Question

4. The difference between the tens and the units digits of
a two-digit number is 3. If the digits are interchanged
and the number is added to the original number, we
get 121. What is the original number?​

Answer 1

Answer :-

★ Hence, the Original Number = 74

★ Concept :

The concept used here is one Linear Equations in two Variables. By using this concept, we find the value of one term using another. This is similar to Trial and Error method.

★ Solution :

• Let the unit place be 'x' and tens place be 'y' in the original number.

Here 'x' and 'y' are variables.

Now, by using the variables 'x' and 'y', the

Original Number = 10y + x

And,

Number after interchanging the digits

= 10x + y

Now according to the question,

✏ y - x = 3 .... (i)

✏ (10x + y) + (10y + x) = 121 .... (ii)

From equation (ii) , we get,

=> 10x + y + 10y + x = 121

=> 11x + 11y = 121

Dividing each term at both sides by 11, we get,

=> x + y = 11

=> x = 11 - y ...(iii)

Now from equation (i) and (iii), we get,

▶ y - x = 3

▶ y - ( 11 - y ) = 3 {from eq. (iii)}

▶ y - 11 + y = 3

▶ 2y - 11 = 3

▶ 2y = 3 + 11

▶ 2y = 14

▶ $y \: \: = \: \: \dfrac{14}{2}$

▶ y = 7

So we get the value of, y = 7.

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

• x = 11 - y = 11 - 7 = 4

So, we get the value of, x = 4

Now,

▶ Original Number = 10y + x = 10(7) + 4

= 70 + 4 = 74

▶No. after interchanging the digits = 10x + y

= 10(4) + 7 = 40 + 7 = 47

Hence, Original Number = 74

★ Verification :-

In order to verify, these values of x and y, we must ensure them if they are correct by applying those values in the equations we formed.

From equation (i), we get,

=> y - x = 3

=> 7 - 4 = 3 {since, y = 7, x = 3}

This equation satisfies with the values of x and y.

From equation (iii), we get,

=> x + y = 11

=> 4 + 7 = 11

This equation satisfies with the values of x and y.

Since both of the equation satisfies with values of 'x' and 'y'. Hence, our answer is correct..

★ More to know :-

• If we draw the graph of these equations, we are going to get intersecting lines since the solution is unique.

• On basis, of variables, the linear equations are classified into One Variable, Two Variable and So on.

• In this type of problems, it is better to use the verification method in order to check your solution. You can directly apply value in rough to see if the answer is correct.

• Also, kindly check the consistency of the solution when you solve any equation like this. Checking gives us proper estimation of answer.

• While verification use both, the equations, not only one. Sometimes, one equation gives other value and another something different. So check using both.

Answer 2

Answer:

$\bigstar{\bold{Original\:number=74}}$

Step-by-step explanation:

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

• Difference between tens and unit digit = 3
• If the digits are interchanged and the number is added to the original number we get 121

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

• The original number

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

→ Let the digit in the unit's place be x

→ Let the digit in the ten's place be y

→ Hence,

  The number = 10y + x

→ By given,

  y - x = 3

  y = 3 + x ---(1)

→ Reversing the digits we get,

  Reversed number = 10x + y

→ By given,

  Original number  + Reversed number = 121

→ Substitute the data,

  10y + x + 10x + y = 121

  11x + 11y = 121

→ Divide the whole equation by 11

   x + y = 11

→ Substitute the value of y from equation 1

    x + 3 + x = 11

    3 + 2x = 11

    2x = 11 - 3

    2x = 8

      x = 8/2

      x = 4

→ Hence the digit in the one's place = 4

→ Now substitute the value of x in equation 1

  y = 3 + 4

  y = 7

→ Hence the digit in the ten's place is 7

→ Therefore,

   The number = 10y + x

   The number = 10 × 7 + 4

   The number = 70 + 4

   The number = 74

   $\boxed{\bold{Original\:number=74}}$

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

→ Difference between tens and unit digit = 3

  y - x = 3

  7 - 4 = 3

  3 = 3

→  10y + x + 10x + y = 121

   10 × 7 + 4 + 10 × 4 + 7 = 121

   70 + 4 + 40 + 7  = 121

   110 + 11 = 121

   121 = 121

→ Hence verified.