The sum of the digits of a two digit number is $8$ and the difference between the number and that formed by reversing the digits is $18$ . Find the number. $\newline$ $-53$ $\newline$ $35$ $\newline$ $53$ $\newline$ $-35$

Full solution

Q. The sum of the digits of a two digit number is

8

and the difference between the number and that formed by reversing the digits is

18

. Find the number.

\newline

-53

\newline

35

\newline

53

\newline

-35

Identify Digits: Let's denote the tens digit of the number as $x$ and the units digit as $y$ . The two-digit number can be represented as $10x + y$ . The number formed by reversing its digits is $10y + x$ . According to the problem, the sum of the digits is $8$ , which gives us the equation: $\newline$ $x + y = 8$
Form Equations: The difference between the original number and the reversed number is $18$ , which gives us the second equation: $\newline$ $(10x + y) - (10y + x) = 18$ $\newline$ Simplifying this, we get: $\newline$ $9x - 9y = 18$ $\newline$ Dividing both sides by $9$ , we get: $\newline$ $x - y = 2$
Solve System: Now we have a system of two linear equations: $\newline$ $1$ ) $x + y = 8$ $\newline$ $2$ ) $x - y = 2$ $\newline$ We can solve this system by adding the two equations together to eliminate $y$ . $\newline$ $(1) + (2)$ gives us: $\newline$ $x + y + x - y = 8 + 2$ $\newline$ $2x = 10$ $\newline$ Dividing both sides by $2$ , we get: $\newline$ $x = 5$
Find Tens Digit: Now that we have the value of $x$ , we can substitute it back into one of the original equations to find $y$ . Let's use the first equation: $\newline$ $x + y = 8$ $\newline$ $5 + y = 8$ $\newline$ Subtracting $5$ from both sides, we get: $\newline$ $y = 8 - 5$ $\newline$ $y = 3$
Substitute and Calculate: We have found that $x$ (the tens digit) is $5$ and $y$ (the units digit) is $3$ . Therefore, the original two-digit number is: $10x + y = 10\times5 + 3 = 50 + 3 = 53$