Question

A two digit number is such that the product of the digits is $14$ . When $45$ is added to the number, then the digits interchange their places. Find the number.
a). $72$
b). $27$
c). $37$
d). $14$

Answer 1

Hint: In this question we have to find a number. So we will assume $x$ and $y$ be the number at tens and unit places.
We will put this value in the form of a number and we have been also given the product of the digits: $14$.
 Then we will use the algebraic identity formula i.e.
 ${(x + y)^2} = {(x - y)^2} + 4xy$ to solve the equation and get the required answer.

Complete step-by-step solution:
Let us assume the digit in the units place be $y$ and the number to be at tens place be $x$ .
So we can say that the form of the number is
$10x + y$ .
The product of the digits can be written as
 $xy = 14$ .
It is given in the question that if $45$ is added to the number, then the digits interchange their places.
We can write the equation as
$(10x + y) + 45 = 10y + x$
We will now arrange the similar terms together:
$10x - x + y - 10y + 45 = 0$
On simplifying we have
 $9x - 9y + 45 = 0$
We can see that there is a common factor in all the three terms i.e.
 $9$
So we will take the common factor out, and it gives us
 $x - y + 5 = 0$, let this be equation one.
 It can also be written as
$x - y = - 5$
Now we will find the value of $(x + y)$ by the
${(x + y)^2} = {(x - y)^2} + 4xy$
By putting the value in the formula we have:
${(x + y)^2} = {( - 5)^2} + 4 \times 14$
On simplifying we have
${(x + y)^2} = 25 + 56 \Rightarrow {(x + y)^2} = 81$
It gives us value
 $(x + y) = \sqrt {81} $
$x + y = 9$ , this is our second equation.
Now we subtract second equation from the first equation i.e.
$(x - y + 5) - (x + y - 9)$
On simplifying we have :
$x - y + 5 - x - y + 9 = 0$
Solving the equation, we get
$ - 2y + 14 = 0$
$ - 2y = - 14$
So we have
$y = \dfrac{{14}}{2} = 7$ .
Now we put the value of $y$ in second equation, so we have
$x + 7 = 9$
It gives
$x = 9 - 7 = 2$
We have
 $x = 2,y = 7$
By putting the value in the number form , we have:
$10 \times 2 + 7$
It gives
 $20 + 7 = 27$
Hence the correct option is (b) $27$

Note: We should note that in this type of question, we can assume any variables instead of $x,y$ but we have to place them carefully. We must know all the algebraic identities to solve this type of question. We know that if we have a two digit number such as $12$ , we can write it as
 \[12 = 10 \times 1 + 2\] .