Speed, Distance and Time

Use distance = speed × time.

So $10\text{ mph}\times 3\text{ h}=30\text{ miles}$.

The cyclist travels 30 miles.

The speed and the time must be multiplied, not added, subtracted or divided. A quick sense check: 10 miles by the end of the first hour, 20 by the second, 30 by the third.

Use speed = distance ÷ time.

So $200\text{ km}÷2\text{ h}=100\text{ km/h}$.

The average speed is $100\text{km}$ per hour.

The distance goes on top and the time underneath. Because $\text{distance}=\text{speed}\times \text{time}$, finding the speed means undoing that multiplication by dividing. Multiplying gives $200\times 2=400$, which is not the speed. And dividing the other way round, $2÷200=0.01$, gives an answer far too small for a car. Always sense-check the size of your answer.

Use time = distance ÷ speed.

So $90\text{ km}÷60\text{ km/h}=1.5\text{ h}$.

Half an hour is 30 minutes, so 1.5 hours is 1 hour 30 minutes.

Three things to watch. Time is not base-10, so the 0.5 is 30 minutes, not 50. The leftover half hour must not be chopped off, so the answer is not 1 hour. And dividing the other way round gives $60÷90=\frac{2}{3}$, which as a fraction of an hour is 40 minutes, the numbers used back to front.

First convert the time. 30 minutes is half an hour, so 1 hour 30 minutes is 1.5 hours (not 1.3 hours: time is not base-10).

Use distance = speed × time. So $12\text{ mph}\times 1.5\text{ h}=18\text{ miles}$.

The cyclist travels 18 miles.

Check the common slips: using 1.3 hours gives $12\times 1.3=15.6$ miles; dividing instead of multiplying gives $12÷1.5=8$ miles; and ignoring the extra half hour gives only $12\times 1=12$ miles.

First convert the time. 40 minutes is $\frac{40}{60}=\frac{2}{3}$ of an hour (not 0.4 of an hour).

Use speed = distance ÷ time. So $10÷\frac{2}{3}=10\times \frac{3}{2}=15\text{ km/h}$.

Check it: the runner covers $5\text{km}$ every 20 minutes, so $15\text{km}$ in 60 minutes.

Average speed uses the totals, not each stage separately.

Total distance is $100+120=220\text{ miles}$ and total time is $2+3=5\text{ h}$.

Use speed = distance ÷ time. So $220\text{ miles}÷5\text{ h}=44\text{ mph}$.

The average speed is 44 miles per hour.

The stage speeds are $100÷2=50$ mph and $120÷3=40$ mph. Stopping at the first of those gives 50 mph, which ignores the rest of the journey; averaging the two gives 45 mph, which is wrong because more time was spent at the slower speed; and adding them gives 90 mph, which is not an average at all.