A formula for Success
The easiest way to understand the use of algebra is to look at some formulas (known by snobbish mathematicians/examiners as algebraic equations).
Formulas loosely speaking are mathematical tools, which help us make something at the end of the day.
A recipe is a very simple formula, such as this one, which makes 30 small cubes of the tasty sweet Coconut Ice:
Two and a half cups of icing sugar
A quarter of a teaspoon of
cream of tartar
One Egg white (lightly beaten)
A quarter of a cup of condensed
milk
One and three quarters of a
Cup of desiccated coconut
This doesn't look too much like a mathematical formula yet, however, what if we simplify each line of the recipe with a shorthand code e.g. "a" can be a cup of icing sugar, "b" can be a teaspoon of cream of tartar and so on. We use "x" to represent a cube of coconut ice. Now we can translate the wordy recipe into a more mathematical formula (which is what we would have to do if we wanted a computer to understand the recipe):
2.5a + .25b + c + .25d + 1.75e = 30x
Notice that when we construct formulas using these shorthand codes, that the multiplication sign x is not used - instead we just miss it out altogether. This means that 2.5a means the same as 2.5 multiplied by (or times if you prefer) a.
The reason for this is that when we mix numbers and letters we can get away with this, but not when we are only dealing with numbers:
For example if we wrote 2 x 5 in the same way we get 2 5 (does this mean 2 x 5 or 25?)
There's also another good reason for doing this and that is that we often use x as a shorthand code. Hence if we included the multiplication sign in 30x, we end up with 30 x x (does this mean 30 times x or 30 times x times x?).
So, you have seen that formulas do not appear by magic or from the minds of insanely intelligent academics. They are based on what we see in real life and a constructed specially to represent real life and to enable us to more easily construct and calculate things in the future.
Creation of a formula helps us solve the problem below (suprisingly, it's actually much easier to create formula than to try to work out the problem using another method - e.g. trail and error)
Railway excursion tickets for a certain journey cost £5 for an adult and £3 for a child. A party of football supporters contains five times as many adults as children; their tickets cost a total of £252.
a) If x children are in the party, give an expression for the number of adults in terms of x.
b) Create a formula which can be used to find the number of children and the number of adults.
a) There are five times as many adults as children hence our first mini-formula is:
No. of Adults = 5x
The expression is "examiner speak" for just write out 5x as the answer.
b) We have x children times £3 plus 5x adults times £5 giving a total of £252. So, our second formula is:
3x + 25x = 252
Using some of the techniques from algebra we can then calculate how many children (shorthand code for this is "x") there were:
28x = 252 (add quantities of x together)
x = 9 (divide both sides by 28)
So, if we have 9 children, from the formula in a) we find that there were 45 adults.