Question

Sum of the digits of a two-digit number
is 5. When we interchange the digits, it is
found that the resulting new number is
less than the original number by 27.
What is the two-digit number?
​

Answer 1

GIVEN:

• Sum of the digits of a two-digit number is 5.
• When we interchange the digits, the resulting new number is less than the original number by 27.

TO FIND:

• What is the original number ?

SOLUTION:

Let the digit at ten's place be 'x' and the digit at unit's place be 'y'

✧ NUMBER = 10x + y

CASE:- 1)

✍ Sum of the digits of a two-digit number is 5.

According to question:-

➾ x + y = 5....❶

➾ x = 5 –y

CASE:- 2)

✍ When we interchange the digits, the resulting new number is less than the original number by 27.

✦ Reversed Number = 10y + x

✦ Reversed Number = Original Number –27

According to question:-

➾ 10y + x = 10x + y –27

➾ 27 = 10x + y –10y –x

➾ 27 = 9x –9y

Take common 9 from both sides

➾ 3 = x –y....❷ 

Put the value of 'x' from equation 1) in equation 2)

➾ 3 = 5 –y –y

➾ 3 = 5 –2y

➾ 3 –5 = –2y

➾ –2 = –2y

➾ y = $\sf{\cancel\dfrac{-2}{-2}}$

❬ y = 1 ❭

Put the value of 'y' in equation 1)

➾ x + 1 = 5

➾ x = 5 –1

❬ x = 4 ❭

◉ NUMBER = 10x + y

◉ NUMBER = 10(4) + 1

◉ NUMBER = 40 + 1

◉ NUMBER = 41

❝ Hence, the number formed is 41 ❞

______________________

Answer 2

Given : Sum of the digits of a two-digit number is 5. When we interchange the digits, it is found that the resulting new number is less than the original number by 27.

$\rule{130}{1}$

Solution :

Let the digits at tens place and ones place be x and 5 - x respectively.

$\therefore\:{\underline{\sf Original\: number\::}}\\\\\\:\implies\sf 10x + (5 - x)\\\\\\:\implies\sf 50 - 9x\\\\\\:\implies\sf 9x + 5$

$\rule{130}{1}$

Now interchange the digits: digit at one's place and tens place be x and 5 - x respectively.

$\therefore\:{\underline{\sf New\: number\::}}\\\\\:\implies\sf 10(5 - x) + x\\\\\\:\implies\sf 50 - 10x + x\\\\\\:\implies\sf 50 - 9x$

$\rule{130}{1}$

☯ $\underline{\boldsymbol{According\: to \:the\: Question\:now :}}$

$:\implies\sf New\: number = Original\: number + 27\\\\\\:\implies\sf 50 - 9x = 9x + 5 + 27\\\\\\:\implies\sf 50 - 9x = 9x + 32\\\\\\:\implies\sf 18x = 18\\\\\\:\implies\sf x = 1$

Hence the digit at tens place and ones place of the number are 1 and 5 - x = 5 - 1 = 4.

$\therefore\:\underline{\textsf{The two digit number is 50 - 9x = 50 - 9 (1) = \textbf{41}}}.$

Peace :)!

$\rule{185}{2}$