Simple Rates Comparisons
We've already looked at how to express the relationship between two different units of measurement as a rate.
Usually, we write the rates as unit rates. A unit rate describes how many units of the first type of quantity corresponds to one unit of the second type of quantity.
Some common unit rates are distance per hour, cost per item and earnings per week. Do you see how in each example the first quantity is related to one unit of the second quantity?
Basically, when we compare unit rates, we are making a judgement about the characteristics of rates, whether it be which item is the best value (ie. the lowest cost per unit of product), which car is faster or which company is more productive. Unit rates make it easier to compare two different rates as they give us a common point of comparison.
For example, if Steve shot $35$35 baskets in $1$1 minute and Tyrell shot $18$18 baskets in $30$30 seconds, who had the faster rate of scoring? To solve this problem, we need a common point of comparison. In other words, we need to know how many baskets both players scored in one minute.
We already know that Steve shot $35$35 baskets/min.
To work out Tyrell's rate, we need to double both quantities (since $30\times2=60$30×2=60 seconds). Then we can say Tyrell's rate was $36$36 baskets/min, so his rate of scoring was faster.
Let's look at some more examples now!
Worked Examples
Question 1
Consider the following three trips:
A) driving $180$180km in $3$3 hours
B) bicycling $132$132km in $4$4 hours
C) a taxi trip that took $15$15 minutes to travel $11$11km
Which option correctly orders the trips from lowest average speed to highest average speed?
$A$A, $B$B, $C$C
A$C$C, $A$A, $B$B
B$B$B, $C$C, $A$A
C$C$C, $B$B, $A$A
D$A$A, $C$C, $B$B
E$B$B, $A$A, $C$C
F
Question 2
Two shearers wanted to work out who was the faster shearer.
Jenny sheared $144$144 sheep over $6$6 days, and Sean sheared $115$115 sheep in $5$5 days.
At what rate per day did Jenny shear sheep?
If Jenny continued at this rate, how many would she be able to shear in $25$25 days?
At what rate per day did Sean shear sheep?
Who sheared sheep at a faster daily rate?
Sean
AJenny
B
Question 3
Peter runs daily and usually covers his $19$19 km in $75$75 minutes.
He also enters half marathon and full marathon events. The last time he ran the Sydney half marathon, an event $21$21 km in length, it took him $126$126 minutes.
What is his running rate during his daily run?
What was his running rate during the half marathon?
In which event was he running faster on average?
On his daily run.
AIn the half marathon.
B