A mathematical formula is a relationship between two or more quantities. The quantities it relates are represented by unknown amounts, called variables. If you know the value of all the variables except one, you can use the formula to calculate that one.

Here is a formula with which you are already familiar. It is a formula which will let you calculate the area of a rectangle. But first you must know the values of the length and width. The formula is  Area = Length x Width


By knowing the values of the variables L and W, you can calculate the quantity A.


There are many kinds of formulas. Some of them are found mathematically, and we know they are correct because we can prove they work. For example, in the picture of the rectangle above, if you made a rectangle with paper that was 8 cm by 4 cm, and cut it into 1 square centimetre pieces, you would discover that there were exactly 32 of them.


Other formulas are discovered by carefully measuring quantities which seem to be related, and looking for a pattern.

Here's an interesting example. Scientists have been studying living creatures for hundreds of years. One thing they noticed is that there seems to be a relation between how fast a mammal's heart beats, and how long it lives. For example, an elephant's heart beats very slowly, and they live a long time. On the other hand, a mouse's heartbeat is very fast, and mice live for only a few years.

The faster the heartbeat, the less time it lives.

A relationship like this, where one quantity gets smaller as the other gets bigger, is called an inverse relationship.


Scientists also made the startling discovery that most mammals live for about the time it takes their heart to beat 1 billion times, regardless of the size of the mammal. When a mouse dies after a few years, its heart will have beaten about a billion times. When an elephant dies after 70 years, its heart will have beaten about a billion times.

Scientists went even further, looking for a formula which would let them predict the lifespan of a mammal, based on the speed of its heartbeat. In this case there are only two variables, the heartbeat speed (in beats per minute) and the lifespan (in years).

Here's the formula they came up with:



... where  a is the mammal's lifespan in years, and  h is its heartrate in beats per minute.

Let's see how it works. Consider the common shrew, which is the tiniest mammal on earth. Its heart beats very fast, about 800 times per minute. Using this number for  h in the formula ...



The answer is pretty close to the expected lifespan for a shrew. Now lets do it for an elephant.
An elephant's heart is huge, and beats only 25 times per minute. Filling in  25 for the variable  h and dividing gives an answer of  76.1 years. Again, the answer is typically correct for the lifetime of an elephant.

People are mammals. Do you think it will work? Let's see what happens.
The standard heartrate for a normal healthy human is about  60 beats per minute. When you fill this value in for  h, it doesn't seem to work. The answer turns out to be  31.7 years. What went wrong? Human lifetimes are around 70-80 years, aren't they?

Actually, the formula works just fine. But it will predict the lifespan of a mammal living in the wild. Humans don't live in the wild; most of us don't have to hunt for or gather our food every day, and we don't have any natural predators. Moreover, we have technology and medicines to keep us healthy and free from disease, and a plentiful diet.
However, we didn't always have these life-extending amenities. Before New Testament biblical times (2000 years ago) or so, humans' lifespans weren't much longer than 30-40 years. So the formula works for humans too, as long as they are living in a situation where they don't have modern technology. Primitive cultures still exist today where there is little or no contact with the outside world, and the average lifetime of their members isn't much more than 30-40 years.


Formulas exist for calculating all sorts of things, and you will learn about more of them as you continue your studies in mathematics, physics, chemistry, and biology.


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