A two digit number is such that product of it's digits is 20. If 9 is added to the number the digits interchange their place. find the number...
Answers
AnswEr :
Let the Unit's Place be y, and Ten's Place be x. Number Formed be : (10x + y)
• According to the Question Now :
⇒ Original No. + 9 = Interchange No.
⇒ (10x + y) + 9 = (10y + x)
⇒ 9 = 10y + x – 10x – y
⇒ 9 = 9y – 9x
⇒ 9 = 9(y – x)
- Dividing both term by 9
⇒ 1 = y – x
⇒ y = x + 1 ⠀⠀— eq. ( I )
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• Product of Digits :
↠ xy = 20 ⠀⠀— eq. ( II )
- putting the value of y from eq.( I )
↠ x(x + 1) = 20
↠ x² + x = 20
↠ x² + x – 20 = 0
↠ x² + 5x – 4x – 20 = 0
↠ x(x + 5) – 4(x + 5) = 0
↠ (x – 4)(x + 5) = 0
↠ x = 4 and, x = - 5
we will consider value of x as 4, Because it's Positive Integer.
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• Putting the value of x in eq. ( II ) :
↠ xy = 20
↠ 4 × y = 20
- Dividing both term by 4
↠ y = 5
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• N U M B E R⠀F O R M E D :
↠ Number = (10x + y)
↠ Number = [10(4) + 5]
↠ Number = [40 + 5]
↠ Number = 45
∴ Therefore, Number Formed will be 45.
Question :-- A two digit number is such that product of it's digits is 20. If 9 is added to the number the digits interchange their place. find the number...
Solution :-
Let the Two digit number be (10a+b) .
it has been said that now, when we add 9 to the original number , its digit interchange .
So,
→ interchanging Digit will be = (10b+a) .
A/q,
→ (10a+b) + 9 = (10b+a)
→ 9 = (10b+a) - (10a+b)
→ 9 = 10b + a - 10a - b
→ 9 = 9b - 9a
→ 9 = 9(b-a)
Dividing both sides by 9 ,
→ (b-a) = 1
→ b = (1+a) ------------------- Equation (1)
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Now, it has been said that, when we Multiply both digits we get 20.
So,
→ a * b = 20
Putting value of b now, from Equation (1) , we get,
→ a(1+a) = 20
→ a² + a - 20 = 0
Splitting the Middle term now ,
→ a² + 5a - 4a -20 = 0
→ a(a+5) - 4(a+5) = 0
→ (a+5)(a-4) = 0
Putting both Equal to zero now, we get,
→ a + 5 = 0. or,. (a-4) = 0
→ a = (-5) → a = 4 .
[ Since, Negative value not Possible ] .
we get, a = 4 .
Putting this value in Equation (1) now, we get,
→ b = 1 + a
→ b = 1 + 4
→ b = 5
So,
our Two digit number is = 10a+b = 10*4 + 5 = 40+5 = 45 (Ans).
Hence, The required original Number is 45.
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Similar Question :-
we can solve it by Assume Method also, as it was a two digit Number only .
or we can use Concept i told in this Solution :----
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