Question

Sum of the digits of a two-digit number is
8.
When we interchange the digits, it is
found that the resulting new number is
greater than the original number by 36.
What is the two-digit number?
​

Answer 1

Answer:

$\bigstar{\bold{The\:original\:number=26}}$

Step-by-step explanation:

$\Large{\underline{\underline{\it{Given:}}}}$

• The sum of the digits of a two digit number is 8
• When the digits are interchanged, the reversed number is greater than the orignal number by 36

$\Large{\underline{\underline{\it{To\:Find:}}}}$

• The two digit number

$\Large{\underline{\underline{\it{Solution:}}}}$

➟ Let the ten's digit of the number be x

➟ Let the unit's digit of the number be y

➟ By given,

    x + y = 8

    x = 8 - y------(1)

➟ Hence,

    The number = 10x + y

➟ Reversing the digits,

    Reversed number = 10y + x

➟  Also by given,

    Reversed number = Original number + 36

➟ Substitute the value,

     10y + x = 10x + y + 36

➟ Substitute the value of x from equation 1

     10y + 8 - y = 10 (8 - y) + y + 36

     9y + 8 = 80 - 10y + y + 36

     9y + 8 = 80 - 9y + 36

     9y + 9y = 80 + 36 - 8

     18y = 108

         y = 108/18

         y = 6

➟ Hence the unit's digit of the number is 6

➟ Substitute the value of y in equation 1

    x = 8 - y

    x = 8 - 6

    x = 2

➟ Hence the ten's digit of the number is 2

➟ Therefore,

    The number = 10x + y

    The number = 10 × 2 + 6

    The number = 20 + 6

     The number = 26

➟ Hence the original number is 26

    $\boxed{\bold{The\:original\:number=26}}$

$\Large{\underline{\underline{\it{Verification:}}}}$

➟ By given,

    x + y = 8

    2 + 6 = 8

    8 = 8

➟ 10y + x = 10x + y + 36

    10 × 6 + 2 = 10 × 2 + 6 + 36

    60 + 2 = 20 + 6 + 36

    62 = 26 + 36

    62 = 62

➟ Hence verified.

Answer 2

Answer :-

★The original number is = 26

________________________________

» Concept :-

Here the concept Linear Equation in Two Variables is used. According to this concept, we make the value of one variable depend on other, so that we can get value of both of them. The standard form of a Linear Equation in Two Variables is :-

ax + bx + c = 0

and

px + qy + r = 0

________________________________

★ Solution :-

Given

• The sum of two digit number = 8

• The difference between the New Number and Orginal Number = 36

» Let the unit place digit be 'x'.

» Let the tens place digit be 'y'.

Then,

▶Orginal Number = 10y + x

▶ New Number after interchanging the digits = 10x + y

________________________________

Now according to the question :-

~ Case I -

Sum of digits = 8

=> x + y = 8

=> x = 8 - y ... (i)

~ Case II -

New number is greater than the Orginal number by 36. So,

=> 10x + y = 10y + x + 36 ....(ii)

________________________________

From equation (i) and (ii) , we get,

▶10(8 - y) + y = 10y + 8 - y + 36

▶ 80 - 10y + y = 9y + 44

▶ 80 - 9y = 9y + 44

▶ 9y + 9y = 80 - 44

▶ 18y = 36

▶ $y \: \: = \: \: \dfrac{36}{18}$

▶ y = 2

Hence we get, y = 2.

Now by applying the value of y in equation (i), we get,

✒ x = 8 - y

✒ x = 8 - 2 = 6

Hence, we get, x = 6

_________________________________

✳ Value of unit digit in Orginal number = x = 6

✳ Value or tens digit in original number = y = 2

So, Orginal Number = 10y + x = 10(2) + 6 = 26

New Number = 10(6) + 2 = 62

• Hence, the orginal number = 26

___________________________

★ Verification :-

In order to verify that if our answer is wrong or right, we must just apply the value of x and y in the equations we formed.

• Case I -

=> x + y = 8

=> 6 + 2 = 8

=> 8 = 8

LHS = RHS

• Case II -

=> 62 = 26 + 36

=> 62 = 62

LHS = RHS

Clearly, LHS = RHS,

Hence, verified.

________________________________

★More to know :-

• Linear Equation in Two Variable can be solved by the following ways :-

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

• Also, in order to get correct answer, verification is necessary using both the formed equations. Even, we should get a unique solution for our equations.