1.3.4 New Vehicle
It’s time for another math activity. If we are given only partial information, then sometimes our logic and reasoning skills can help us recover the missing pieces. In other words, we often rely, perhaps unknowingly, on mathematics to fill in the gaps. Let’s consider an example where we can retrieve information despite not having the entire picture.
Activity: New Vehicle
You’ve recently been shopping for a new vehicle. You don’t know at the moment if you would like a motorbike or a car. Because you would like to get the best price possible and have the most options available, you would like to start by visiting the dealership that has the greatest number of vehicles of each type.
When you call the dealership to inquire about its inventory, the assistant manager, Paul, jokingly tells you that they have 21 vehicles available, with a total of 54 wheels. Just as you ask him how many of each type of vehicle he has, the phone call is inadvertently disconnected.
Do you have enough information to answer your own question? Can you figure out how many motorcycles and how many cars the dealership currently has?
There are many ways to solve this problem. You might consider trying using pictures (visualization can be very helpful) or select a starting point, such as assuming half are motorbikes and half are cars, and then revising your first guess.
Method 1: Diagrams
Since there are different methods that can be used to tackle this problem, it’s possible you might have tried a different (but perfectly valid!) approach than the ones presented here.
Let’s suppose we have all motorbikes. Draw a picture (you don’t need to be an artist) that shows 21 motorbikes. Be sure to use a representation that will allow you to clearly distinguish between two and four wheels.
Since there are 21 motorbikes with two wheels each, your drawing shows a total of 42 wheels. Paul said that there were 54 wheels, so we need an additional 12 wheels, because . In other words, some of the motorbikes we drew need to be turned into cars.
We could just start adding two wheels to each motorbike until we’ve counted up to 12, or we could be clever. If we change a motorbike into a car, we gain two additional wheels. Since we know we need 12 more wheels to reach the required total, we need to change six motorbikes over to cars, because . Can you see how that worked?
From our picture, we observe that there are six cars and 15 motorbikes, for a total of 21 vehicles. We can check that the total number of wheels adds up to 54.
- Each car has four wheels, which makes wheels.
- Each motorbike has two wheels, which makes wheels.
- Together, this is wheels.
Thus, the dealership has 15 motorbikes and six cars in stock.
Method 2: Educated Guess
You can make any guess you think is reasonable. For example, you might assume that about half of the vehicles were motorbikes—say, ten of the vehicles. To keep track of your guesses, a table is quite useful. As you make adjustments to your guesses, remember that the number of motorbikes plus the number of cars must equal 21.
|Motorbikes||Cars||Total Number of Wheels|
|10||11||(too many wheels → need fewer cars)|
|12||9||(too many wheels → need fewer cars)|
|14||7||(too many wheels → need fewer cars)|
|15||6||This matches with what Paul told you.|
Once again, we come to the conclusion that there are 15 motorbikes and six cars at the dealership.
Method 3: Pairs of Wheels
Another way to solve this problem is to consider the pairs of wheels. Because the assistant manager told you that there are a total of 54 wheels, this means there are 27 pairs of wheels, because . If all of the vehicles were motorbikes, you would have 27 motorbikes, wthis hich is too many: remember, Paul told you there are only 21 vehicles.
Next, determine how many extra pairs of wheels there are. Because , we know there are six extra pairs of wheels. This indicates that six of the vehicles have to have an extra pair of wheels (beyond the pair for each vehicle that you’ve already counted). In other words, six vehicles must have two pairs of wheels, or four wheels each. Thus, there are six cars. To find the number of motorbikes, we need to take the cars away from the total number of vehicles: .
Consequently, there are 15 motorbikes and six cars.