Question

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

Answer 1

Hint: Write both the given numbers in a simpler way i.e. in a way where we add or subtract numbers from nearest multiple of 10. We use the distributive property of multiplication over subtraction \[a \times (b - c) = a \times b - a \times c\] to open the term formed.
* Distributive property is the property that helps to break the multiplication of large numbers into the sum of multiplication of smaller numbers. We distribute the large number into two or more parts and then multiply the number outside the bracket to each of the distributed parts.

Complete step-by-step answer:
We have to evaluate \[9 \times 99\] … (1)
We know the second number is near to 100 i.e. a multiple of 10
We can write the second number as number subtracted from 100
\[ \Rightarrow 99 = 100 - 1\] … (2)
Substitute the values from equation (2) in equation (1)
\[ \Rightarrow 9 \times 99 = 9 \times (100 - 1)\]
Now use the distributive property of multiplication over subtraction to open the terms in right hand side of the equation
\[ \Rightarrow 9 \times 99 = \left( {9 \times 100} \right) - \left( {9 \times 1} \right)\]
Multiply each of the products given inside the bracket
\[ \Rightarrow 9 \times 99 = 900 - 9\]
Use BODMAS rule and calculate the difference in right hand side of the equation
\[ \Rightarrow 9 \times 99 = 891\]

\[\therefore \]The value of \[9 \times 99\]using distributive property is 891.

Note:
Many students make mistake of calculating the answer wrong in the last step where they add and subtract values in given order, keep in mind we always apply BODMAS rule and proceed in that manner for any arithmetic operation i.e. we solve any equation in stepwise manner of bracket, order, Division, multiplication, addition and then subtraction.
Alternate method:
Many students break the product in a way that both numbers are written as differences.
\[ \Rightarrow 9 \times 99 = (10 - 1) \times (100 - 1)\]
Now apply distributive property
\[ \Rightarrow 9 \times 99 = 10 \times (100 - 1) - 1 \times (100 - 1)\]
Multiply the values on right hand side of the equation one by one
\[ \Rightarrow 9 \times 99 = \left( {10 \times 100} \right) - \left( {10 \times 1} \right) - \left( {1 \times 100} \right) - \left( {1 \times - 1} \right)\]
Calculate product in each bracket
\[ \Rightarrow 9 \times 99 = 1000 - 10 - 100 + 1\]
Use BODMAS rule and add positive terms
\[ \Rightarrow 9 \times 99 = 1001 - 110\]
Subtract the terms on right hand side of the equation
\[ \Rightarrow 9 \times 99 = 891\]
\[\therefore \]The value of \[9 \times 99\] using distributive property is 891