Question

the sum of a number of two digits and of the number formed by reversing the digit is 110, and the difference of the digit is 6. find the number.​

Answer 1

Answer-

Number is 82.

$\rule{100}2$

Explanation-

Let the ten's digit number be M and one's digit number be N.

Difference between the digit is 6.

=> M - N = 6

=> M = 6 + N ---- [1]

The sum of a number of two digits and of the number formed by reversing the digit is 110.

We have ten's digit number is M and one's digit number is N.

Means, Number = 10M + N

Revered number = 10N + M

Sum of two digit number i.e. 10M + N and reversed number i.e. 10N + M is 110.

=> 10M + N + 10N + M = 110

=> (10M + M) + (10N + N) = 110

=> 11M + 11N = 110

Take 11 common on both sides

=> 11(M + N) = 11(10)

11 throughout cancel

=> M + N = 10

Substitute value of M above

=> 6 + N + N = 10

=> 2N = 4

=> N = 2

One's digit = N = 2

Substitute value of N = 2 in equation [1]

=> M = 6 + 2

=> M = 8

Ten's digit = M = 8

Now,

Number = 10M + N

From above calculation, we have -

• M = 8 (ten's digit)
• N = 2 (one's digit)

=> 10(8) + 2

=> 82

Answer 2

$\textbf{ \underline{ \: \: \underline{Solution : } \: \: \: }}$

$\bigstar \: \text{Let the two digit numbers be} \: \tt{ xy } \\ \\ \small{ \text{tens \: digit \: no. \: is \: x }} \\ \small{ \text{ons \: digit \: no. \: is \: y }} \\ \\ \text{Difference of these two numbers is } \tt{6} \\ \rightarrow \: \tt{x - y = 6} \\ \rightarrow \: \boxed{\tt{x = 6 + y} } - - - (1)$

$\textsf{ \underline{According to the question : }}$

The sum of a number of two digits and of the number formed by reversing the digit is 110.

$\textsf{First no. will be }\tt{10x + y} \\ \textsf{reversed no. will be} \: \tt{10y + x}$

Sum of these two numbers is 110.

$\bf{10x + y + 10y + x = 110} \\ \\ \rightarrow \tt{11x + 11y = 110} \\ \\ \rightarrow \tt{11(x + y) = 110} \\ \\ \rightarrow \tt{x + y = \cancel{ \frac{110}{ 11} }} \\ \\ \rightarrow \tt{x + y = 10} \\ \\ \sf{putting \: the \: value \: of \: x \: from \: eq \: (1)} \\ \\ \bf{x + y = 10} \\ \\ \rightarrow \tt{6 + y + y = 10} \\ \\ \rightarrow \tt{2y = 10 - 6} \\ \\ \rightarrow \tt{y = \cancel{ \frac{4}{2} } } \\ \\ \rightarrow \boxed{\tt{y = 2}}$

Ones digit number is 2.

$\textsf{putting \: the \: value \: of \: y \: in \: eq \: (1)} \\ \\ \bf{x = 6 + y} \\ \\ \rightarrow \tt{x = 6 + 2} \\ \\ \rightarrow \boxed{ \tt{x = 8}}$

Tens digit number is 8.

$\underline{ \boxed{ \red{ \textbf{The \: number \: is \: 82}}}}$