Question

A two digit number is 3 more than 4 times the sum of its digits. If 18 is added to this  number, the sum is equal to the number obtained by interchanging the digits. Find the  number.

Answer 1

Let the two digit number be 10x + y. 

A two digit number is 3 more than 4 times the sum of its digits.

10x + y = 4 (x + y) + 3

⇒ 10x + y = 4x + 4y + 3

⇒ 2x − y = 1   ...(I)

If 18 is added to this number, the sum is equal to the number obtained by interchanging the digits.

10x + y + 18 = 10y + x

⇒ 9x − 9y = −18

⇒ x − y = −2    ...(II)

(I) − (II)

2x − y = 1
x − y = − 2 
−   +       +   
x  =  3

Put x = 3 in equation (II)

∴ x − y = − 2

∴ 3 − y = − 2

∴ 3 + 2 = y

∴ y = 5

∴ The required number = 10x + y

= 10 × 3 + 5

= 30 + 5

= 35

Thus, the number obtained is 35. 

Answer 2

Step 1: Form the first equation

10a + b = 4(a + b) + 3

10a + b = 4a + 4b + 3

10a − 4a + b − 4b = 3

6a − 3b = 3

2a − b = 1     ...(1)

Step 2: Form the second equation

10a + b + 18 = 10b + a

10a − a + b − 10b = −18

9a − 9b = −18

a − b = −2     ...(2)

Step 3: Solve equations (1) and (2)

a = b − 2

2(b − 2) − b = 1

2b − 4 − b = 1

b − 4 = 1

b = 5

a = b − 2

= 5 − 2

= 3

10a + b

= 10(3) + 5

= 35