Identifying and writing rates
A ratio is a comparison of the relative size of two or more quantities. For example, if a bag of marbles contains $3$3 blue marbles and $2$2 red marbles, we can describe the comparison between the number of red and blue marbles as a ratio like red:blue = $2:3$2:3.
A rate is a ratio between two quantities that are measured in different units. For example, the rate a tap leaks may be $30$30 mL every $5$5 minutes. Rates are often expressed as unitary rates (or a unit rate) where the second quantity in the rate has a measure of $1$1. The unitary rate for the leaking tap would be $6$6 mL for every $1$1 minute. To abbreviate this we drop the $1$1 and use the word per or the notation '/' to separate the two units, thus we could write $6$6 mL/min. This compound unit, mL/min, represents the division of one measurement by another to obtain the rate.
Examples of unit rates:
$60$60 km/h - read as $60$60 kilometres per hour. Many rates are generated with time as the second quantity. Such as speeds, (distances per time), or rates of flow (capacity per time), or rates of growth (growth per time).
$10$10 g/m2 - read as $10$10 grams per square metre. These sorts of rates are common in agricultural settings where seeds, fertilisers or weed control measures are based on an application per square metre of land.
Examples of non-unit rates:
$20$20 km per $3$3 h - read as $20$20 kilometres per $3$3 hours. This might describe how the speed a person walked on the weekend.
$1$1 tablet/$20$20 kg - read as $1$1 tablet per $20$20 kilograms. Might describe how many tablets are required based on the body weight of a person.
Sometimes the term rate is used for quantities like tax rates or literacy rates, which are usually given as a percentage. A percentage, is a ratio of two quantities that have the same units, and so isn't a rate in the formal sense. However, the term rate is still used.
Writing a rate from a description requires you to be able to identify the two quantities and their corresponding units.
Worked examples
Example 1
A person walks $25$25 steps every $30$30 seconds. Write this information as a walking rate.
Think: The quantities are $25$25 steps and $30$30 seconds. In maths, words like 'every' and 'for each' means the same as 'per'. Let's use the symbol $/$/ for 'per'.
Do: We translate the written statement $25$25 steps every $30$30 seconds, to $25$25 steps / $30$30 seconds.
Example 2
A fruit picker fills $10$10 crates of fruit every hour. What is the rate of fruit picking?
Think: The quantities are $10$10 crates and $1$1 hour.
Do: We translate the written statement $10$10 crates of fruit every hour, to $10$10 crates /$1$1 h, but we don't need to write the $1$1. So we end up with $10$10 crates/h.
Practice questions
Question 1
Which of the following scenarios can be described by a rate of hours/book?
Select all that apply.
The number of hours taken to read a given number of books.
AWe cannot answer this question without numbers.
BThe amount of time in an hour spent reading.
CThe number of books read this week compared to last week.
DThe number of books read in a given number of hours.
E
Question 2
Quentin is making jewellery. Which of the following rates would have the units necklaces/hour?
cannot answer this question without numbers
Anumber of necklaces made in 1 hour
Bnumber of hours taken to make 1 necklace
C
Simplifying rates
A rate is usually considered simplified when it is represented as a unit rate. Remember that a unit rate is a rate where the second quantity is just $1$1 of the unit prescribed. To calculate a rate, divide one quantity by another. This will give us two components, the numeric value and the compound unit. The numeric component can often be further simplified just as we simplify fractions.
Worked example
Example 3
Write the speed $24$24 km per $4$4 hrs as a unit rate.
Think: Calculate the rate by writing it as a division of distance divided by time.
Do:
| Rate | $=$= | $\frac{24}{4}$244 |
Divide the distance in km by the time in hrs. |
| $=$= | $6$6 km/h |
Simplify the division and include the compound unit. |
Once we have a simplified rate it's easier to compare and interpret situations that use rates.
Not all compound units are written using a slash and instead use the letter 'p' to represent 'per'.
For example, heart rate has units beats per minute, which uses the compound unit bpm and frame rates for animation commonly uses the unit frames per second, which is abbreviated to fps. Units for download rates are another common example, such as Mbps which is an abbreviation for megabits per second.
Practice questions
Question 3
Luke earned $\$1134$$1134 in $6$6 hours. What is the rate he earns per hour?
Question 4
A tap fills up a $240$240 litre tub in $4$4 hours.
Which of the following is the compound unit for the rate of water flow?
L
AL/hr
Bhrs/L
Chrs
DWhat is the rate of water flow of the tap in litres per hour?