Unit Rates

A rate is a special kind of comparison: it compares two different kinds of thing. Not "cats to dogs" (those are both animals — that is a plain ratio), but things measured in different units — miles and hours, pounds and apples, words and minutes. "120 miles in 2 hours" is a rate. "£3 for 2 apples" is a rate.

A rate is most useful when we squash it down to one of something. The unit rate is the amount for exactly one unit: miles per one hour, cost per one apple. To find it you do the same simple thing every time — divide the top quantity by the bottom quantity:

\text{unit rate} \;=\; \frac{\text{how much}}{\text{how many}}

So £3 for 2 apples becomes 3 \div 2 = 1.50 — that is £1.50 per apple. And 120 miles in 2 hours becomes 120 \div 2 = 60 — that is 60 miles per hour (60 mph). The little word "per" is the giveaway: it always means "for every one".

a car A car covers 120 miles in 2 hours. That pair of numbers, on its own, is hard to picture. But share the distance out evenly across the hours — 120 \div 2 = 60 — and suddenly it means something you can feel: the car eats up 60 miles every single hour. That is the unit rate, its speed: 60 mph. Now you can predict anything — in 3 hours it goes 60 \times 3 = 180 miles; in half an hour, 30 miles. Finding the "per one" turns a one-off fact into a rule.

See it: a double number line

A double number line stacks the two quantities so they march along together. The top line counts the miles, the bottom line counts the hours, and each rung ties them. Look at the very first step — one hour lines up with 60 miles. That one step is the unit rate, and every rung after it is just another 60 stacked on.

Because every hour adds the same 60 miles, the two lines never drift apart — that steady "60 miles for every 1 hour" is the whole story of the rate. This is the same idea as direct proportion: distance grows in perfect step with time.

Worked examples

1. Cost per apple. A bag of 4 apples costs £6. One apple costs 6 \div 4 = 1.50 — £1.50 each. Now any number is easy: 10 apples cost 1.50 \times 10 = 15 pounds.

2. Reading speed. You read 300 words in 5 minutes. Your unit rate is 300 \div 5 = 60 — 60 words per minute. So a 200-word page takes about 200 \div 60 \approx 3.3 minutes.

3. The better buy. A big bottle: 6 litres for £9. A small bottle: 2 litres for £4. Which is cheaper per litre? Big: 9 \div 6 = 1.50 per litre. Small: 4 \div 2 = 2.00 per litre. The big bottle wins — £1.50 a litre beats £2.00 a litre. Unit rates let you compare deals that look nothing alike.

The traps that catch everybody with unit rates:

apples Shop A sells 4 apples for £8. Shop B sells 3 apples for £9. Shop B's bag is cheaper overall — but you get fewer apples, so that is not a fair fight! Bring them both down to one apple: Shop A is 8 \div 4 = 2 pounds each, Shop B is 9 \div 3 = 3 pounds each. Now it is a fair fight, and Shop A wins at £2 an apple. The only honest way to compare prices is per one.

See it: a ratio table normalised to 1

Here is a ratio table: the apple row on top, the cost row below. The left column is what you are told — some apples for some pounds. To find the unit rate we shrink that column down to a single apple by dividing both rows by the same number. Press Play to watch it normalise, and Refresh for a fresh rate.

Whatever you do to the top row you must do to the bottom row — divide both by the same number and the rate stays the same, now measured per one apple. That "cost of one" in the right-hand column is the unit rate.