Say you wanted to install a wooden surface onto your bedroom floor that is in the shape of a rectangle. The length and width of your floor measure \(5\) metres by \(4\) metres long. Given these dimensions, is there a way for you to determine how many wooden panels you would need to cover your floor?
Well, since you have the length and width of your floor, you could simply use the area of a rectangle formula to identify the amount of material that you need. The area of a rectangle is given by the product of its length and width. In this case, you would need a total of \(20\) square metres worth of wooden panels to cover your floor. This is an example of a math formula.
In this article, you will look at math formulas and ways in which you can express them in order to use them to solve Number problems.
Definition of a Formula in Math
A formula in mathematics is a helpful tool used to determine solutions through a given expression. By knowing the general recipe needed to solve a particular problem, you would be able to replicate the same style of working if you encounter a similar situation. This process is done through various mathematical operations.
A math formula is a rule in the form of a statement expressed as symbols to help solve problems easily.
Formulas consist of different quantities connected together with the equal sign. They contain variables and sometimes constants. This means that if you have the values of certain variables in a formula you can find the value of the remaining variables.
Example of a Math Formula
To give us a better gist of what a math formula is, let's demonstrate it with an example.
Consider that a rectangle is a plot of land owned by Mr Parker. He wants to make it into a park where children can come around from the neighbourhood to play. He wants to know the exact measurement around this land particularly, the total of all the lengths and widths. This measurement is known as the perimeter.
One way to measure the perimeter of the rectangle above is to manually measure the whole plot of land. However, this can be done mathematically if some sides are known. If you know that the length is \(100\) metres and the width is \(55\) metres, you could simply use a math formula that gives you a general recipe that calculates the perimeter of a rectangle.
Carefully examining the properties of the rectangle, you will notice that the two opposite sides are equal. This means that if the length below is \(100\) metres, the length above will also be \(100\) metres. By this, you can write the formula for finding its perimeter. Let the letter \(l\) represent the length, and \(w\) represent the width:
\[ \text{Perimeter of rectangle } = l + l + w+w.\]
This can further be simplified by adding the like terms
\[ \text{Perimeter of rectangle } = 2l + 2w.\]
You can factor out \(2\) to get
\[ \text{Perimeter of rectangle } = 2(l + w).\]
Having this as the formula for finding the perimeter of a rectangle, you can go ahead to substitute numbers in it to see if it helps Mr Parker deal with his problem efficiently.
\[ \begin{align} \text{Perimeter of rectangle } &= 2(l + w) \\ &= 2(100 + 55) \\ &= 2(155) \\ &= 310 \, m. \end{align}\]
With the use of the formula, Mr Parker can simply know the perimeter of his plot of land without having to manually measure it all.
Across several mathematic fields, different formulas are being applied. To know where and how formulas can be applied, you must understand the problem you are dealing with and know which variables are significant.
How to Write a Math Formula
As mentioned earlier, formulas are in the form of Equations or identities. They consist of variables and sometimes constants. The fundamental task of writing formulas is knowing what to represent as a relevant variable.
For example, if you want to write a formula for the perimeter of a rectangle you should know that the length has a close relationship to the perimeter. You can take an example of how formulas are written.
Suppose you know that \(3\) cats eat as much food as one large dog. Write a formula to determine the volume of food you will need to feed \(27\) cats and \(10\) large dogs in terms of the Number of dogs you have.
Solution
It is a good idea decide what you are trying to do first! You are looking to find a formula for volume given number of cats and number of dogs. So let's give these things some variables.
- \(c\) is the number of cats
- \(d\) is the number of dogs
- \(V\) is the volume of food
You are asked to find a formula for the volume of food for \(27\) cats and \(10\) dogs. What do you know? You know that \(3\) cats eat as much as one large dog. So
\[3c = 1d.\]
You want the formula for \(27\) cats and \(10\) dogs, or in other words the formula for
\[ V = 27c + 10d,\]
but you want it in terms of dogs, not dogs and cats! What to do? Well, you haven't made use of the fact that \(3c = d\). You can do a little factoring to get
\[ \begin{align} V &=27c + 10d \\ &= 9(3c) + 10d, \end{align}\]
and then substitute in \(3c = d\) to get
\[ \begin{align} V &=9(3c) + 10d \\ &= 9d+ 10d \\ &= 19d, \end{align}\]
which is a formula for the amount of food you need to feed \(27\) cats and \(10\) large dogs in terms of the number of dogs you have.
Most Important Math Formulas
The term "most important" is a bit misleading, since it really depends on who you ask! However in this section, you will discuss some common formulas that are used across mathematics.
Areas of Shapes
The area of a shape is defined by a two-dimensional region bounded by the given shape.
Concept | Formula |
Area of rectangle | Area = length \(\times\) width |
Area of parallelogram | Area = base \(\times\) height |
Area of triangle | Area = \( \dfrac{1}{2} \times\) base \(\times\) height |
Area of circle | Area = \(\pi\times \) radius\(\times\) radius |
Volumes of Solids
The volume of a solid is the amount of three-dimensional space occupied by an object, container or closed surface.
Concept | Formula |
Cuboid | Volume = length \(\times\) base\(\times\) height |
Triangular prism | Volume =\( \dfrac{1}{2} \times\) length \(\times\) base\(\times\) height |
Cylinder | Volume = \(\pi\times \) radius\(\times\) radius\(\times\) height |
Compound Measure
Compound measures are expressions that contains more than one quantity.
Concept | Formula |
Speed | \( \text{Speed } = \dfrac{ \text{ Distance}}{\text{time}}\) |
Density | \( \text{Density } = \dfrac{ \text{ Mass}}{\text{ Volume}}\) |
Pressure | \( \text{Pressure } = \dfrac{ \text{ Force}}{\text{Area}}\) |
Algebra of Rewriting Formulas
It is useful to know how to rewrite formulas as you may be given the area of a rectangle and be asked to find its length. When you rewrite a formula the aim is to create an equation that is equivalent to the formula but with the missing variable by itself.
The fundamental rule used to do this is the golden rule of manipulating Equations. It says that do unto the side of an equation what you do to the other. This means that if the manipulation requires that you add values to one side of the equation, do the same addition on the left side of the equation. Here is an example.
If the values for mass and density were given, what will be the formula for volume?
Solution
A formula where all the quantities mentioned are present is the formula for density.
\[\text{Density } = \dfrac{ \text{ Mass}}{\text{ Volume}}\]
To find the formula for volume, you will have to make volume the subject of the equation. This will mean that any form of manipulation on any side of the equation will require it to be replicated on the other side. To do this, you will first need to multiply both sides of the equation by volume,
\[\text{Density }\times \text{ Volume } = \dfrac{ \text{ Mass}}{\text{ Volume}} \times \text{ Volume}\]
and then cancel to get
\[\text{Density }\times \text{ Volume } = \text{ Mass}.\]
Now you can divide both sides by Density
\[\dfrac{\text{Density }\times \text{ Volume }}{\text{Density } } = \dfrac{\text{ Mass}}{\text{Density} }\]
and cancel again to get
\[\text{ Volume } = \dfrac{\text{ Mass}}{\text{Density} }.\]
Let us look at another example.
Find the length of a rectangle given the area to be \(42\, cm^2\) and its width to be \(6\, cm\).
Solution
First of all, you can write the formula for Finding the Area of a rectangle down:
\[A = lw.\]
To find the length, you will have to make it the subject of the equation. This means that you have a few manipulations to perform. What you do on one side will require that it be done on the other. To isolate length to be alone on one side of the equation, you will have to divide both sides of the equation by width
\[ \frac{A}{w} = \frac{lw}{w}\]
and then cancel to get
\[ l = \frac{A}{w}.\]
You now have a formula for finding length in this scenario. You can go ahead to find the solution to the problem by substituting into the formula:
\[ \begin{align} l &= \frac{A}{w}\\ &= \frac{42}{6} \\ &= 7.\end{align}\]
Don't forget the units! The length is \(7\, cm\).
Substitution in Formulas
Substitution into formulas is the process of replacing a variable with its value into a formula. In this section, the use of formulas becomes extremely evident. Given the right values of variables, the unknown variables can be found.
The whole process of substituting into formulas is replacing the letter (variables) with their values given. You will take lots of examples to see how the different types of possible situations can be approached.
Find \(z\) when \(x=7\) in the given formula
\[z = x+2.\]
Solution
All you have to do here is to replace \(x\) in the formula with \(7\) since the problem says \(x\) is the same as \(7\).
\[ \begin{align} z &= x+2 \\ &= 7 + 2 \\ &= 9.\end{align}\]
Here is another example for you!
Find \(l\) when \(m=5\) in the given formula
\[ l = 7m.\]
Solution
Here you will replace the letter \(m\) with the number \(5\) as given in the problem, then you can go ahead to find \(l\). So
\[ l = 7m.\]
The relationship between \(7\) and \(m\) here is multiplication. This whole formula can fundamentally be written as
\[l = 7 \cdot m,\]
or
\[l = 7(m).\]
Substituting \(5\) in for \(m\), you get
\[ \begin{align} l &= 7(5) \\ &= 35.\end{align}\]
Math Formula - Key takeaways
- A math formula is a rule in a statement form expressed as symbols to help solve problems easily.
- Formulas consist of different quantities connected together with the equal sign.
- The fundamental rule used to rewrite formulas is the golden rule of manipulating equations which says that do unto the side of an equation what you do to the other.
- Substitution into formulas is the process of replacing a variable with its value into a formula.
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