Think of a number, multiply it by 2 and take away 6.
Now halve your answer and add 3. You should end up back at the number you first thought of.
Can you use algebra to explain why this happens?
Hint: start by using a letter for the number that you first thought of.
Try it before you read on...
Okay, so let's call the number that you started with "n".
We're going to start by "dressing it up" to make an algebraic expression.
"Think of a number, multiply it by 2 and take away 6."
If you multiply n by 2 then you get 2n.
Now take away 6 and you have 2n - 6.
Now we're going to "undress" it again to get the n on its own... but by a slightly different route!
"Now halve your answer and add 3."
How can we halve it? Well, 2n - 6 is the same as 2 lots of n and one lot of -6...
... and -6 is the same as 2 lots of -3
... so if you halve 2n - 6 then you get half of 2n and half of -6, giving n - 3.
The last step is easy: we now have n - 3, i.e. 3 less than n. If we add 3 to that then the -3 and the +3 cancel out
and we're left with just n, the number we started with!
How about this one?
Think of a number, add 4 and treble it (i.e. multiply it by 3). Take away 12. Tell me what you got and I'll tell you what number you started with. How do I do it?
Think it through and see if you can work it out yourself first.
Of course, I could do the whole calculation backwards (add 12, divide the answer by 3 and then take away 4) but the numbers are carefully chosen to make it much easier than that. In fact, all I have to do is to divide by 3. It's a bit like going through a labyrinth of side streets and eventually finding yourself just a few steps away from where you started!
Here's why: If you start with n and add 4 then you have n + 4. Treble it then you get 3 lots of n + 4, i.e. 3(n + 4).
This is the same as 3 lots of n and 3 lots of 4, which we can write as 3n + 12 (this is called multiplying out the bracket).
When you take away the 12 you're left with just 3n...
... so to get back to n I just have to divide by 3.
What we're doing here - in a mathematical sense - is building up equations and then solving them.
In the first example, you found the value of 2n - 6 and then "undressed the n" to get back to its original value.
In the second one you found the value of 3(n + 4), multiplied out the bracket and again "undressed the n" to find its value.
Here's a slightly harder one for you to mull over:
Think of a number. Double it, add 2 and multiply your answer by 3. Take away 6 from your answer. Now take away the number you first thought of.
If you were to tell me the number you'd ended up with, how could I work out the number you started with?
Can you make up some more of your own?