A number consists of two digits whose sum is 9. If 27 is subtracted from the number its digits are reversed. Find the number.
ANSWER
Verified
Hint: Here, we will first consider the two digits of the number as x and y. Then, we will try to form equations using the given conditions to obtain the values of x and y and hence, we can find the number.
Complete Step-by-Step solution: We know that if the digits of a two digit number are known then we can easily find the number. If the first digit of the two digit number is x and the second digit is y, then we can write the given number as $10x+y$ . Since, it is given that the sum of the digits of the number is 9. So, we can write the following equation: $x+y=9...........\left( 1 \right)$ It is also given that if we subtract 27 from the number, its digit gets reversed. The reversed number will have now y as its first digit and x as its second digit. So, the number formed after reversing the digits can be written as $10y+x$ . So, on subtracting 27 from the given number and then equating it to its reverse, we get: \[\begin{align} & 10x+y-27=10y+x \\ & \Rightarrow 10x+y-10y-x=27 \\ & \Rightarrow 9x-9y=27 \\ & \Rightarrow 9\left( x-y \right)=27 \\ & \Rightarrow x-y=\dfrac{27}{9}=3 \\ \end{align}\] Therefore, we have another equation as: $x-y=3..........\left( 2 \right)$ On adding equation (1) and equation (2), we get: $\begin{align} & x+y+x-y=9+3 \\ & \Rightarrow 2x=12 \\ & \Rightarrow x=\dfrac{12}{2}=6 \\ \end{align}$ So, the value of x comes out to be = 6. On substituting x = 6 in equation (1), we get: $\begin{align} & 6+y=9 \\ & \Rightarrow y=9-6=3 \\ \end{align}$ So, the value of y is =3. Since, the number is of the form of $10x+y$, we can write that the number is : $\begin{align} & =10\times 6+3 \\ & =60+3 \\ & =63 \\ \end{align}$ Hence, the required number is 63.
Note: Students should keep in mind that a two digit number is always represented in the form of $10x+y$, where x and y are the first and second digits of the number respectively. It is not necessary to find the value of y using equation (1), it can also be found by using equation (2) also.