Unit Rate Formula - Definition, Steps, Examples

Key Points

• The unit rate formula is $\text{unit rate} = \frac{a}{b}$, where $b$ is the quantity being reduced to one unit.

• A unit rate always names its units, as in miles per hour or cost per kilogram, and the word "per" marks the denominator.

• Unit price is the unit rate of money against quantity, which is what makes shelf prices comparable between different package sizes.

• Inverting the division gives a valid but different rate, so kilometres per litre and litres per kilometre are both correct and not interchangeable.

• NIST's Office of Weights and Measures runs national legal metrology programmes covering price verification, packaging and labelling, and retail pricing laws.

What Is the Unit Rate Formula?

The unit rate formula compares two quantities by reducing the second to a single unit:

$$\text{unit rate} = \frac{a}{b}$$

Here $a$ is the quantity being measured and $b$ is the quantity reduced to 1. If 12 pens cost 96, dividing 96 by 12 gives 8 per pen.

The word "per" tells you which quantity is the denominator. In "cost per kilogram", kilograms go underneath.

Why Reducing to One Unit Makes Rates Comparable

Two prices cannot be compared while they describe different quantities. A 750 g box at 180 and a 1.2 kg box at 270 give no obvious winner, because both numbers differ.

Reducing each to a price per kilogram fixes one side of the comparison and leaves only one number to judge. The first works out at 240 per kg and the second at 225 per kg, so the larger box is better value.

That is the whole purpose of a unit rate. It removes one of the two varying quantities so that a single figure carries the comparison.

Variable Key

Examples of the Unit Rate Formula

Example 1

12 pens cost 96. Find the unit rate.

$$\frac{96}{12} = 8$$

Final answer: 8 per pen

Example 2

A car travels 150 km on 12 litres of fuel. Find the fuel economy in km per litre by dividing litres by kilometres.

Both numbers are given, so the division could go either way:

$$\frac{12}{150} = 0.08$$

That 0.08 is a real quantity, litres per kilometre, but it is not what was asked. The question says kilometres per litre, and "per litre" puts litres in the denominator.

$$\frac{150}{12} = 12.5$$

Final answer: 12.5 km per litre. The inverted figure of 0.08 litres per km is correct for a different question, and the two are reciprocals.

Example 3

A 750 g box costs 180 and a 1.2 kg box costs 270. Which is better value?

Convert to a common unit first, then divide:

$$\frac{180}{0.75} = 240, \qquad \frac{270}{1.2} = 225$$

Final answer: The 1.2 kg box, at 225 per kg against 240 per kg.

Example 4

A worker assembles 84 units in 7 hours. Find the unit rate, then the time per unit.

$$\frac{84}{7} = 12 \text{ units per hour}$$ $$\frac{7}{84} = 0.0833 \text{ hours per unit, about 5 minutes}$$

Final answer: 12 units per hour, or about 5 minutes per unit.

Both are unit rates of the same situation, each reducing a different quantity to 1.

Example 5

A recipe uses 300 ml of stock for 4 servings. Find the stock needed for 7 servings.

Find the unit rate, then scale:

$$\frac{300}{4} = 75 \text{ ml per serving}, \qquad 75 \times 7 = 525$$

Final answer: 525 ml

Going through the unit rate is why this method works for any number of servings, rather than only for multiples of 4.

Don't Confuse a Unit Rate With These Three

A ratio normally compares quantities of the same kind and can be written without units, while a rate compares different kinds and always carries them. Unit price is simply the most common unit rate in daily life.

Unit Rate, Where Things Go Sideways

Mistake 1: Dividing the wrong way round

Where it slips in: Two numbers invite division in either order, and nothing in the arithmetic flags the wrong one.

Don't do this: $\frac{12 \text{ litres}}{150 \text{ km}}$ when kilometres per litre was asked.

The correct way: Whatever follows "per" goes in the denominator. For km per litre, divide kilometres by litres.

Students who write the units into the fraction before any numbers almost never invert a rate, because the units themselves show whether the answer will read as km/litre or litre/km. The habit carries straight into physics, where compound units do the same job.

Mistake 2: Comparing rates in different units

Where it slips in: Package sizes are given in grams and kilograms, or distances in metres and kilometres.

Don't do this: Divide 180 by 750 and 270 by 1.2 and compare the results.

The correct way: Convert to a common unit first, as in Example 3. Comparing 0.24 per gram against 225 per kilogram is meaningless even though both are correctly computed.

Mistake 3: Leaving the second quantity above 1

Where it slips in: The rate is computed but not reduced, so it stays as a rate rather than a unit rate.

Don't do this: Report "150 km per 12 litres" as the unit rate.

The correct way: Divide until the second quantity is 1, giving 12.5 km per litre. Only then can two rates be compared by a single number.

Bottom Line

• The unit rate formula is $\frac{a}{b}$, with the second quantity reduced to one unit.

• The word "per" identifies the denominator, which is the quickest guard against dividing the wrong way.

• Rates in different units must be converted to a common unit before they can be compared.

• Inverting a rate gives a valid but different quantity, and the two are reciprocals.

• To work through rates and ratios with a teacher, explore a 6th grade math tutor or math tutoring.

Three Problems to Cement Unit Rate

Find the unit rate for 18 apples costing 234. Then decide which is better value: 2.5 kg at 450 or 400 g at 76. Finally, convert 20 km per litre into litres per 100 km.

Want a live Bhanzu trainer to walk through more rates and ratios problems? Book a free demo class.

Read More

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• What Is Ratio

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• Average Speed Formula

• What Is a Unit