Pop Video: How Do You Solve Number Puzzles?

In this video we look at two very different ways of solving a maths puzzle about ratio and proportion. Find out whether you prefer the concrete, or the abstract approach, and consider trying the other one the next time you try to solve a puzzle.


Video Transcript

Do you like to visualize your number problems or does nothing make your pulse race more quickly with excitement than a nice bit of good old-fashioned algebra? In this video, we’re gonna take a simple number puzzle and look at two very different ways of solving it. One way is something called a bar model, which is increasingly being used in schools around the world after its success in Singapore and around Asia, and the other uses algebra to set up and solve some equations.

Okay, here’s the puzzle.

James is eight years old. His uncle, John, is 32 years old. How old will James be when John is twice James’s age?

So it’s a ratio and proportion question. And it jumps from a very concrete statement about James’s and John’s ages now to a bit of an abstract question about a future time when John will be twice as old as James is. You may be up to think of many different ways to tackle the question. But let’s start by using the bar method. If we don’t quite know how to approach things, we can draw out what we do know using a diagram so we can start to get our brain working on the problem and visualize the situation.

Well, here’s James aged eight and John aged 32 — it hasn’t really given us a great deal of insight. But we can see that John is four times as old as James at the moment and we’re looking for the time when John will be twice as old as James. Okay, so maybe that picture wasn’t terribly helpful. But if we represent their ages using bars on a consistent scale, James is eight and John is 32 and the difference in their ages is 32 minus eight, which is 24. And the difference in their ages will always be 24 years because each year that James gets one year older, John will also get one year older. For example, in a year’s time, James will be nine and John will be 33. The difference will still be 24.

Now, we need to think about the point in time when John is twice as old as James. So I’ve drawn this on a different diagram with a different scale. The length of the bar representing James’s age and this difference between the bars are the same because John is twice as old as James or James is half as old as John. Well, we know that the difference in their ages is always 24. So I can write that here. Because we know these two lengths are the same, we could also write 24 here. At this time then, James will be 24 and John will be two times 24, which is 48.

Now, remember the question was actually how old will James be when John is twice James’s age. So the answer is James will be 24 when John is twice James’s age. This visual representation of the ages with the bars created a real aha moment, where you can see that one bar here was half the length of the other bar and you knew that the difference was 24. With this approach, anyone can solve the problem.

Okay, now let’s rewind and do the puzzle again using algebra.

James is eight years old. His uncle John is 32 years old. How old will James be when John is twice James’s age?

First, let’s define some variables. Let 𝑎 be James’s age in years and let 𝑏 be John’s age in years. Now, we can solve a general form of our problem. Think that James is 𝑎 years old and his uncle John is 𝑏 years old, how old will James be when John is twice James’s age? Now, we can let 𝑛 be the number of years that must pass for John to be twice James’s age. Then, we know that in 𝑛 years time, James will be 𝑎 plus 𝑛 years old and John will 𝑏 plus 𝑛 years old. But we also know that in 𝑛 years time, John will be twice James’s age. So we can also say that two times James’s age in 𝑛 years equals John’s age in 𝑛 years. Algebraically, two times 𝑎 plus 𝑛 is equal to 𝑏 plus 𝑛.

Now, we can carry out some algebraic manipulation to work out how many years must pass before John is twice James’s age and how old James will be at that time. So we know that in 𝑛 years, two times 𝑎 plus 𝑛 is equal to 𝑏 plus 𝑛. So multiplying that two by the parentheses on the left-hand side of the equation using the distributive property gives us two 𝑎 plus two 𝑛 is equal to 𝑏 plus 𝑛. Then, subtracting 𝑛 from both sides of the equation leaves us with two 𝑎 plus one 𝑛 is equal to just 𝑏. And finally, subtracting two 𝑎 from each side gives us an expression for 𝑛 in terms of 𝑎 and 𝑏.

But we know that James was eight to start off with. So 𝑎 is equal to eight. And we know that John was 32 to start off with. So 𝑏 is 32. So let’s substitute those numbers into this equation. That gives us 𝑛 is equal to 32 minus two times eight. And two times eight is 16. So 𝑛 is 32 minus 16 and 32 minus 16 is just 16. So 𝑛 is 16. It’ll be 16 years’ time when James is half of John’s age. And this means James will be eight plus 16. That’s 24 years old at that time.

And here, we have it. We got the same answer. But we made quite a different journey. Each method has its own advantages and disadvantages. For example, using algebra, you’ve developed a general solution which could be easily adapted if you were given different starting numbers and you’re developing skills that would help you to build spreadsheets or write computer programs. With the bar method, you can easily visualize, solve, and verify the problem without having to think about abstract formulae.

So we’ve probably all got our own preferred ways of solving problems. Some of us like to take a concrete approach and visualize a specific problem, while others prefer to work in an abstract and generalized way using algebra. Either way, it can be useful to take the time to try solving problems using another method to give us a different way of looking at and analyzing the world around us.

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