Question

The sum of digits of a two-digit number is 15. The original number is reversed. The reversed number is
greater than original number by 9. Find the original number.
​

Answer 1

Answer:

$\bigstar{\bold{The\:original\:number=78}}$

Step-by-step explanation:

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

• Sum of the digits = 15
• Reversed number = Original number + 9

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

• The original number

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

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

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

↠ Hence,

    The number = 10x + y

↠ By given,

    x + y = 15

    x = 15 - y-------(1)

↠ Reversing the number we get,

    Reversed number = 10y + x

↠ Also by given,

    Reversed number = Original number + 9

↠ Substitute the data,

    10y + x = 10x + y + 9

↠ Substitute the value of x from equation 1

   10y + 15 - y = 10 (15 - y) + y + 9

   9y + 15 = 150 - 10y + y + 9

   9y + 15 = 150 -9y + 9

   9y + 9y = 159 - 15

   18y = 144

       y = 144/18

       y = 8

↠ Hence the unit's digit of the number is 8

↠ Substitute the value of y in equation 1

    x = 15 - 8

    x = 7

↠ Hence the ten's digit of the number is 7

↠ Therefore,

    The number = 10x + y

    The number = 10 × 7 + 8

    The number = 78

↠ Therefore the number is 78

    $\boxed{\bold{The\:original\:number=78}}$

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

↠ By first case,

    x + y = 15

    7 + 8 = 15

    15 = 15

↠ By second case,

    10y + x = 10x + y + 9

    10 × 8 + 7 = 10 × 7 + 8 + 9

    80 + 7 = 70 + 17

    87 = 87

↠ Hence verified.

Answer 2

Answer :-

✳ The Original Number = 78

__________________________

★Concept :-

Here the concept of Linear Equations in Two Variables is used. According to this, the value of one variable is made to depend on other in order to find their both values. The standard form is given as :-

ax + by + c = 0

and

px + qy + r = 0

______________________________

★ Solution :-

• Let the value of unit place be 'x'

• Let the value of unit place be 'y'

Then,

▶ Original Number = 10y + x

▶ New Number after reversing digits = 10x + y

______________________________

~ Case I -

• Sum of the digits of two digit number = 15

This can be formulated by using the variables as,

✏ Unit Place Digit + Tens Place Digit = 15

✏ x + y = 15

✏ x = 15 - y ... (i)

~ Case II -

• The new number formed after reversing the digits is greater than the original number by 9.

So,

✏ New Number = Original Number + 9

✏ 10x + y = 10y + x + 9 ... (ii)

______________________________

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

✒ 10(15 - y) + y = 10y + (15 - y) + 9

✒ 150 - 10y + y = 10y + 15 - y + 9

✒ 150 - 9y = 9y + 24

✒ 9y + 9y = 150 - 24

✒ 18y = 126

✒ $y \: \: = \: \: \dfrac{126}{18}$

✒ y = 7

Hence, the value of y = 7.

From equation (i) we get,

✒ x = 15 - y

✒ x = 15 - 8

✒ x = 7

Hence, we get the value of x = 7

_____________________________

So,

• Original Number = 10y + x = 10(7) + 8 = 78

• New Number = 10x + y = 10(8) + 7 = 87

» Hence, we get Original Number = 78

__________________________

★ Verification :-

For verification, we just need to apply the values we got into the equations we formed. If the values satisfy the equation, we are correct else our answer is wrong.

~ Case I -

=> x + y = 15

=> 8 + 7 = 15

=> 15 = 15

Clearly, LHS = RHS

~ Case II -

=> 10x + y = 10y + x + 9

=> 10(8) + 7 = 10(7) + 8 + 9

=> 80 + 7 = 70 + 8 + 9

=> 87 = 87

Clearly, LHS = RHS.

Since both the conditions are satisfied, our answer is correct.

_______________________________

★ More to know :-

• Linear Equations are the set of equations that a variable as well as consonant term which are solved using algebraic methods. On basis of variables, Linear Equations are :-

• Linear Equation in One Variable
• Linear Equation in Two Variable
• Linear Equation in Three Variable

• Note* while checking the equation kindly check the satisfactory terms with both the equations formed because sometimes the results from both the equations while verification doesn't match and if so happens kindly recheck your answer.

Here the answer is correct so both the conditions are satisfied.