In today’s post, we are going to work on proportionality by looking at some examples of proportionality problems.
Before we begin, you can take a look at other exercises with proportional numbers where we review the concepts of reason and proportionality constant.
Direct Proportionality and Inverse Proportionality
Proportional magnitudes can be directly proportional or inversely proportional.
When are they directly proportional? When increasing one of the magnitudes proportionally increases the other. That is if you multiply or divide one by a number, the other also multiplies or divides by the same number.
However, they are inversely proportional when increasing one of the magnitudes decreases the other proportionally. In other words, when multiplying one by a number, the other is divided by the same number, or vice versa, dividing one by a number and the other is multiplied by the number.
We’ve previously seen what direct and inverse proportionality are but it is very important to remember that there are situations that seem proportional but are not because they only fulfill some, and not all, of the conditions. To see things more clearly, let’s look at some examples to learn why they are not proportional.
Example 1 of No Proportionality
John’s personal best for running the 100m dash is 17 seconds. How long will it take him to run 1km?
This seems like a problem that can be solved by multiplying 17 by 10 since a kilometer is 10 times 100 meters, right? In reality, this is not proportional because this is his best time for the 100m and although it is not impossible, we cannot say for sure if he would be able to maintain this speed for a kilometer. Therefore the answer to this problem is “There is no way to know with this information.”
Example 2 of No Proportionality
A train engine measures 12 meters and with four train cars attached it measures 52 meters in length. How long would the train measure with 8 train cars attached?
It seems like a problem dealing with direct proportionality – more train cars means longer in length – and with double the number of train cars it should double the length right? Well, no. The same engine, the only one mentioned in the problem, will be pulling the four extra train cars that have been added. This means that the total length would be 92 meters because the engine measures 12 meters and with 4 train cars the train measures 52 meters, meaning each train car measures 10 meters.
Example 3 of No Proportionality
Today it took me 4 hours to fertilize a patch of land. How long would it take to fertilize a patch of land twice as big?
Another problem that seems to be directly proportional right? You may think that a larger surface area means it will take a longer time but beware. If the area is doubled, the proportionality ratio will not be two but four because the surface of an area that is twice the size is four times greater! If you are a bit confused about this it is best to draw two squares, with one being twice the size of the other.
Now we are going to look at some problems by first confirming that they are proportional, then seeing if they are directly or inversely proportional, and finally, solving them.
First Proportionality Problem
Upon arriving at the hotel we were given a map with the local sights and were told that 5 centimeters on the map represented 600 meters in reality. Today we want to go to a park that is located 8 centimeters from the hotel on the map. How far is the park from the hotel?
In order to solve this problem, we must first determine its proportionality. Maps, if they are well made, always fulfill the greater distance on the map in reality. If instead of 5 centimeters we talked about twice as many centimeters on the map (10 cm) would it be more or less than 10 meters in reality?
It would be more: double the number of meters in reality.
If doubling one magnitude (centimeters) also doubles the other (meters) then we are talking about direct proportionality.
Therefore, we are going to solve the problem:
Since 5 centimeters represent 600 meters, 1 centimeter will represent…
600 : 5 = 120 meters
Since 1 centimeter represents 120 meters, 8 centimeters will represent…
120 x 8 = 960 meters
Solution: The park can be found 960 meters from the hotel.
Second Proportionality Problem
Yesterday 2 trucks transported merchandise from the port to the warehouse. Today, 3 trucks will have to make 6 trips in order to transport the same amount of merchandise as yesterday from the warehouse to the mall. How many trips did the trucks have to make yesterday?
We wonder if it is proportional, and what kind. To determine this we can think…
If instead of 3 trucks we talked about twice as many trucks (6), would they have to make more or fewer trips?
The more trucks with merchandise, the fewer trips needed to move everything: they would only need to make half the number of trips.
If doubling one magnitude (trucks) divides the other (trips) in half, then we are dealing with inverse proportionality.
Therefore, we’re going to solve the problem:
Since 3 trucks need to make 6 trips, 1 truck would need to make…
3 x 6 = 18 trips
Since 1 single truck would need to make 18 trips, 2 trips had to make…
18 / 2 = 9 trips
Solution: Yesterday 2 trucks made 9 trips.
What did you think of this post? Has it helped you to better understand proportionality problems?
Remember to register with Smartick and try it for free. You can practice exercises and problems about proportionality and other math content adapted to your level.
The examples for ‘no proportionality’ were taken from an article by Dooren Bock, and Verschaffel. They are Dutch math researchers and we highly recommend reading the article if you are interested in math education.