Question

Evaluate the following using identities:
\[98 \times 99\]

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.

Step-By-Step answer:
We have to evaluate \[98 \times 99\] … (1)
We know both the numbers are near to 100 i.e. a multiple of 10
We can write both numbers as numbers subtracted from 100
\[ \Rightarrow 98 = 100 - 2\] and \[99 = 100 - 1\] … (2)
Substitute the values from equation (2) in equation (1)
\[ \Rightarrow 98 \times 99 = (100 - 2) \times (100 - 1)\]
We now open the multiplication bracket with respect to the difference in first bracket
\[ \Rightarrow 98 \times 99 = 100 \times (100 - 1) - 2 \times (100 - 1)\]
Now use the distributive property of multiplication over subtraction to open the terms in right hand side of the equation
\[ \Rightarrow 98 \times 99 = \left( {100 \times 100} \right) - \left( {1 \times 100} \right) - \left( {2 \times 100} \right) - \left( { - 2 \times 1} \right)\]
Multiply each of the products given inside the bracket
\[ \Rightarrow 98 \times 99 = 10000 - 100 - 200 - \left( { - 2} \right)\]
Write product of two negative signs as a positive sign
\[ \Rightarrow 98 \times 99 = 10000 - 100 - 200 + 2\]
Use BODMAS rule and add the terms first and then subtract the terms from that sum
\[ \Rightarrow 98 \times 99 = (10000 + 2) - (100 + 200)\]
\[ \Rightarrow 98 \times 99 = 10002 - (300)\]
Calculate the difference in right hand side of the equation
\[ \Rightarrow 98 \times 99 = 9702\]

\[\therefore \]The value of \[98 \times 99\] using the identities is 9702.

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.