Apply and compare rates and unit rates in context

Alberta Grade 11 Mathematics

Measurement

Use units to describe, compare, and judge how quantities change

A rate compares two quantities that use different units. For example, litres per minute compares a volume with a time. A unit rate tells how much of one quantity corresponds to one unit of the other. Unit rates help us compare choices fairly, but the context matters: a faster rate is not always the better choice. In this lesson, you will use units to calculate, compare, and interpret rates.

What you will learn

Explain what a rate tells you in a real situation.
Calculate a unit rate and state its units.
Compare rates by expressing them in the same units.
Decide whether a calculated rate makes sense in its context.

Review: quantities, division, and units

A quantity is an amount that can be measured or counted. Examples include distance, time, volume, and number of pages. A unit tells what is being measured, such as kilometres, hours, or litres.

Division can compare two quantities. If a cyclist travels 24 kilometres in 2 hours, dividing distance by time gives the distance travelled for each hour. The order matters: distance divided by time gives kilometres per hour, while time divided by distance gives hours per kilometre. These rates answer different questions.

Keep the units attached to the numbers as you work. They show what the answer means and help you catch errors. A rate is often written using the word per, a slash, or a fraction bar.

\text{rate}=\frac{\text{first quantity}}{\text{second quantity}}

A rate compares quantities with different units.
The order of the quantities affects the meaning and units.
Units are part of the answer, not decoration.

Unit rates and fair comparisons

A unit rate is a rate for one unit of the second quantity. For example, kilometres per hour is distance for one hour. To find it, divide the first quantity by the number of units in the second quantity. If a bus travels 150 kilometres in 3 hours, its average rate is 50 kilometres per hour.

The word average means that the rate summarizes the whole trip. It does not say the bus travelled at exactly the same speed at every moment. The calculation is still useful when the question asks for the overall distance per hour.

To compare two rates, express them in the same units and compare the values. If one rate is in kilometres per hour and another is in metres per second, convert one rate before deciding which is larger. Do not compare the bare numbers while ignoring their units.

A larger unit rate means more of the first quantity for each one unit of the second quantity. Whether that is preferable depends on the situation. A larger filling rate may be useful when speed matters. A smaller cost per item may be preferable when comparing prices.

\text{unit rate}=\frac{\text{quantity}}{1\text{ unit}}

Divide to find the amount per one unit.
Convert rates to matching units before comparing them.
Interpret which quantity is on top and which is below.

Use a rate model and check the meaning

A constant rate means the same amount is added or used for each equal unit of time or another quantity. If a tap fills a container at a steady 3.6 litres per minute, the total volume after several minutes can be found by multiplying the rate by the time. This model is suitable only while the rate remains steady and the container can hold the added water.

A rate graph can show how an amount changes as the other quantity changes. Time is often placed on the horizontal axis, and the accumulated amount on the vertical axis. The rate is the change in the vertical quantity for each one unit of horizontal quantity. A steeper line represents a greater rate when the axes use the same scales.

After calculating, check the units and the situation. For distance divided by time, the answer should have distance units per time unit. Also ask whether the size is reasonable. A person walking at 80 kilometres per hour is not a sensible walking rate, so such an answer would suggest a unit or calculation error.

\text{total change}

=

\text{rate}

×number of units

Multiplying a constant rate by elapsed units gives the total change.
Read a graph using both axis labels and their scales.
Use units and real-world context to check an answer.

Student try: compare before choosing

A garden hose fills a 28-litre container in 7 minutes. Another hose fills a 45-litre container in 10 minutes. Find each filling rate in litres per minute, then decide which hose fills faster.

Before calculating, predict which hose may be faster. Then divide litres by minutes for each hose. If your results do not have litres per minute as their units, revisit the order of your division. Finally, describe the difference in context rather than reporting only two numbers.

Write the requested units before calculating.
A comparison is clearest when both rates use the same units.
A rate difference describes the difference in amount per unit of time.

Compare the filling hoses

Hose	Volume	Time	Rate
First	28 L	7 min	4 L/min
Second	45 L	10 min	4.5 L/min

Worked example

Find a steady filling rate

A tap adds 18 litres of water to a tank in 5 minutes. Find its average filling rate in litres per minute. If the rate stays constant, how much water does it add in 8 minutes?

Water added at a steady rate

The line represents 3.6 litres added per minute, for up to 8 minutes.

Divide to find the unit rate
The rate compares litres with minutes, so divide the volume by the time. This gives the amount added for one minute.
$18\div5=3.6\text{ L/min}$
Use the rate for 8 minutes
At a constant rate, multiply litres per minute by the number of minutes. The minute units cancel, leaving litres.
$3.6\text{ L/min}\times8\text{ min}=28.8\text{ L}$

Answer: The average filling rate is 3.6 litres per minute. At that constant rate, the tap adds 28.8 litres in 8 minutes.

Check: The result is greater than 18 litres because 8 minutes is longer than 5 minutes. The units in the second calculation reduce to litres.

Worked example

Compare two delivery rates

Two delivery riders travel on separate routes. Rider A travels 36 kilometres in 1.5 hours. Rider B travels 50 kilometres in 2 hours. Find and compare their average speeds in kilometres per hour.

Rider B's average distance rate

This rate line shows Rider B's average of 25 kilometres per hour over 2 hours.

Calculate Rider A's rate
Divide the distance by the time. The answer is the distance Rider A travels in one hour on average.
$36\text{ km}\div1.5\text{ h}=24\text{ km/h}$
Calculate Rider B's rate
Use the same order and units for Rider B so the rates can be compared directly.
$50\text{ km}\div2\text{ h}=25\text{ km/h}$
Compare and interpret
Both rates are in kilometres per hour. Rider B's average speed is 1 kilometre per hour greater, so Rider B covered more distance per hour on average.
$25\text{ km/h}-24\text{ km/h}=1\text{ km/h}$

Answer: Rider A's average speed is 24 kilometres per hour. Rider B's is 25 kilometres per hour. Rider B's average speed is greater by 1 kilometre per hour.

Check: Rider B travelled farther, but also spent more time travelling. The unit-rate comparison accounts for the different travel times.

Worked example

Compare rates with different time units

A small pump moves 900 millilitres in 2 minutes. A larger pump moves 8 litres in 15 minutes. Which pump moves more liquid per minute? Give both rates in litres per minute.

Larger pump's average flow rate

The rate line shows about 0.533 litres per minute for up to 15 minutes.

Convert the small pump's volume
The rates must use the same volume unit. Since 1000 millilitres make 1 litre, convert 900 millilitres to litres before dividing.
$900\text{ mL}=0.9\text{ L}$
Find the small pump's rate
Divide its volume in litres by its time in minutes to get litres per minute.
$0.9\text{ L}\div2\text{ min}=0.45\text{ L/min}$
Find the larger pump's rate
The second volume is already in litres. Divide by its time and compare the results, which now have matching units.
$8\text{ L}\div15\text{ min}\approx0.533\text{ L/min}$

Answer: The small pump moves 0.45 litres per minute. The larger pump moves about 0.533 litres per minute, so the larger pump moves more liquid per minute.

Check: The larger pump moves 8 litres in 15 minutes, which is more than 0.45 litres for each minute. Its calculated rate is therefore reasonably a little above 0.45 litres per minute.

Common mistakes and how to avoid them

Dividing time by distance when asked for distance per hour.

Correction: Put distance first and time second. Check that the resulting units are distance units per time unit.

Comparing 900 millilitres per 2 minutes directly with 8 litres per 15 minutes.

Correction: Convert the volumes to the same unit and calculate both rates per minute before comparing.

Reporting a rate as only a number, such as 25.

Correction: Include the units. In the delivery example, 25 kilometres per hour tells what the number measures.

Assuming a constant rate when the situation does not support it.

Correction: Use a rate to predict a later total only when the question states or reasonably assumes that the rate stays constant.

Lesson summary

A rate compares two quantities with different units.
A unit rate gives the amount for one unit of the second quantity.
Divide in the order that matches the question, and keep units in the calculation.
Convert rates to matching units before comparing them.
Check whether the answer and any constant-rate assumption make sense in context.

Check your understanding

Question 1

A printer produces 84 labels in 6 minutes. What is its average rate in labels per minute?

14 labels per minute
78 labels per minute
0.071 labels per minute
504 labels per minute

Show answer and explanation

14 labels per minute

Divide labels by minutes: 84 divided by 6 is 14 labels per minute.

Question 2

Which rate is greater: 2.4 litres per minute or 1300 millilitres per minute?

2.4 litres per minute
1300 millilitres per minute
They are equal
There is not enough information to compare them

Show answer and explanation

2.4 litres per minute

Convert 1300 millilitres to 1.3 litres. Since 2.4 litres is greater than 1.3 litres, the first rate is greater.

Question 3

A cyclist travels 42 kilometres in 2 hours at a steady average rate. How far would that rate predict in 3 hours?

21 kilometres
47 kilometres
63 kilometres
84 kilometres

Show answer and explanation

63 kilometres

The rate is 42 divided by 2, or 21 kilometres per hour. Multiply by 3 hours to get 63 kilometres.

Key terms

Quantity: An amount that can be measured or counted.
Unit: The named measure used for a quantity, such as minutes or litres.
Rate: A comparison of two quantities with different units.
Unit rate: A rate that gives the amount of one quantity for one unit of another quantity.
Average rate: A rate that summarizes the quantities over a whole interval, such as total distance divided by total time.
Constant rate: A rate that stays the same for equal amounts of the other quantity.

Continue through Math 20-2

View the complete Math 20-2 Alberta Grade 11 Mathematics curriculum and lessons

Tutor support for M1 · Apply and compare rates and unit rates in context

Working on M1 · Apply and compare rates and unit rates in context in Math 20-2? Bring the question and your own attempt. In a live session, you can point to evidence in the text, explain how it supports your claim, and revise one sentence together. Ask about the first step that does not make sense, then try a similar question yourself.

Explore math tutoring · Request an eligible free 30-minute introduction. Your request is matched by subject and language; a tutor must accept before a time is confirmed.

برای مبحث M1 · نرخ‌ها و نرخ‌های واحد را در مسئله‌های کاربردی به کار ببرید و مقایسه کنید در درس Math 20-2، سؤال و تلاش خود را همراه بیاورید. در جلسه می‌توانید شواهد متن را نشان دهید، ارتباط آن را با ادعای خود توضیح دهید و یک جمله را با هم بازنویسی کنید. راهنمای معلم ریاضی را ببینید یا جلسهٔ معارفهٔ رایگان ۳۰ دقیقه‌ای درخواست کنید. زمان جلسه پس از پذیرش معلم قطعی می‌شود.

About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Alberta Grade 11 Mathematics (Math 20-2), expectation M1. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

Official curriculum reference

Report a correction or ask a question