutes.
Think: This time, it will be easier to substitute our values into the formula when $T$T is the subject.
Do:
| $T$T | $=$= | $\frac{D}{S}$DS |
|
| $T$T | $=$= | $\frac{875}{74}$87574 |
substitute $D=875$D=875 and $S=74$S=74 |
| $T$T | $=$= | $11.824\ldots$11.824… |
|
| $T$T | $=$= | $11$11 hours $49$49 minutes (nearest minute) |
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Reflect: In the final steps of the calculation, notice that our value for time was equal to $11.82432\ldots$11.82432… hours. To convert this time into hours and minutes, we simply multiplied the decimal portion by $60$60. To isolate the decimal portion, subtract $11$11 first, then multiply by $60$60. This gives $49.4594\ldots$49.4594… minutes, or $49$49 minutes, rounded to the nearest minute.
Practice questions
Question 1
Maria travels by car for $420$420 km. The trip takes $10$10 hrs. What is the average speed of the trip?
Question 2
A dog travels $42$42 km in $1.4$1.4 hours.
At what speed is it travelling?
Question 3
If a sugar glider possum travels $6$6 km at a speed of $15$15 km/hr, how long will it take the animal to cross the whole distance?
Question 4
If a sugar glider possum travels for $1.5$1.5 hours at speed of $42$42 km/hr, how far will the sugar glider possum travel?
Question 5
The table below shows the distances between various locations.
How long does it take to travel from Paris to Madrid via bus then Madrid to Athens via bike, if a bus has an average speed of $80$80 km/h and a bike has an average speed of $31$31 km/h?
| London | Paris | Rome | Madrid | Berlin | Athens | |
|---|---|---|---|---|---|---|
| London | $0$0 | $456$456 | $841$841 | $1715$1715 | $1099$1099 | $3151$3151 |
| Paris | $456$456 | $0$0 | $1431$1431 | $1271$1271 | $1055$1055 | $2915$2915 |
| Rome | $841$841 | $1431$1431 | $0$0 | $1963$1963 | $1519$1519 | $1349$1349 |
| Madrid | $1715$1715 | $1271$1271 | $1963$1963 | $0$0 | $2322$2322 | $3290$3290 |
| Berlin | $1099$1099 | $1055$1055 | $1519$1519 | $2322$2322 | $0$0 | $2346$2346 |
| Athens | $3151$3151 | $2915$2915 | $1349$1349 | $3290$3290 | $2346$2346 | $0$0 |
First, calculate your answer in hours, to two decimal places.
Using the result of the previous part, give your answer in hours and minutes, rounding the number of minutes to the nearest whole number.
$\editable{}$ hours and $\editable{}$ minutes.