Framing Formulas - MathsTips.com

Definition:

A formula is a relation between certain quantities.

For example, if $a=$ side of a square then its perimeter $p$ can be expressed by the following formula which we already know, i.e., $p=4a$ . This formula shows the relationship between the side of a square and the perimeter of a square. If we know the values of all the quantities except one in a formula, we can find the unknown quantity.

Now let us have a look at certain formulas that we already know:

The perimeter $p$ of a rectangle is twice the sum of its length $l$ and breadth $b$ . This can be represented by the formula, $p=2(l+b)$ .
The volume $V$ of a cube is the cube of its side $a$ . This is expressed as $V=a^3$
The force $F$ of an object is the product of the mass $m$ of the object and acceleration $a$ of the object. This is expressed as $F=ma$ .
Suppose the sum of two numbers is 85. The required formula is $x+y=85$ , where $x$ and $y$ are the unknown quantities.

Framing a formula:

Step 1: We first need to choose variables such as $a, p, A, P$ etc. for the quantities we are dealing with. There are certain symbols traditionally used to denote certain variables. Also, the same symbol can be used to denote different variables in different contexts. For example, $P$ is used to denote principal in Arithmetic and it is also used to denote power in Physics.

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

Subject of a formula:

When one quantity is expressed in terms of other quantities, the quantity thus expressed is called the subject of the formula. The subject of a formula is written on the left-hand side of the equality sign, while the other variables and constants are usually written on the right-hand side of the equality sign in a formula.

For example, $I=\dfrac{PRT}{100}$ . 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.

Conversely, if we write $P=I \times \dfrac{100}{RT}$ . Here, P(principal) is expressed in terms of I(simple interest), R(rate of interest) and T(time). In this case, P is the subject of this formula. We can change the subject of a formula by using such transformations.

Substitution in a formula:

When the variables in an algebraic expression are assigned certain values, the expression gets a particular value. This process is called substitution.

To find the value of an unknown variable from a formula when the values of the other variables are known, we need to follow the following steps.

1) Make the unknown quantity the subject of the formula.

2) Substitute the values of the known quantity in the formula and find the value of the subject.

Examples:

Example 1: Frame a formula for each of the following statements.

i) Pooja’s age is 4 years more than thrice that of Riya.

ii) A distance d is travelled by a body moving with a speed s in time t.

Solution:

i) Let Pooja’s age be $x$ years and Riya’s age be $y$ years.

Then, according to the question, $x=3y+4$ .

This is the required expression.

ii) We know that $distance= speed \times time$

So, the required expression is $d=st$ .

Example 2: In the formula $9C+160=5F$ , make $C$ the subject.

Solution:

$9C+160=5F$

$\Rightarrow 9C=5F-160$

$\Rightarrow C=\dfrac{5F-160}{9}$

$\therefore$ $C=\dfrac{5F}{9}-32$ .

This is the required formula.

Example 3: Frame a formula for the statement: Angle $P$ is a supplement of angle $Q$ .

Solution:

If angle $P$ and $Q$ are supplementary, then their sum is $180 \textdegree$

i.e. $P+Q=180 \textdegree$

$\therefore$ $P=180 \textdegree -Q$ .

This is the required formula.

Example 4: The average (A) of two numbers $m$ and $n$ is $\dfrac{m+n}{2}$ , find $m$ when $A =25, n=20$ .

Solution:

Here, $A=\dfrac{m+n}{2}$

$\Rightarrow 2A=m+n$

$\Rightarrow m=2A-n$

$\Rightarrow m=2 \times 25-20$

$\Rightarrow m=50-20$

$\therefore m=30$ .

Exercise:

Form a formula for each of the following statements:
1. The volume (V) of a cuboid is the product of its length (l), breadth (b) and height (h)
2. The area (A) of a triangle is half the product of its base (b) and height (h).
3. Angle A is a complement of angle B.
4. The average (A) of three numbers p,q and r is one-third of their sum.
5. The area (A) of a square is equal to the square of the length (a) of its side.
6. The diameter (d) of a circle is twice its radius (r).
7. Maya is thrice as old as Reeta.
8. A number (x) multiplied by itself is equal to another number (y).
9. The sum of a number (p) and 50 is five times another number (q).
10. A number (a) exceeds another number (b) by 15.
Make the given literal the subject of the formula.
1. $A=P+I, I$
2. $s=ut, u$
3. $Z=\dfrac{x+y}{2}, x$
4. $C=\dfrac{5}{9} (F-32), F$
5. $v=u+at, u$
6. $v^2=u^2+2as, s$
7. $x=a+(n-1)d, d$
8. $W=F \times S, S$
Find the values of the following expressions.
1. $13x^2-15xy$ , where $x=-2, y=-5$
2. $12ab+3a^2+5$ , where $a=2, b=-8$
3. $m^2-mn+n^2$ , where $m=-1, n=3$
4. $14xy-6x^2+5y^2-7x+2$ , where $x=-2, y=0$
Simplify $(2x-y)(x+y)-(2x+y)(x-y)$ and find its value, where $x=5, y=4$ .
Solve the following:
1. Use the formula $P=2(l+b)$ to find $l$ , where $P=32m, b=6m$ .
2. If $A=\pi r^2$ , find $A$ , where $r=7cm$ and $\pi=\dfrac{22}{7}$
3. If ,
  1. Find $v$ , when $u=40, a=2, t=4$
  2. Find $u$ , when $v=60, a=5, t=6$
4. If $C=\dfrac{5}{9} (F-32)$ , find $F$ , when $C=30$
5. If $a=1, b=-2, c=3$ , find the value of $a^2+4b-\dfrac{c}{a}-2b^2+4c$

Definition:

Framing a formula:

Subject of a formula:

Substitution in a formula:

Examples:

Exercise:

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