Question

the sum of the digits of a two-digit number is 15 if the number formed by reversing the digit is less than the original number by 27 find the original number

Answer 1

The sum of two digits - 10x + y

The sum of digits is 15.so.
x + y= 15 __(i) __

No. forming by reversing the digits is (10y +x)

=>(10x + y) - (10y+x) = 27
=>(10x - x) -(10y - y) =27
=>9x - 9y=27
=>x - y = 3. (27/9=3) __(ii) __

From i and ii we get-

y=9 and x=6

Solving from the eqn.

10(9)+6= 96. Required original no.

Answer 2

Let the unit's place = x

The ten's place = 15

$\bull \:  \sf{Original \:  Number  =10(15−x)+x}$

$\:  \:  \:   \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \: \:   \:  \sf   =150−10x+x$

$\:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \: \sf \: =150−9x$

By reversing the digits, we get

$\sf {New \: number=10x+(15−x)}$

$\:  \: \:  \:  \:  \: \:  \:  \:  \sf=10x+15−x$

$\:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:    \:  = \boxed{ \sf 9x−15} \green\bigstar$

According to the question

$\sf \: Original \:  number−New \:  number=27$

$: \implies \sf \: 150−9x−9x+15=27$

$: \implies \sf{−18x+165=27}$

$: \implies \sf{−18x=27−165=(−108)}$

$: \implies \sf{x= \frac{−18}{−108}=6}$

$\sf \: original  \: number=150−9x$

$\:  \: \:  \:  \:  \:  \: \:  \:  \:  \:  \: \:  \:  \sf  = 150−9×6$

$\:  \: \:  \:  \:  \:  \: \:  \:  \:  \:  \: \:  \: \sf  = 150- 54$

$\:  \: \:  \:  \:  \:  \: \:  \:  \:  \:  \: \:  \:  = \underline{\boxed{\frak{96}}} \: \red{ \bigstar}$

• Hence, the original number 96.

Thank you!

@itzshivani