Question

The sum of a two digit number and the number formed by interchanging its digits is 110. If 10 is subtracted from the first number, the new number is 4 more than 5 times the sum of the digits in the first number. Find the first number.

Answer 1

Hint:
Here, we need to find the first number. A two-digit number can be written as 10 \[ \times \] the digit at ten’s place \[ + \] the digit at unit’s place. We will write the original number and the reversed number. Then, using the given information, we can form two linear equations in two variables. We will solve these equations to find the values, and use these values to find the first number.

Complete step by step solution:
We will use two variables \[x\] and \[y\] to form a linear equation in two variables using the given information.
A two-digit number can be written as 10 \[ \times \] the digit at ten’s place \[ + \] the digit at unit’s place.
For example, 28 can be written as \[2 \times 10 + 8\].
Let the digit at ten’s place be \[x\] and the digit at unit’s place be \[y\].
Therefore, we get the first number as
\[10 \times x + y = 10x + y\]
When the digits are interchanged, the digit at ten’s place becomes \[y\] and the digit at unit’s place becomes \[x\].
We can write the number when the digits are interchanged as
\[10 \times y + x = 10y + x\]
Now, it is given that the sum of the first number, and the number formed by interchanging the digits is 110.
Thus, we get
\[ \Rightarrow 10x + y + 10y + x = 110\]
Adding the like terms, we get
\[ \Rightarrow 11x + 11y = 110\]
Factoring the number 11, we get
\[ \Rightarrow 11\left( {x + y} \right) = 110\]
Dividing both sides by 11, we get
\[\begin{array}{l} \Rightarrow \dfrac{{11\left( {x + y} \right)}}{{11}} = \dfrac{{110}}{{11}}\\ \Rightarrow x + y = 10 \ldots \ldots \ldots \left( 1 \right)\end{array}\]
It is given that if 10 is subtracted from the given first number, the new number is 4 more than 5 times the sum of the digits in the given first number.
Hence, the new number will be \[10x + y - 10\].
The new number is 4 more than 5 times the sum of digits of the first number.
Thus, we get
\[ \Rightarrow 10x + y - 10 = 5\left( {x + y} \right) + 4\]
Multiplying the terms using the distributive law of multiplication, we get
\[ \Rightarrow 10x + y - 10 = 5x + 5y + 4\]
Rewriting the expression, we get
\[ \Rightarrow 10x - 5x + y - 5y - 10 - 4 = 0\]
Adding and subtracting the like terms, we get
\[\begin{array}{l} \Rightarrow 5x - 4y - 14 = 0\\ \Rightarrow 5x - 4y = 14 \ldots \ldots \ldots \left( 2 \right)\end{array}\]
We can observe that the equations \[\left( 1 \right)\] and \[\left( 2 \right)\] are a pair of linear equations in two variables.
We will solve the equations to find the values of \[x\] and \[y\].
Rewriting equation \[\left( 1 \right)\], we get
\[ \Rightarrow x = 10 - y\]
Substituting \[x = 10 - y\] in equation \[\left( 2 \right)\], we get
\[ \Rightarrow 5\left( {10 - y} \right) - 4y = 14\]
Multiplying the terms using the distributive law of multiplication, we get
\[ \Rightarrow 50 - 5y - 4y = 14\]
Adding and subtracting the like terms, we get
\[ \Rightarrow 50 - 9y = 14\]
Rewriting the equation, we get
\[ \Rightarrow 9y = 50 - 14\]
Subtracting the terms in the expression, we get
\[ \Rightarrow 9y = 36\]
Dividing both sides by 9, we get
\[\begin{array}{l} \Rightarrow \dfrac{{9y}}{9} = \dfrac{{36}}{9}\\ \Rightarrow y = 4\end{array}\]
Substituting \[y = 4\] in the equation \[x = 10 - y\], we get
\[ x = 10 - 4\]
Subtracting 4 from 10, we get
\[ \Rightarrow x = 6\]
Now, we will use the values of \[x\] and \[y\] to find the original number.
Substituting \[x = 6\] and \[y = 4\] in the expression \[10x + y\], we get
\[ \Rightarrow \]First Number \[ = 10\left( 6 \right) + 4\]
Simplifying the expression, we get
\[ \Rightarrow \] First Number \[ = 60 + 4 = 64\]

Therefore, the first number is 64.

Note:
We have formed two linear equations in two variables and simplified them to find the number. A linear equation in two variables is an equation of the form \[ax + by + c = 0\], where \[a\] and \[b\] are not equal to 0. For example, \[2x - 7y = 4\] is a linear equation in two variables.
We can verify our answer by using the given information.
The number obtained by reversing the digits of 64 is 46.
We can observe that \[64 + 46 = 110\].
Thus, the sum of the number and the number formed by interchanging the digits is 110.
Subtracting 10 from the first number, we get \[64 - 10 = 54\].
The sum of the digits of the number 64 is 10.
We can observe that \[5\left( {10} \right) + 4 = 54\].
Thus, if 10 is subtracted from the number 54, the result is equal to 4 more than 5 times the sum of the digits.
Hence, we have verified our answer.