Sum of the digits of a two-digit number is 11. The given number is less than the
number obtained by interchanging the digits by 9.Find the number.
Answers
Given :
- Sum of the digits of a two digit number = 11
- The given number is less than the number obtained by interchanging the digits by 9.
To find :
- The number
Solution :
Let the two digit number be 10x + y.
where,
- x and y are the two numbers
According to the first condition,
Sum of the digits of a two digit number :-
- x + y = 11 ------(1)
Numbers obtained after interchanging the digits :-
→ 10y + x
According to the second condition,
The given number is less than the number obtained by interchanging the digits by 9.
→ 10x + y = 10y + x - 9
→ Grouping the like terms and solving them :
→ 10x - x = 10y - y - 9
→ 9x = 9y - 9
→ Taking 9 common from both the sides.
→ 9(x) = 9(y - 1)
→ 9 will get cancelled.
→ x = y - 1
→ x - y = - 1 ---------(2)
Solving equation 1 and 2 by using the method of making the coefficients same.
⠀⠀⠀⠀⠀⠀⠀⠀ x + y = 11
⠀⠀⠀⠀⠀⠀⠀⠀⠀x - y = - 1
⠀⠀⠀⠀⠀⠀ ━━━━━━━━
⠀⠀⠀⠀⠀⠀⠀⠀2x ⠀ = 10
⠀⠀⠀⠀⠀⠀ ━━━━━━━━
→ 2x = 10
→ Transposing 2 to the other side.
→ x = 10 ÷ 2
→ x = 5
The value of x = 5
Substituting the value of x in equation (1) :
→ x + y = 11
→ 5 + y = 11
→ y = 11 - 5
→ y = 6
The value of y = 6
Therefore,
- The two numbers are 5 and 6.
And the two digit number is :-
→ 10x + y
→ 10(5) + 6
→ 50 + 6
→ 56
Therefore,
- The two digit number = 56
━━━━━━━━━━━━━━━━━
Verification :
To verify the value of two digit number, substitute the value of x and y in the expression "10x + y = 10y + x - 9"
Taking LHS :
→ 10x + y
→ 10(5) + 6
→ 50 + 6
→ 56
LHS = 56
Taking RHS :
→ 10y + x - 9
→ 10(6) + 5 - 9
→ 60 + 5 - 9
→ 65 - 9
→ 56
RHS = 56
LHS = RHS
Hence, verified.
Step-by-step explanation:
Let x be the unit digit of the number.
Then, the ten's digit = 11−x
Therefore, Number = 10(11−x)+x=110−10x+x=110−9x
When digits are interchanged ,
unit digit=11−x and tens digit =x
So, number obtained by interchanging the digits= 10x+(11−x)=9x+11
As per the question-
9x+11−(110−9x)=9
⇒9x+11−110+9x=9
⇒18x+11−110=9
⇒18x=9−11+110
⇒18x=108
⇒x=6
Hence, unit's digit = 6 and ten's digit = 11−6=5.
Therefore, the required number is 56. mark my answer as Brainallist