Distributive Law And Taking Common

Multiplication is distributive over addition. That is we can say that:

$a(b + c) = ab + ac$

Here, when we open the bracket, the multiplication with $a$ distributes over all the terms inside the bracket.

Any operation which is distributive over addition, is also distributive over subtraction.

Therefore, we can say that:

$p(a - b - c) = pa - pb -pc$

This lead to the very important concept of taking common. Using the same example from above we can say that:

$pa - pb - pc = p(a - b - c)$

In other words we are taking $p$ common from the 3 terms by introducing a bracket.

As we will see, this has very important use in making our calculation easy.

Let us take an example to understand this.

We will simplify the expression $\dfrac{22\times14 - 22\times11}{44\times17 - 44\times15}$

If we do this it will involve performing four large multiplications, like $22 \times 14,\ 22 \times 11,\ 44 \times 17\ and\ 44 \times 15$. Then again we will need to calculate the difference of their results.

That would take a lot of time and the chances of committing mistakes are very high.

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But, if we solve this using the technique of taking common, then the problem becomes so simple that it can be done mentally. Let's see how.

$\dfrac{22 \times 14 - 22 \times 11}{44 \times 17 - 44 \times 15} \phantom{0000} \Leftarrow \hbox{take 22 common in the numerator and 44 in the denominator}$

$= \dfrac{22(14 - 11)}{44(17 - 15)} \phantom{0000} \Leftarrow \hbox{cancel out 22 in both the numerator and denominator}$

$= \dfrac{3}{2 (2)}$

$= \dfrac{3}{4}$

Therefore, the answer is $\dfrac{3}{4}$