Question

How do you use the distributive property to find a product mentally?

Answer 1

Hint: At first we break down the larger of the two multiplicands into a sum of very simple numbers, such that the product gets drastically simpler. Thereafter, we apply the distributive property and perform the additions and multiplications mentally.

Complete step-by-step solution:
There are various properties which are associated with mathematical operations. Primarily, these are the commutative property, the associative property and the distributive property. The commutative property simply means that if we change the sequence of operation between two numbers, then the outcome does not change. For example, $3\times 4$ and $4\times 3$ both give the same result which is $12$ . Thus, multiplication and addition are commutative operations. Associative property means that we can perform the operation regardless of how the numbers are grouped. For example, $2+3+5$ can be done either by $2+\left( 3+5 \right)$ or $\left( 2+3 \right)+5$ as both give the same result $10$ . Distributive property means that if a term is multiplied with a bracket containing some terms, then this is equivalent to addition of all the terms multiplied with the common term. For example, $3\times \left( 2+3 \right)$ is equal to $3\times 2+3\times 3$ as both give the same answer $15$ .
This was all about pen and paper. To apply the distributive property mentally, no rocket science is required. Instead of pen and paper, we do it in mind. Suppose if a small number is to be multiplied with a large number, then the large number can be broken down into sum of small,, easily multipliable numbers. For example, we have to multiply $4$ with $1256$ . Solving this using pen and paper does not require mental maths, but if we have to do it mentally, distributive property is necessary. Therefore, we write $1256$ as $1000+200+50+6$ . We can see that the product of $4$ with all these numbers is quite easy. So, the multiplication goes as $4\times 1256=4\times \left( 1000+200+50+6 \right)$ . We now apply the distributive property as
$\Rightarrow 4\times \left( 1000+200+50+6 \right)=4\times 1000+4\times 200+4\times 50+4\times 6$
All of the products being simple, we then proceed as,
$\Rightarrow 4\times 1000+4\times 200+4\times 50+4\times 6=4000+800+200+24$
Addition also being simple, we do it mentally as
$\Rightarrow 4000+800+200+24=5024$
Thus, we have seen how to apply distributive property to solve a product mentally.

Note: As distributive property is to be applied mentally, we do not break down the number into such numbers, whose multiplication is tedious and doing them mentally will be prone to errors. Thus, we should generally break the larger number into decreasing multiples of $10$ such as the above case, which makes the multiplication and final addition easier. In case of very large numbers. We should not prefer the mental method.