11.06 Convert rates

Let's review how to identify and use rates to make useful comparisons. Recall the following definition and distinction between a rate and a ratio:

• A ratio is a comparison of the relative size of two or more quantities using the same units. 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.

Common examples of rates include:

• Speeds, for example $60$60 km/h, $20$20 m/s, $400$400 m every $5$5 minutes
• Growth rates, for example $20$20 cm/year, $15$15 mm/month, $2$2 kg per week
• Cost of groceries, for example $\$5$$5 per kg, $\$1.20$$1.20 per $100$100 g, $\$10$$10 per box

 

Converting 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. 

Converting rates allows us to compare rates given in different units and to also obtain a rate in units suitable for a particular application. Common applications are comparing unit prices to find the best deal and comparing speeds of different objects.

 

Worked example

Holly runs a $42$42 kilometre marathon in $3$3 hours and $30$30 minutes.

(a) Find Holly's simplified running rate (speed) in kilometres per hour.

Think: To obtain a rate in km/h, we need to divide the distance in kilometres by the time in hours. ($3$3 hours and $30$30 minutes$=3.5$=3.5 hours)

Do:

$\text{Speed}$Speed	$=$=	$\frac{42\text{ km}}{3.5\text{ h}}$42 km3.5 h​	Divide distance in kilometres by time in hours
	$=$=	$12\text{ km/h}$12 km/h	Simplify the fraction and don't forget units

 

(b) Convert her speed to metres per second.

Think: Both the distance unit and time unit are being converted. Let's first change the distance unit. In this case, we are converting distance from kilometres to metres. The number of metres travelled in a given amount of time is $1000$1000 times greater than the number of kilometres, so we want to multiply the rate by $1000$1000. 

Then we want to convert the new rate in m/h to m/s. There are $60$60 seconds in a minute, and then $60$60 minutes in an hour, so the number of metres travelled in a second will be the rate in m/h divided by $60^2$602.

Do:

Speed	$=$=	$12$12 km/h	Write the given rate including units
	$=$=	$12\times1000$12×1000 m/h	Convert the kilometres to metres
	$=$=	$12000$12000 m/h	Simplify the rate
	$=$=	$\frac{12000}{60^2}$12000602​ m/s	Convert the rate from per hours to per second
	$=$=	$3\frac{1}{3}$313​ m/s	Simplify the rate

 

 

 

Practice questions

Question 1

Question 2

Question 3

Question 4