Succeed with maths – Part 1

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# 3.1 New vehicle

Just as with the Sudoku puzzle, in the next problem you need to use logic and reasoning to recover missing pieces of information.

## Activity 4 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

Suppose that all the vehicles are motorbikes. Draw a diagram (you don’t need 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.

Figure 8 Picture representing 21 motorbikes

Since there are 21 motorbikes with 2 wheels each, your drawing shows a total of 42 wheels. Paul said that there were 54 wheels, so you need an additional 12 wheels, because 54 – 42 = 12. In other words, some of the motorbikes drawn need to be turned into cars.

You could just start adding 2 wheels to each motorbike until you’ve counted up to 12, or you could be clever. If you change a motorbike into a car, you gain 2 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 ÷ 2 = 6. Can you see how that worked?

Figure 9 Picture representing 21 motorbikes altered to show the first 6 with 2 extra wheels added to make 4

From the picture, you can observe that there are 6 cars and 15 motorbikes, for a total of 21 vehicles. You can check that the total number of wheels adds up to 54.

• Each car has 4 wheels, which makes 6 × 4 = 24 wheels.
• Each motorbike has 2 wheels, which makes 15 × 2 = 30 wheels.
• Together, this is 24 + 30 = 54 wheels.

Thus, the dealership has 15 motorbikes and 6 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 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.

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 54 ÷ 2 = 27. If all the vehicles were motorbikes, you would have 27 motorbikes, which is too many: remember, Paul told you there are only 21 vehicles.

Next, determine how many extra pairs of wheels there are. Because 27– 21 = 6, you know there are 6 extra pairs of wheels. This indicates that 6 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, 6 vehicles must have 2 pairs of wheels, or 4 wheels each. Thus, there are 6 cars. To find the number of motorbikes, you need to take the cars away from the total number of vehicles: 21 – 6 = 15.

Consequently, there are 15 motorbikes and 6 cars.

So, remember when you come across a problem that 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.