Framing of a Formula: A formula denotes the relationship between two or more variables. Formulas are usually in the form of an algebraic equation. In this article, we will discuss in detail the concepts of formula and framing the formula. Once a formula is framed, it is easy to find the relationship between the variables. Also, we will discuss the subject of formula and substitution in a formula.

We have learned the framing of an algebraic equation and forming linear equations. We will frame various formulae and express the relation between the unknown variables and learn more about it. 

Formula Definition

A formula is a relation between two or more quantities. It can be an equation that expresses an association between two or more quantities using literals and symbols. Letters are used to symbolise words. In many cases, letters that represent words are standardised so that the formula will be written with the same letters in a different context. When we look into the formula, it must be stating how a variable is related to another variable.

Let us look at the formula for the perimeter of a square; if \(a\) is the side and \(p\) is the perimeter, then the relationship between perimeter and sides of a square is given by \(p=4 a\) If we know the perimeter, we can easily find the sides of the square. And if the sides are known, then the perimeter can be found out.

Now let us recall some of the formulae that we already know:

1. The perimeter \(p\) of a rectangle is equal to two times the sum of its length \(l\) and breadth \(b\). The formula to represent the relation between perimeter, length, and breadth of a rectangle is \(p=2(l+b)\).
2. If \(V\) is the volume of a cube and \(a\) is its side, then the volume of a cube is expressed as \(V=a^{3}\).
3. The force \(F\) of an object is the product of the mass \(m\) of the object, and acceleration \(a\) of the object is expressed as \(F=m a\).
4. Suppose the sum of any two numbers is \(25\). Let us say the two numbers are \(x\) and \(y\), then \(x+y=25\) is the formula that expresses the statement.

Framing of an Equation

An unknown number is represented by a letter such as \(a, b, c, x, y, z\).

Framing of formula and framing of an equation are similar. The framing of the formula is standardised for certain situations. In contrast framing of the equation is not standardised. It is situational. Every situation has a different condition and is not related to anything in particular. For example, a statement says the cost of \(10\) oranges and \(12\) apples is \(250\). Here the number of oranges and apples bought is not fixed nor the cost of the fruits. Thus, we cannot standardise the equations like formulae.

Let’s frame the equation for the above example,

Assume the cost of oranges \(₹x\) and the cost of apple \(₹y\).

The required equation will be \(10 x+12 y=250\)

Framing of a Formula

Steps to be Followed in Framing of a Formula

Step 1: Choose the variables with suitable letters for the quantities we are handling. There are few fixed symbols used to denote some variables. In a different context, the same symbols can also be used to denote different variables. For example, \(P\) is used to denote the principal in the amount formula, and it is also used to denote the perpendicular side in the Pythagoras theorem formula.

Step 2: Next, use the conditions relevant to the context to frame the formula.

In framing the formula, we frame the formula by translating mathematical statements using literals and symbols. Some examples are given below:

1. Mathematical statement: Amount \((A)\) is the sum of the principal \((P)\) and interest \((I)\).

Formula: \(A=P+I\)

2. Mathematical statement: The area of the rectangle \((A)\) is equal to the product of the length \((L)\) and breadth \((B)\) of the rectangle.

Formula: \(A=L \times B\)

3. Mathematical statement: The sum of the angles \((\angle x, \angle y, \angle z)\) of a triangle is equal to two right angles \(\left(2 \times 90^{\circ}=180^{\circ}\right)\).

Formula: \(\angle x+\angle y+\angle z=180^{\circ}\)

4. Mathematical statement: In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Here \(H\) is used to denote the hypotenuse and \(P, B\) for denoting the other two sides.

Formula: \(H^{2}=P^{2}+B^{2}\)

Subject of a Formula

When one variable (quantity) is expressed in terms of another variable (quantity), the quantity expressed is the subject of the formula. Usually, we write the subject of the formula on the left-hand side of the equal sign. At the same time, other variables and constants are written on the right-hand side of the equal sign when we frame a formula.

Example: In the formula, \(I=\frac{P T R}{100}\), make \(P\) as the subject of the formula.

Here, \(I\) (simple interest) is expressed in terms of \(P\) (principal), \(R\) (rate of interest), and \(T\) (time). In this case, \(I\) is the subject of this formula.

Again, if we write \(P=I \times \frac{100}{R T}\)

Here, \(P\) (principal) is expressed in terms of \(I\) (simple interest), \(R\) (rate of interest), and \(T\) (time). So \(P\) becomes the subject of this formula. We can change the subject of a formula by using such transpositions.

Substitution in a Formula

When the formula’s unknown quantity (variable) is assigned a certain value, the given formula or expression gets a particular value. This process is known as substitution. 

When the values of other variables are known, the following steps are followed to find the value of an unknown variable in a formula.

1. Write the formula such that the unknown quantity becomes the subject of the formula. You can even substitute the values directly and then transpose all the known quantities on one side of the equation and the unknown variable on the other side.
2. Substitute the given value for which you want to find the value of an unknown quantity. Thus, the subject of the formula can be found.

Example: A rectangle has a length of \(2 \mathrm{~cm}\), and its area is \(8 \mathrm{~cm}\). Find the breadth of the rectangle.
Given the length, \(l=2 \mathrm{~cm}\), area, \(A=8 \mathrm{~cm}\)
We know that area of a rectangle, \(A=l \times b\)
Since breadth is to be found out and we make breadth as the subject:
\(\Rightarrow b=\frac{A}{l}\)
We substitute the given values to find the breadth of the rectangle:
\(\Rightarrow b=\frac{8}{2}=4 \mathrm{~cm}\)
Thus, the breadth of the rectangle is \(4 \mathrm{~cm}\).

Solved Examples – Framing of a Formula

Q.1. In the formula, \(3 C+150=7 F\), make \(C\) as the subject of the formula.
Ans: Given, \(3 C+150=7 F\)
Transpose all the constants and coefficients on the other side of the formula:
\(\Rightarrow 3 C=7 F-150\)
\(\Rightarrow C=\frac{7 F-150}{3}\)
\(\Rightarrow C=\frac{7 F}{3}-50\)
The required formula with \(C\) as the subject is \(C=\frac{7 F}{3}-50\).

Q.2. Avni’s father’s age is \(4\) years more than \(3\) times Avni’s age. Father age is \(39\) years. Express the following as an equation.
Ans: Given that Avni’s father’s age is \(4\) years more than \(3\) times Avni’s age. Father’s age is \(39\) years.
Let Avni’s age be \(x\) years.
Three times her age \(=3 x\)
Father’s age \(=3 x+4\) Given father’s age \(=39\) years
Equating the father’s age in terms of Avni’s age and given father’s age:
\(3 x+4=39\)
Thus, the final equation is \(3 x+4=39\).

Q.3. A rectangular box is of height \(h \mathrm{~cm}\). Its length is \(7\) times its height, and the breadth is \(5 \mathrm{~cm}\) less than the length. Express the relation between length, breadth, and height.
Ans: Given, a rectangular box is of height \(h \mathrm{~cm}\).
Let, height \(=h\), length \(=l\), breadth \(=b\)
The length of the rectangle is \(7\) times the height.
The length of the rectangle \(=7 h\)
The breadth of the rectangle is \(5 \mathrm{~cm}\) less than the length.
The breadth of rectangle \(=l-5\) As \(l=7 h, b=7 h-5\)
Thus, height \(h\), length \(l=7 h\), and breadth \(b=7 h-5\).

Q.4. Frame a formula for the statement: Angle \(G\) is a supplement of angle \(H\).
Ans: If angle \(G\) and \(H\) are supplementary, then their sum is \(180^{\circ}\).
i.e., \(G+H=180^{\circ}\)
\(\therefore G=180^{\circ}-H\)
This is the required formula.

Q.5. The average \((A)\) of two numbers \(p\) and \(q\) is \(\frac{p+q}{2}\), find \(p\) when \(A=25, q=20\).
Ans: Here, \(A=\frac{p+q}{2}\)
Making \(p\) as the subject of the formula:
\(\Rightarrow 2 A=p+q\)
\(\Rightarrow p=2 A-q\)
Substituting the given values:
\(p=2 \times 25-20\)
\(\Rightarrow p=50-20\)
\(\Rightarrow p=30\)

Summary

In this article, we studied the framing of a formula. A formula is an equation that expresses the relationship between two or more quantities using literals and symbols. We learned the definition of formula and examples of some standard formulae. We discussed the framing of a linear equation and how it is different from a formula.

The subject of the formula is expressing the formula in terms of an unknown quantity. And the substitution of the formula is substituting the known quantities to find the unknown one. Many examples have been solved to make the students understand the concept of framing the formula, thus making the article very helpful.

Maths Formulas for Class 12

Frequently Asked Questions (FAQs)

Q.1. What are the steps in the framing of the formula?
Ans: Choose the variables with suitable letters for the quantities we are handling. There are few fixed symbols used to denote some variables. Next, use the conditions relevant to the context to frame the formula.

Q.2. How do you frame a linear equation?
Ans: Steps involved in the framing of linear equation in one variable from the given word problem are as follows
Step I: Read the given problem statement carefully and note down the given and required quantities separately.
Step II: The unknown quantities may be denoted as \(^{\prime} x^{\prime},{ }^{\prime} y^{\prime},{ }^{\prime} z^{\prime}\), etc.
Step III: Form the equation connecting the variables.

Q.3. What is a mathematical formula? 
Ans: A mathematical formula is the problem statement expressed in unknown values (variables) and constants. The formula contains an equality sign similar to the equation. 

Q.4. What does it mean to establish a formula? 
Ans: In a given context, the relationship between variables expressed by equality (or inequality) is called a formula. This equality (or inequality) between the variables is a formula for the variables. In the case of equality, we get an equation.

Q.5. How do you frame a linear equation in two variables?
Ans: When there are two unknown values and are directly proportional to one another, we can frame linear equations in two variables.
Steps involved in the framing of linear equation in two variables from the given word problem are as follows
Step I: Read the given problem statement carefully and note down the given and required quantities separately.
Step II: The unknown quantities may be denoted as \(^{\prime} x^{\prime},{ }^{\prime} y^{\prime},{ }^{\prime} z^{\prime}\), etc.
Step III: Form the equation connecting the variables.

We hope this detailed article on the framing of a formula helped you in your studies. If you have any doubts, queries or suggestions regarding this article, feel to ask us in the comment section and we will be more than happy to assist you. Happy learning!