Question

A two digits number becomes $\frac{5}{6}$th of itself when its digits are reversed. If the difference of the digits is 1, then the number is

• A. 45
• B. 54
• C. 56
• D. 65

Answer 1

The correct option is B 54

Let the units digit be 'y'
Let the tens digit be 'x'
Two digit no = 10x+y
Reversed no = 10y + x
According to the question,
Reversed no = $\frac{5}{6}$(Two digit no)
$\Rightarrow 10y+x=\frac{5}{6}(10x+y)$
$\Rightarrow 6(10y+x)=5(10x+y)$
$\Rightarrow 60y+6x=50x+5y$
$\Rightarrow 60y-5y=50x-6x\Rightarrow 55y=44x$
$5y=4x\Rightarrow 5y-4x=0$......(1)

Case 1
$x-y=1$
$\Rightarrow x=1+y$ .........(2)
Substituting x = 1+y from (2) in (1)
$5y-4x=0$
$\Rightarrow 5y-4(1+y)=0$
$\Rightarrow 5y-4-4y=0$
$\Rightarrow y=4$
$\therefore x=1+y$ (from (2))
$x=1+4=5$

Case 2
$y-x=1$
$\Rightarrow y=1+x$........(3)
Substituting $y=1+x$ from (3) in (1)
$5y-4x=0$
$\Rightarrow 5(1+x)-4x=0$
$\Rightarrow 5+5x-4x=0$
$\Rightarrow 5+x=0$
$\therefore x=-5$
Required two digit number $=10x+y$
$=10\left(5\right)+4$
$=54$