Converting Compound Units

A unit with a unit underneath it

A car travels at 72 km/h. The unit km/h is read "kilometers per hour", and it means the car covers 72 kilometers in every 1 hour. A unit like this is called a compound unit, because it is made of two units: a distance on top and a time underneath.

The slash in km/h works like the line in a fraction. 72 km/h is 72 km over 1 h: the numerator is a distance and the denominator is a time. To write the same speed in other units, you change the numerator and the denominator separately, one at a time.

At 72 km/h the car covers 72 km in the first hour, 144 km in two hours and 216 km in three.

Change the numerator, then the denominator

Suppose we want the speed in meters per second, $\frac{m}{s}$m/s. Start with the numerator. 1 km = 1000 m, so 72 km = 72 × 1000 = 72000 m. The car covers 72000 m in 1 hour.

Now the denominator. An hour is 60 minutes, and each minute is 60 seconds, so 1 hour = 60 × 60 = 3600 seconds. The car covers 72000 m in 3600 s. To find the distance in one second, divide: 72000 ÷ 3600 = 20. The car covers 20 m every second, so its speed is $20\frac{m}{s}$20 m/s.

You can also take the hour apart in two steps. In 1 minute the car covers 72000 ÷ 60 = 1200 meters, and in 1 second it covers 1200 ÷ 60 = 20 meters. Dividing by 60 and then by 60 again is the same as dividing by 3600.

The car covers 72000 m in 60 minutes, and 72000 ÷ 60 = 1200, so it covers 1200 meters in each minute.

The same speed, measured another way

$20\frac{m}{s}$20 m/s means the car covers 20 m in every second: 20 m after 1 second, 40 m after 2 seconds, 60 m after 3 seconds. Nothing about the car has changed. In 3600 seconds it covers 20 × 3600 = 72000 m, which is the 72 km it covers in an hour. 72 km/h and $20\frac{m}{s}$20 m/s are the same speed written in different units.

The same car measured in meters and seconds: 20 m in each second, $20\frac{m}{s}$20 m/s.

Why km/h to $\frac{m}{s}$m/s is divide by 3.6

Every speed in km/h goes through the same two steps: multiply by 1000 to change kilometers into meters, then divide by 3600 to change hours into seconds. Multiplying by 1000 and dividing by 3600 is the same as dividing by 3600 ÷ 1000 = 3.6. So to change km/h into $\frac{m}{s}$m/s, divide by 3.6. 36 km/h is $10\frac{m}{s}$10 m/s, because 36 ÷ 3.6 = 10, and 90 km/h is $25\frac{m}{s}$25 m/s, because 90 ÷ 3.6 = 25.

To go back from $\frac{m}{s}$m/s to km/h, do the opposite and multiply by 3.6. A cyclist at $5\frac{m}{s}$5 m/s covers 5 × 3600 = 18000 m in an hour, which is 18 km, so $5\frac{m}{s}=5\times 3.6=18$5 m/s = 5 × 3.6 = 18 km/h.

Check the direction every time: the same speed always has a smaller number in $\frac{m}{s}$m/s than in km/h. Changing kilometers into meters makes the number 1000 times bigger, but changing hours into seconds makes it 3600 times smaller, and 3600 is more than 1000.

The usual mistakes

Changing only the numerator. 72 km is 72000 m, but the car covers 72000 m in an hour, not in a second. $72000\frac{m}{s}$72000 m/s changes the distance and forgets to change the time.

Dividing by 60 only once. 72000 ÷ 60 = 1200 is the distance in one minute. A second is a sixtieth of a minute, so divide by 60 again: 1200 ÷ 60 = 20, and the speed is $20\frac{m}{s}$20 m/s.

Multiplying by 3.6 when you should divide. 72 × 3.6 = 259.2, a bigger number than 72, but a speed in $\frac{m}{s}$m/s is always the smaller number. From km/h to $\frac{m}{s}$m/s, divide by 3.6.

Meters per minute

The same two steps work for any pair of units. In the next problem, a speed in km/h is used with a time in minutes, and then a second speed is asked for in meters per minute: change the kilometers into meters, and the hours into minutes.