Using Letters as Numbers

You already know what to do with a sum like this:-

4 x 3 = __

You would normally write the answer (12) into the space on top of the line (assuming you were given it to do on paper). However, the same sum can also be written out as follows, using a letter instead of a line:-

4 x 3 = a

In this case you should realise that a=12 and that is what you would write down as your answer. But why, you might wonder, would anyone want to write out sums in this silly way? Well, when it's such a simple sum, no one normally does write it out like that, but take a close look at this one:-

2 x a = b

Now do you get the point? This time there is more than one possible answer, because if you try making a=1, then b must be 2, whereas if you try making a=2, then b must be 4. There are actually lots of possible answers, including ones like a=1.5 when b=3, and a=0.1 when b=0.2. What we actually have here is an equation which describes a line, and you can soon see this if we just plot a few of the possible answers on a graph:-

You can draw a straight line through all the dots, and every point on that line will fit the equation 2 x a = b. Now, here's the equation for another line:-

2 x b = a + 6

If a=0, then 2 x b = 6, in which case b=3. If a=1, then 2 x b = 7, in which case b=3.5. If a=2, then 2 x b = 8, in which case b=4. If a=3, then 2 x b = 9, in which case b=4.5. Let's now put these dots on the graph as well (in red), plus a few more:-

You can now see that all of these dots form another straight line through the graph, and you can also see that the two lines cross at the point where a=2 and b=4: that is the only point found on both of these lines. If you are given the equations of both lines as a pair of sums to be solved, the only correct solutions for the pair are a=2 and b=4. You can obviously work these out by drawing out the graph and looking to see where the two lines cross, but you can also work them out without drawing anything at all, and this is an important skill in maths, so let me show you how it's done. You don't need to learn the method at this point: we are just looking ahead to the kind of maths that is done in secondary/high schools. The trick is to try to eliminate either all the a's or all the b's from both of the sums. It is actually normal to miss out the "x" symbol when doing this kind of maths, so we can change the sum 2 x a = b into 2a = b, and the sum 2 x b = a + 6 into 2b = a + 6, so we now have:-

2a = b
2b = a + 6

The first thing we need to do is re-arrange one of the sums to get the a's on one side of the "=" and the b's on the other side in each sum, so I'm going to put the a's on the left and the b's on the right:-

2a = b
a + 6 = 2b

That was easy to do: it's fairly obvious that if 2b = a + 6, then a + 6 = 2b. We could actually eliminate the b's more easily than the a's at the moment, but I'm keen to get rid of the a's first, so my next step will be to clear the 6 in the second sum out of the way by moving it to the other side of the "=", and that's not hard to do:-

2a = b
a = 2b - 6

That change isn't hard to understand: it is no different from changing 2 + 3 = 5 into 2 = 5 - 3. Our next step is to try to eliminate the a's, but we can't do that until we've made them both the same size. We can achieve this by making all the numbers in the second sum twice as big:-

2a = b
2a = 4b - 12

We are allowed to do this because the new bottom sum is every bit as true as it was before, just as we can turn 2 + 3 = 5 into 4 + 6 = 10 by doubling all the numbers in it. So long as we double every number in the sum, it will still be correct. We now do something quite wonderful, because we're going to take one whole sum away from the other in a most unlikely manner, but I'm going to move the one that was underneath onto the top because it looks as if it will be easier to take the other one away from it:-

2a = 4b - 12
2a =  b       
 0 = 3b - 12 

So we have 2a - 2a = 0, then we have 4b - b = 3b, and finally 12 - nothing = 12. We have now created a new sum to solve which only has one letter in it to work out: 0 = 3b - 12, and we can rearrange it to 3b - 12 = 0 to make things easier. Again we can move the number 12 to the other side of the "=", thereby changing the sum to 3b = 0 + 12, which can then be simplified to 3b = 12. Now we can see that b must be 4, and having worked out what b is, we can go back to one of the original sums and replace b in it with 4, so 2a = b becomes 2a = 4, and that means that "a" must be 2. We have now worked out where the two lines will cross on the graph without having to draw it: it is the point where a=2 and b=4. You'll be working with equations and graphs a lot in the future, but they will be more meaningful.


Another reason for using letters instead of numbers is for working things out using a formula. Here is an interesting formula which you can try out for yourself:-

h = 5 x t x t

The letter "h" stands for height in metres, and "t" stands for time in seconds. If you drop a tennis ball off the top of a high building, you can time how many seconds it takes to reach the ground, and then you can replace each of the t's in the formula with that number to work out the height of the building. Suppose it takes exactly four seconds for the ball to reach the ground, then you must put 4 in place of each of the t's, and that gives you h = 5 x 4 x 4. When you multiply 5 by 4 you get 20, and when you multiply that by the other 4 you get 80, so that means h = 80: the building would have to be 80 metres tall.
indentWhat I want you to do now is use this formula to work out how high up in the air you can throw a tennis ball. You will need a stopwatch and a calculator for this, and whatever time you record between throwing the ball and it hitting the ground will have to be divided by 2 to get the time either for the journey up or the journey back down (they will both take more or less the same length of time as each other). You can then put that number into the formula in place of each of the t's and calculate the height. If the ball stays in the air for three seconds, then each half of the trip will last 1.5 seconds (that's 1.5 seconds to go up and another 1.5 seconds to come back down), so you will have h = 5 x 1.5 x 1.5, and that means h = 11.25. Eleven and a quarter metres: can you beat that? Feel free to cheat by using a tennis racquet or cricket bat if you have one, and whack the ball upwards as hard as you can. If you don't have a tennis ball, please use something similar which is reasonably soft: I don't want you to risk injuring anyone, including yourself. It doesn't matter if the ball goes sideways as well as upwards: the formula will still tell you how high it went, so you can work out how high every ball goes in a baseball game just by timing how long it stays in the air, as long as it lands at ground level. You can try this out while watching a game on TV.
indentThe formula h = 5 x t x t can also be written as h=5tt, and whenever a number or letter is to be multipled by itself, the second one can be replaced with a little "2" instead, which should be pronounced as "squared", so we now have h=5t2. Letters are often used instead of numbers in computer programs, and they are then called "variables" because the value that the letters represent can vary. If fact, whole words are often used rather than single letters, so the formula h= 5 x t x t could be written as height = 5 x time x time. Unlike in normal maths, in computer programming languages the "x" symbol can't be left out, and there is no symbol for squared either, so the word "time" has to be repeated. Because the symbol "x" may be used as a letter rather than as the times symbol, the times symbol actually has to be turned into an asterisk (*): this means that you would actually write the formula as height = 5 * time * time when using it in a computer program. When the program is run, it might then go through a range of possible values for "time" and work out the height that goes with each of those time values before plotting all the results on a graph. I hope you now understand the purpose of using letters (and words) to represent numbers in maths, because it's an extremely useful idea.