Question

(v) The sum of the digits of a two-digit number is 8. If its digits are reversed, the
new number so formed is increased by 18. The number is
(a) 37
(6) 48
(c) 35
(d) 46​

Answer 1

Given : -

The sum of the digits of a two digit number is 8 .

If it's digits are reversed , the new number which is formed is increased by 18 .

Required to find : -

• The original number ?

Solution : -

The sum of the digits of a two digit number is 8 .

If it's digits are reversed , the new number which is formed is increased by 18.

So,

Let's solve this question ;

Let the unit digit be x

Ten's digit be y

This implies ;

Number formed =

=> 10 ( y ) + 1 ( x )

=> 10y + x

• No. formed = 10y + x

Similarly,

On reversing the digits

Let the unit digit be y

Ten's digit be x

No. formed ( reversed digits ) =

=> 10 ( x ) + 1 ( y )

=> 10x + y

• No. formed ( reversed digits ) = 10x + y

It is mentioned that ;

If it's digits are reversed , the new number which is formed is increased by 18 .

So,

According to problem ;

No. formed = No. formed ( reversing digits ) + 18

=> 10y + x = 10x + y + 18

=> 10y - y + x - 10x = 18

=> 9y - 9x = 18

=> Take 9 as a common on the LHS part

=> 9 ( y - x ) = 18

=> y - x = 18/9

=> y - x = 2

=> - x + y = 2

Multiply with - ( minus ) on both sides

=> - ( - x + y ) = - ( 2 )

=> x - y = - 2 $\longrightarrow{\tt{\pink{Equation - 1 }}}$

Consider this as equation - 1

However,

It is also mentioned that ;

The sum of the digits of a two digit number is 8 .

So,

y + x = 8

This implies ;

x + y = 8 $\longrightarrow{\tt{\pink{Equation - 2 }}}$

Consider this as equation 2

Add equation 1 and equation 2

$\tt x - y = - 2 \\ \tt \underline{x + y = \: \: \: 8 \: } \\ \tt \underline{ 2x + 0 = 6} \\ \\ : \implies \tt \: 2x = 6 \\ \\ : \implies \tt \: x = \frac{6}{2} \\ \\ \implies \tt \: x = 3$

Substitute the value of x in Equation 1

=> x - y = - 2

=> 3 - y = - 2

=> - y = - 2 - 3

=> - y = - 5

=> y = 5

Hence,

The original number can be 53 or 35

Let's find out whether which of these is the original number .

Verification : -

In order to find the original number let's verify whether which of the 2 is original number .

For this let's consider one of the statement mentioned in the question .

If it's digits are reversed , the new number which is formed is increased by 18 .

This implies ;

Let's assume that 53 is the original number ;

53 = 35 - 18

53 = 17

LHS ≠ RHS

Now,

Let's assume that 35 is the original number

35 = 53 - 18

35 = 35

LHS = RHS

Hence,

• 35 is the original number !

Logic used in the verification process : -

The reason behind subtracting the 18 from the number formed by reversing the digits is , since it is mentioned that the reversed number will be 18 more than the original number . So, subtracting 18 from reversed number will give original number .

Therefore,

35 is the original number

Option - c is correct ✓

Answer 2

◘ Given ◘

$\bigstar$ The sum of the digits of a two-digit number is 8.

$\bigstar$ If its digits are reversed, the  new number so formed is increased by 18.

◘ To Find ◘

If the original number is

(a) 37

(6) 48

(c) 35

(d) 46​

◘ Solution ◘

Let the digit in the units place be y and the digit in the tens place be x.

The number is like : 10x + y

A/q,

x + y = 8        . . . (i)

_______________

If the digits are reversed,

10y + x = 10x + y + 18

→ 10y - y + x - 10x = 18

→ 9y - 9x = 18

→ 9(y - x) = 18

→ y - x = 18/9

→ y - x = 2        . . . (ii)

_______________

Solving equations (i) and (ii) :-

From (ii), we get -

y = 2 + x          . . . (iii)

• Putting this value in equation (i) -

x + y = 8

→ x + 2 + x = 8

→ 2x + 2 = 8

→ 2x = 6

→ x = 6/2

→ x = 3

• Putting the value of x in equation (iii) :-

y = 2 + x

→ y = 2 + 3

→ y = 5

Therefore, the required answer is 10x + y = 10(3) + 5 = 30 + 5 = 35.

Hence, the answer is (c) 35.