Question

# A certain two digits number is equal to five times the sum of its digits. If $$9$$ were added to the number, its digits would be reversed. The sum of the digits of the number is:

A
6
B
7
C
8
D
9

Solution

## The correct option is D $$9$$Let the ones and tens digit of the number be $$y$$ and $$x.$$According to question, we have$$10x+y=5\left( x+y \right)$$$$\Rightarrow 5x-4y=0\quad\quad\quad\dots(i)$$And,$$10x+y+9=10y+x$$$$\Rightarrow 9x-9y=-9$$$$\Rightarrow x-y=-1\quad\quad\quad\dots(ii)$$Multiply $$(ii)$$ by $$-4$$ we get,$$\Rightarrow -4x+4y=4\quad\quad\quad\dots(iii)$$Add equations $$(i)$$ and $$(iii),$$$$\left( {5x - 4y} \right) + \left( { - 4x + 4y} \right) = 0 + 4$$$$\Rightarrow 5x - 4x - 4y + 4y = 4$$$$\Rightarrow x = 4$$Substitute $$x=4$$ in equation $$(ii),$$$$4 - y = - 1$$$$\Rightarrow - y = - 5$$$$\Rightarrow y=5$$Thus, the sum of the digits of the number is equal to $$x+y=9.$$.Mathematics

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