Rate in Mathematics: Definition and Examples

What is a rate in math?

A rate is a mathematical comparison of two related quantities that have different units of measurement. While a standard ratio might compare the number of apples to oranges, a rate compares completely different types of measurements, such as distance and time.

A rate always describes how one quantity changes in relation to another quantity.

Whenever you measure how fast something moves, how quickly a task is completed, or how much an item costs per unit, you are using rates.

Recognise different units

To identify a rate, look for two entirely different units of measurement working together. If both quantities share the same unit, it is not considered a rate in standard mathematics.

Common rate examples include:

• Speed, which compares distance and time (kilometers and hours).
• Typing speed, which compares text volume and time (pages and minutes).
• Flow rates, which compare liquid volume and time (liters and days).
• Earning rates, which compare money and time (dollars and hours).

Because the units are different, they must always be written alongside the numbers. Writing "$15$ to $3$" is incomplete for a rate; it must be written as "$15$ kilometers per $3$ hours".

Read rate language

The most important word used in rate language is "per". When you see or hear the word "per", it translates to "for every" or "in each".

For instance, saying "$30$ pages per minute" means that for every single minute that passes, exactly $30$ pages are completed. The word "per" is represented mathematically by the division symbol or a fraction bar.

When reading a rate aloud, you will typically state the first quantity and its unit, say the word "per", and then state the second unit.

Rate versus ratio

While all rates are ratios, not all ratios are rates. The key difference in ratio versus rate lies in the units being compared.

A standard ratio compares quantities with the exact same units, meaning the units can be simplified and removed from the final ratio. A rate compares different units that cannot be simplified away.

Because the units in a rate are different, they must remain in the final mathematical expression.

Represent a rate

The clearest way to represent a rate is by writing it as a fraction. The first quantity becomes the numerator, and the second quantity becomes the denominator.

You must include the measurement units in both the numerator and the denominator. For example, if a machine pumps $400$ liters of water every $8$ hours, the rate is represented as $\dfrac{400 \text{ liters}}{8 \text{ hours}}$.

By writing the rate as a fraction, you can easily divide the numerator by the denominator to simplify the measurement.

Worked examples

The following examples demonstrate how to identify and calculate rates using different real-world contexts.

Example 1: Calculating pages per minute

Question: A printing press produces $240$ pages in $6$ minutes. What is the printing rate?

Method:

1. Identify the two different quantities and their units. The quantities are $240$ pages and $6$ minutes.
2. Write the quantities as a fraction, keeping the units attached.

Rate = $\dfrac{240 \text{ pages}}{6 \text{ minutes}}$

3. Divide the numerator by the denominator to simplify the rate for one minute.

Answer: The rate is $40$ pages per minute.

Check: Multiply the simplified rate by the time: $40 \times 6 = 240$ pages. The rate is correct.

Example 2: Calculating liters per day

Question: A large reservoir loses $4{,}500$ liters of water over $9$ days due to a leak. What is the rate of water loss per day?

Method:

1. Identify the volume and the time. Volume is $4{,}500$ liters, time is $9$ days.
2. Set up the rate fraction.

Rate = $\dfrac{4{,}500 \text{ liters}}{9 \text{ days}}$

3. Perform the division: $4{,}500 \div 9$.

Answer: The rate of water loss is $500$ liters per day.

Check: Over $9$ days, $500 \times 9 = 4{,}500$ liters. The total matches the given volume.

Example 3: Calculating kilometers per hour

Question: A train covers a distance of $320$ kilometers in exactly $4$ hours. What is the rate of speed distance time for the train?

Method:

1. Identify distance and time. Distance is $320$ kilometers, time is $4$ hours.
2. Write the fraction.

Speed = $\dfrac{320 \text{ kilometers}}{4 \text{ hours}}$

3. Divide $320$ by $4$.

Answer: The speed is $80$ kilometers per hour.

Check: $80 \text{ kilometers/hour} \times 4 \text{ hours} = 320 \text{ kilometers}$. The answer is verified.

Common mistakes

A frequent mistake is dropping the units when writing a rate. If you write the rate of a car simply as "$50$", it has no mathematical meaning because it could represent $50$ miles per hour, $50$ kilometers per hour, or even $50$ meters per second.

Another mistake is confusing a rate with a standard ratio. Remember that comparing "$5$ blue marbles to $3$ red marbles" is just a ratio because both items are marbles (the same unit type). A true rate must compare fundamentally different properties, like mass and volume or distance and time.

Frequently asked questions

What is a unit rate?

A unit rate is a specific type of rate where the second quantity (the denominator) is exactly one unit. For example, "$50$ kilometers per $1$ hour" is a unit rate, while "$100$ kilometers per $2$ hours" is a general rate.

How do you convert a rate into a unit rate?

The method for finding a unit rate involves dividing the numerator by the denominator. By calculating the division, you automatically scale the denominator down to exactly one unit.

Are all ratios considered rates?

No, all ratios are not rates. Ratios that compare quantities measured in the identical unit (like centimeters to centimeters) are not rates. Rates specifically compare quantities with different units of measurement.