Question

Sum of the two digit number is less by 54 from the number. The number obtained by interchanging the digits exceeds the given number by 27 . find the number

Answer 1

Answer :-

$\: \: \boxed{\boxed{\rm{\mapsto \: \: \: Firstly \: let's \: understand \: the \: concept \: used}}}$

Here the concept of Linear Equations has been used. Its used to find two unknown quantities without knowing their exact values if constant and dependable terms are known. Here we will take the digits of original number as variables.

Let's do it !!

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★ Question :-

Sum of the two digit number is less by 54 from the number. The number obtained by interchanging the digits exceeds the given number by 27 . Find the number

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★ Solution :-

Given,

» Original Number = 54 + (sum of the digits)

» New number after changing digits = Original number + 27

• Let the unit place digit of the number be 'x'

• Let the tens place digit of the number be 'y'.

So,

❐ Original number = 10y + x

❐ New number after changing digits = 10x + y

Then, according to the question :-

~ Case I :-

➺ 10y + x = 54 + (x + y)

➺ 10y + x = 54 + x + y

➺ 10y - y + x - x = 54

➺ 9y = 54

$\: \qquad \qquad \large{\bf{\longmapsto \: \: y \: = \: \dfrac{54}{9} \: = \: \underline{6}}}$

$\: \\ \huge{\boxed{\tt{y \: = \: 6}}}$

~ Case II :- Let's apply the value of y here.

➺ 10x + y = 10y + x + 27

➺ 10x + 6 = 10(6) + x + 27

➺ 10x - x = 60 + 27 - 6

➺ 9x = 81

$\: \qquad \qquad \large{\bf{\longmapsto \: \: y \: = \: \dfrac{81}{9} \: = \: \underline{9}}}$

$\: \\ \huge{\boxed{\tt{x \: = \: 9}}}$

• Hence, the original number = 10y + x

= 10(6) + 9

= 69

• Hence, the new number after changing the digits = 10x + y

= 10(9) + 6

= 96

$\large{\boxed{\sf{Thus, \: the \: original \: number \: is \: \boxed{\bf{\underline{69}}}}}}$

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$\: \: \underline{\underline{\rm{\Longrightarrow \: \: Confused? \: Don't \: worry \: let's \: verify \: it \: :-}}}$

For verification, we need to simply apply the values we got into the equations we formed. Then,

~ Case I :-

➣ 10y + x = 54 + (x + y)

➣ 10(6) + 9 = 54 + (9 + 6)

➣ 60 + 9 = 54 + 15

➣ 69 = 69

Here LHS = RHS.

~ Case II :-

➣ 10x + y = 10y + x + 27

➣ 10(9) + 6 = 10(6) + 9 + 27

➣ 90 + 6 = 60 + 9 + 27

➣ 96 = 96

Clearly, LHS = RHS

Here both the conditions satisfy, so our answer is correct.

Hence, Verified.

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$\: \: \boxed{\boxed{\rm{\mapsto \: \: Reference \: as \: Supplementary \: Counsel \: }}}$

• Methods to solve Linear Equations in Two Variables :-

• Substitution Method
• Elimination Method
• Cross Multiplication Method
• Reducing the pair Method

• Different types of Polynomials :-

• Linear Polynomial
• Quadratic Polynomial
• Cubic Polynomial
• Bi - Quadratic Polynomial

Answer 2

Answer:-

Let the digit at ten's place be x and digit at one's place be y.

⟶ The number = 10x + y

Given:

Sum of the digits is 54 less than the number.

⟶ 10x + y - (x + y) = 54

⟶ 10x + y - x - y = 54

⟶ 9x = 54

⟶ x = 54/9

⟶ x = 6

Also given that,

The number formed by reversing the digits exceeds the number by 27.

⟶ Reversed number = Original number + 27

⟶ 10y + x = 10x + y + 27

⟶ 10y + 6 - 10(6) - y = 27

⟶ 9y = 27 - 6 + 60

⟶ 9y = 81

⟶ y = 81/9

⟶ y = 9

∴ The required number = 10(6) + 9 = 69.