Question

Using identities, evaluate \[1.05 \times 9.5\].

Answer 1

Hint: We shall first begin by rewriting \[9.5\] as a multiple of 10. Then we will write \[0.95\] and \[1.05\] as the sum and difference of the same set of numbers so that we will get an expression in the form of some identity. Then, we will apply an appropriate identity to simplify the expression and get the required value.

Formula used:
 We will use the formula \[\left( {a + b} \right)\left( {a - b} \right) = {a^2} - {b^2}\].

Complete step-by-step answer:
Let us first rewrite \[9.5\] as
\[9.5 = 0.95 \times 10\]
Therefore, the given equation becomes
\[1.05 \times 9.5 = 1.05 \times 0.95 \times 10\]
We have expressed \[9.5\] as a product of \[0.95\] and \[10\] because we want to have \[1.05\] and \[0.95\] to be written as a sum and difference of the same set of numbers respectively.
We can write \[1.05{\rm{\, as\, }}1 + 0.05\] and \[0.95{\rm{\, as\, }}1 - 0.05\]. That is, we are writing \[1.05\] as a sum of \[1{\rm{\, and\, }}0.05\] and \[0.95\] as a difference of \[1{\rm{ \,and\, }}0.05\]. We do this so that we can apply the identity easily. So,
\[ \Rightarrow 1.05 \times 9.5 = \left( {1 + 0.05} \right) \times \left( {1 - 0.05} \right) \times 10\]
We will now apply the identity \[\left( {a + b} \right)\left( {a - b} \right) = {a^2} - {b^2}\]. Thus, we get,
\[ \Rightarrow 1.05 \times 9.5 = \left( {{1^2} - {{\left( {0.05} \right)}^2}} \right) \times 10\]
We know that \[{1^2} = 1{\rm{\, and\, }}{\left( {0.05} \right)^2} = 0.0025\].
Applying the exponent, we get
\[ \Rightarrow 1.05 \times 9.5 = \left( {1 - 0.0025} \right) \times 10\]
Subtracting the terms, we get
\[ \Rightarrow 1.05 \times 9.5 = 0.9975 \times 10\]
Multiplying the terms, we get
\[ \Rightarrow 1.05 \times 9.5 = 9.975\]
Hence, we get \[1.05 \times 9.5 = 9.975\].

Note: While applying the identity, care must be taken to resolve the numbers in such a form that they are expressed as a sum and difference of the same set of numbers. We can also solve the problem without using identities as follows:
Let us write \[1.05\] as \[\dfrac{{105}}{{100}}\]. We are dividing \[105{\rm{ \,by\, }}100\] because the number of digits after the decimal point is 2. Similarly, we can write \[9.5{\rm{ \,as\, }}\dfrac{{95}}{{10}}\], since there is only one digit after the decimal point in \[9.5\] and so we divide \[95{\rm{ \,by\, 10}}\].
Now we can directly multiply.
\[1.05 \times 9.5 = \dfrac{{105}}{{100}} \times \dfrac{{95}}{{10}}\]
Multiplying \[105{\rm{ \,by\, }}95\] in the numerator and \[100{\rm{ \,by\, }}10\] in the denominator, we get
\[ \Rightarrow 1.05 \times 9.5 = \dfrac{{9975}}{{1000}}\]
Since \[9975\] is divided by \[1000\], we will place the decimal point 3 places from the right. Therefore,
\[ \Rightarrow 1.05 \times 9.5 = 9.975\]
This is the required answer.