Question

Evaluate the following products without multiplying directly:
(i) \[103 \times 107\]
(ii) \[95 \times 96\]
(iii) \[104 \times 96\]

Answer 1

Hint:According to the given question, we will convert the numbers in the equations in the form of \[\left( {100 - x} \right)\] or \[\left( {100 + x} \right)\] .Then we will use different algebraic identities to get the desired result.
Formula used:
Here, we use the algebraic identities that is \[\left( {x + a} \right)\left( {x + b} \right) = {x^2} + \left( {a + b} \right)x + ab\] and \[\left( {a + b} \right)\left( {a - b} \right) = \left( {{a^2} - {b^2}} \right)\] .

Complete step by step solution:
(i) \[103 \times 107\]
Firstly we will rewrite both numbers in the form of \[\left( {100 - x} \right)\] or \[\left(
{100 + x} \right)\] depending on what is the best approach for that number.
\[ \Rightarrow \left( {100 + 3} \right)\left( {100 + 7} \right)\]
Here, we will use the algebraic identity \[\left( {x + a} \right)\left( {x + b} \right) = {x^2} + \left( {a + b} \right)x + ab\]
Putting \[a = 3\] , \[b = 7\] and \[x = 100\] .
On substituting the values in the identity we get,
\[ \Rightarrow \left( {100 + 3} \right)\left( {100 + 7} \right) = {\left( {100} \right)^2} + \left( {3 + 7} \right)100 + 3 \times 7\]
After opening the squares and simplifying the above equation of right hand side we get,
\[ \Rightarrow \left( {100 + 3} \right)\left( {100 + 7} \right) = 10000 + 1000 + 21\]
On adding the right hand side we get,
\[ \Rightarrow \left( {100 + 3} \right)\left( {100 + 7} \right) = 11021\]
Hence, \[103 \times 107 = 11021\]
(ii) \[95 \times 96\]
Firstly we will rewrite both numbers in the form of \[\left( {100 - x} \right)\] or \[\left( {100 + x}
\right)\] depending on what is the best approach for that number.
\[ \Rightarrow \left( {100 - 5} \right)\left( {100 - 4} \right)\]
Here, we will use the algebraic identity \[\left( {x + a} \right)\left( {x + b} \right) = {x^2} + \left( {a + b} \right)x + ab\]
Putting \[a = - 5\] , \[b = - 4\] and \[x = 100\] .
On substituting the values in the identity we get,
\[ \Rightarrow \left( {100 - 5} \right)\left( {100 - 4} \right) = {\left( {100} \right)^2} + \left( { - 5 - 4}
\right)100 + \left( { - 5 \times - 4} \right)\]
After opening the squares and simplifying the above equation of right hand side we get,
\[ \Rightarrow \left( {100 - 5} \right)\left( {100 - 4} \right) = 10000 - 900 + 20\]
After solving the right hand side we get,
\[ \Rightarrow \left( {100 - 5} \right)\left( {100 - 4} \right) = 9120\]
Hence, \[95 \times 96 = 9120\]
(iii) \[104 \times 96\]
Firstly we will rewrite both numbers in the form of \[\left( {100 - x} \right)\] or \[\left( {100 + x} \right)\] depending on what is the best approach for that number.
\[ \Rightarrow \left( {100 + 4} \right)\left( {100 - 4} \right)\]
Here, we will use the algebraic identity \[\left( {a + b} \right)\left( {a - b} \right) = \left( {{a^2} - {b^2}} \right)\]
Putting \[a = 100\] and \[b = 4\] .
On substituting the values in the identity we get,
\[ \Rightarrow \left( {100 + 4} \right)\left( {100 - 4} \right) = {\left( {100} \right)^2} - {\left( 4
\right)^2}\]
Now we will open the squares and simplify on right hand side we get,
\[ \Rightarrow \left( {100 + 4} \right)\left( {100 - 4} \right) = 10000 - 16\]
After solving the right hand side we get,
\[ \Rightarrow \left( {100 + 4} \right)\left( {100 - 4} \right) = 9984\]
Hence, \[104 \times 96 = 9984\]

Note: To solve these types of questions, you must remember the algebraic identities and convert the equations according to the requirement of the identity. So, carefully observe the value of a and b while substituting in the formula.