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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
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B
7
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C
8
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D
9
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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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