2 Problem solving

When two people tackle a problem there is a good chance they will solve it in a different way to each other – that is absolutely fine! Some approaches may just be more efficient and get you to an answer more quickly than others. The more experience you have with problem solving the better you will become at it.

Depending on your previous experiences and how you initially take in a particular problem, the way you decide to begin to solve the problem will vary. Solutions to everyday problems may just need the use of common sense and organisation to work your way through them.

As you work through any problem, remember that there are usually alternative methods for reaching the solution. If you get stuck using your initial approach, try a different one. Keep in mind that using pictures and staying level-headed will carry you far, and most likely help you finish solving an exercise. So try not to panic!

With these thoughts in mind, try the next activity. Just as with the dinner party puzzle in Activity 1, you will need to use logic and reasoning to recover missing pieces of information.

Activity 2 New vehicle

Timing: Allow approximately 10 minutes

Imagine you’ve recently been looking 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.

When you call the dealership to enquire about its stock, 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.

Can you work out how many motorcycles and how many cars the dealership currently has?

As with any problem there are different ways of approaching this problem, so when you’ve got your answer take a look at some other possible methods.

Remember, if you need a hint to help you get started click on ‘Reveal comment’ below.

Comment

There are many ways to solve this problem. You might consider trying to use pictures (visualisation 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

Answer

Download this video clip.Video player: swm_1_s1_new_vehicle_diagrams_animation.mp4

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Transcript

NARRATOR:

Suppose that all the vehicles are motorbikes. Draw a diagram-- you don't have to be an artist-- that shows these 21 motorbikes. Be sure to use a representation that will allow you to clearly distinguish between 2 and 4 wheels. How many wheels do you have? There were 21 motorbikes, each with 2 wheels. So in total, there are 42 wheels.

However, Paul said that there were 54 wheels in total. So you need to add an additional 12 wheels because 54 take away 42 is 12. To work out how many motorbikes and cars the dealership has from here, you could just start adding two wheels to each motorbike until you've counted up to the 12, or you could change a motorbike into a car to gain two additional wheels.

Since you know you need 12 more wheels to reach the required total, you need to change 6 motorbikes to cars because 12 divided by 2 is 6. You can now see, there are six cars and 15 motorbikes, giving a total of 21 vehicles. You can check that the total number of wheels adds up to 54.

Each car has four wheels, which makes 6 times 4 equals 24 wheels. Each motorbike has 2 wheels, which makes 15 times 2 equals 30 wheels. Together, that is 24 plus 30 equals 54 wheels. So the dealership has 15 motorbikes and 6 cars in stock.

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Method 2: Comparing to ‘all cars’

Answer

Download this video clip.Video player: swm_1_s1_new_vehicle_comparing_all_cars_animation.mp4

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Transcript

NARRATOR:

Suppose the dealership had all cars. That would mean there were 84 wheels, because 21 times 4 is 84. But you've been told by Paul there are only 54 wheels. That means there are 30 wheels too many, as 84 take away 54 is 30.

To solve this, you need to change some cars into motorbikes. When a car changes into a motorbike, it loses 2 wheels. Therefore, you need to divide the number of wheels you have too many, that's 30, by 2. 30 divided by 2 is 15. So, 15 cars to change into motorbikes.

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Method 3: Educated guess

Answer

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.

Table 1 Calculating the number of motorbikes and cars using an educated guess

Motorbikes	Cars	Total number of wheels
10	11	10 × 2 + 11 × 4 = 64	(too many wheels → need fewer cars)
12	9	12 × 2 + 9 × 4 = 60	(too many wheels → need fewer cars)
14	7	14 × 2 + 7 × 4 = 56	(too many wheels → need fewer cars)
15	6	15 × 2 + 6 × 4 = 54	This matches with what Paul told you.

Once again, the conclusion is that there are 15 motorbikes and 6 cars at the dealership.

So, remember that when you come across a problem there will usually be more than one way to approach it. If you don’t get anywhere with your first method, see if you can come at the problem a slightly different way.

You are going to leave problem solving behind for a moment and move onto subjects that you might think feel like proper maths! It is important to have a good understanding of how numbers are put together so that they make sense to us and what they represent. You will turn your attention to this in the next section.