Question

How do you use the distributive property to find \[9 \times 99\] ?

Answer 1

Hint: In order to multiply the given terms using distributive property, we break the two digit number into its two simple whole number constituent parts and then multiply the single digit number with each one of them. On doing some simplification we get the required answer.

Complete step-by-step solution:
In this solution, we need to multiply $9 \times 99$ using the distributive property.
We need to break each number into its add ents and then multiply the other with each of them.
As we know that $99$ can also be written as $90 + 9$
Thus we have: $9 \times \left( {90 + 9} \right)$
We multiply using the distributive property:
$ \Rightarrow \left( {9 \times 90} \right) + \left( {9 \times 9} \right)$
We further multiply to get:
$ \Rightarrow \left( {810} \right) + 81$
On simplifying we get:
$ \Rightarrow 891$
Thus, \[9 \times 99 = 891\]

The required solution for the given problem is 891.

Additional Information: Distribution means to divide something in parts. It can either be equal or unequal. We all are familiar with the traditional forms of multiplying, but multiplying using the distributive property gives us a clearer idea of how we generate the traditional multiplying process. According to the distributive property, multiplying the sum of two or more addends by a number will give the same result as multiplying each addend individually by the number and then adding the products together.
Even though division is the inverse or opposite of multiplication, the distributive property only holds true in case of division when the dividends can be broken down.

Note: An alternate way of doing this sum: We also know that $9$ can be written as $\left( {10 - 1} \right)$
Also, $99$ can also be written as $90 + 9$
Thus accordingly, our expression will become: $\left( {10 - 1} \right) \times \left( {90 + 9} \right)$
Using distributive property, we multiply each term with each other:
$ \Rightarrow \left( {10 \times 90} \right) + \left( {9 \times 10} \right) - \left( {1 \times 90} \right) - \left( {1 \times 9} \right)$
Thus, on multiplying the value, we get:
$ \Rightarrow 900 + 90 - 90 - 9$
On cancel the same term and subtract the remaining we get,
$ \Rightarrow 891$