Rates
What a rate is
A rate is a way to compare two quantities that are measured in different units. We write a rate with the word per or with a slash, and you meet rates every day: a speed in kilometres per hour (km/h), a price in dollars per kilogram ($/kg), or a flow in litres per minute (L/min). The key idea is that the two quantities use different units, one on top and one on the bottom. This is what separates a rate from a ratio. A ratio compares two quantities of the same kind in the same unit, such as mixing paint in the ratio 3 : 2, and a ratio has no units attached. A rate always carries its pair of units, and that pair is part of the answer.
Finding a unit rate
A unit rate is a rate that has been written as an amount per one of something: per one hour, per one kilogram, per one minute. To find a unit rate you divide the first quantity by the second, so that the bottom quantity becomes 1. Suppose a car travels 240 km in 3 hours. Dividing the distance by the time gives 240 / 3 = 80, so the car covers 80 km in one hour, written 80 km/h. The unit rate strips the rate down to a single, tidy number, and that makes it far easier to use and to compare with other rates. Whenever a rate looks awkward, finding the unit rate is usually the first move.
Using a unit rate to scale
Once you have the per-one value, you can scale it to any amount by multiplying. If a car holds a steady 80 km/h, then in 5 hours it travels 80 x 5 = 400 km, because each hour adds another 80 km. The unit rate is the key number: multiply it by the new amount of the bottom quantity to scale up, or divide to go the other way and find how long a journey takes. Working from the unit rate keeps the arithmetic simple, since you only ever multiply or divide by the single per-one value rather than juggling the original pair of numbers.
Comparing rates for best value
Rates make it possible to compare deals fairly, even when the packs are different sizes. The trick is to put both deals into the same unit rate, usually dollars per kilogram, and then simply read off which is smaller. Take two bags of rice: 2 kg for $9 works out at 9 / 2 = $4.50/kg, while 5 kg for $20 works out at 20 / 5 = $4.00/kg. The second bag costs less for each kilogram, so it is the better value, even though it has the larger total price. Comparing unit rates turns a vague sense of best value into an objective check that anyone can repeat.
Converting the units of a rate
A rate can be rewritten in different units by converting each part of it in turn. Take a speed of 10 m/s and change it to km/h. There are 3600 seconds in an hour, so multiply by 3600 to move from per second to per hour, and there are 1000 metres in a kilometre, so divide by 1000 to move from metres to kilometres. Those two steps together multiply the number by 3.6, giving 10 x 3.6 = 36 km/h. The safe habit is to track which unit sits on top and which sits on the bottom, and to convert them one at a time, so the final rate carries exactly the units you want.
Why this matters
Rates sit quietly behind a great deal of ordinary life: the speed of a trip and the time it will take, the best value on a supermarket shelf, wages paid per hour, fuel used per 100 km, and the download speed of a file. The core skills are always the same few moves: find a unit rate by dividing, scale with it by multiplying, compare options by their unit rates, and convert the units when they do not match. The two slips to guard against are dividing the wrong way round and losing track of which unit is on top. This unit stays with the rate idea and its arithmetic; the full modelling of an applied problem with ratios and rates comes in a later unit.