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Equations Concept and Formulas



Let us write some equations for practice:
1$A$ exceeds $B$ by 100
$\Rightarrow A - B = 100$  (or) $ A = B + 100$
2$A$ is 50 more than $B$ 
$\Rightarrow A - B = 50$  (or) $ A = B + 50$
3$A$ is 75 less than $B$ 
$\Rightarrow B - A = 75$  (or) $ A + 75 = B$
4$A$ is twice that of  $B$ 
$\Rightarrow$ \(\dfrac{A}{B} = 2\) (or) $A = 2B$
5$A$ is one third of  $B$
$\Rightarrow$ \(A = \dfrac{1}{3}B\)
6$A$ is 25% less than $B$
$
\begin{array}{lrl}
\Rightarrow & A & = 75\% (B) \\[4px]
&A & =\left( {1 - \dfrac{25}{100}} \right)B\\[4px]
&A & =\left( {1 - \dfrac{1}{4}} \right)B\\[4px]
\end{array}
$
7$A$ is 20% more than $B$
$
\begin{array}{lrl}
\Rightarrow & A & = 120\% (B)\\[4px]
&A & =\left( {1 + \dfrac{20}{100}} \right)B\\[4px]
&A & =\left( {1 + \dfrac{1}{5}} \right)B\\[4px]
\end{array}
$

Let us learn how to take variables for the unknowns while solving equations:
1 $A$ exceeds $B$ by 100
$\Rightarrow B = x$ , $A = x + 100$
2 $A$ is 50 more than $B$
$\Rightarrow B = x$ , $A = x + 50$
3 $A$ is twice that of $B$
$\Rightarrow B = x, A = 2x$
4 Consecutive numbers:
$x$, $x +1$, $x + 2$, . . .
5 Consecutive even numbers:
$2n$, $2n +2$, $2n + 4$, . . .
6 Consecutive odd numbers:
$2n + 1$, $2n +3$, $2n + 5$, . . .

General Formulas:
1 \({a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right)\)
2 \({\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}\)
3 \({\left( {a - b} \right)^2} = {a^2} - 2ab + {b^2}\)
4 \({\left( {a + b} \right)^2} = {\left( {a - b} \right)^2} + 4ab\)

Formulas related to Series:
Formula: 1
Sum of the first $n$ natural numbers = $\dfrac{{n\left( {n + 1} \right)}}{2}$
Formula: 2
Sum of the first $n$ natural number squares = $\dfrac{{n\left( {n + 1} \right)\left( {2n + 1} \right)}}{6}$
Formula: 3
Sum of the first $n$ natural number cubes = ${\left( {\dfrac{{n\left( {n + 1} \right)}}{2}} \right)^2}$
Formula: 4
Sum of the first $n$ odd numbers = ${n^2}$
Formula: 5
Sum of the first $n$ even numbers = ${n\left( {n + 1} \right)}$
Formula: 6
Sum of the terms in an Arithmetic Progression (A.P) =
$\begin{array}{rl}
{S_n} & =\dfrac{n}{2}\left( {2a + \left( {n - 1} \right)d} \right) \\[5px]
&=\dfrac{n}{2}\left( {a + l} \right) \\[4px]
\end{array}
$
Here,
$a$ = First term
$l$ = Last term
$n$ = Number of terms
$d$ = Difference between consecutive terms
Formula: 7
Number of terms in an Arithmetic Progression (A.P) = $n = \dfrac{{l - a}}{d} + 1$